multiscale transforms with application to image processing

Signals and Communication Technology

Aparna VyasSoohwan YuJoonki Paik

Multiscale Transforms with Application to Image Processing

Signals and Communication Technology

More information about this series at http://www.springer.com/series/4748

http://www.springer.com/series/4748

Aparna Vyas • Soohwan YuJoonki Paik

Multiscale Transformswith Application to ImageProcessing

123

Aparna VyasImage Processing and Intelligent SystemsLaboratory, Graduate School of AdvancedImaging Science, Multimedia and Film

Chung-Ang UniversitySeoulSouth Korea

Soohwan YuImage Processing and Intelligent SystemsLaboratory, Graduate School of AdvancedImaging Science, Multimedia and Film


Joonki PaikImage Processing and Intelligent SystemsLaboratory, Graduate School of AdvancedImaging Science, Multimedia and Film


ISSN 1860-4862 ISSN 1860-4870 (electronic)Signals and Communication TechnologyISBN 978-981-10-7271-0 ISBN 978-981-10-7272-7 (eBook)https://doi.org/10.1007/978-981-10-7272-7

Library of Congress Control Number: 2017959155

© Springer Nature Singapore Pte Ltd. 2018This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer Nature Singapore Pte Ltd.The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Digital image processing is a popular, rapidly growing area of electrical andcomputer engineering. Digital image processing has enabled various intelligentapplications such as face recognition, signature recognition, iris recognition,forensics, automobile detection, and military vision applications. Its growth isleveraged by technological innovations in the fields of computer processing, digitalimaging, and mass storage devices. Traditional analog imaging applications arenow switching to digital systems to utilize their usability and affordability.Important examples include photography, medicine, video production, remotesensing, and security monitoring. These sources produce a huge volume of digitalimage data every day. Theoretically, image processing can be considered as theprocessing of a two-dimensional image using a digital computer. The outcome ofimage processing could be an image, a set of features, or characteristics related tothe image. Most image processing methods treat an image as a two-dimensionalsignal and implement standard signal processing techniques.

Many image processing techniques were of only academic interest becauseof their computational complexity. However, recent advances in processing andmemory technology made image processing a vital and cost-effective technology ina host of applications. Multi-scale image transformations, such as Fourier trans-form, wavelet transform, complex wavelet transform, quaternion wavelet transform,ridgelet transform, contourlet transform, curvelet transform, and shearlet transform,play an extremely crucial role in image compression, image denoising, imagerestoration, image enhancement, and super-resolution. Fourier transform is apowerful tool that has been available to signal and image analysis for many years.However, the problem with using Fourier transform is frequency analysis cannotoffer high frequency and time resolution at the same time. To overcome thisproblem, windowed Fourier transform or short-time Fourier transform was intro-duced. Although the short-time Fourier transform has the ability to provide timeinformation, a complete multiresolution analysis is not possible. Wavelet is asolution to the multiresolution problem. A wavelet has the important property of nothaving a fixed-width sampling window. The wavelet transform can be classifiedinto (i) continuous wavelet transform and (ii) discrete wavelet transform. The

v

discrete wavelet transform (DWT) algorithms have a firm position in processing ofimages in many areas of research and industry.

The main focus of classical wavelets includes compression and efficient repre-sentation. Important features which play a role in analysis of functions in twovariables are dilation, translation, spatial and frequency localization, and singularityorientation. Singularities of functions in more than one variable vary in dimen-sionality. Important singularities in one dimension are simply points. In twodimensions, zero- and one-dimensional singularities are important. A smooth sin-gularity in two dimensions may be a one-dimensional smooth manifold. Smoothsingularities in two-dimensional images often occur as boundaries of physicalobjects. Efficient representation in two dimensions is a hard problem. To overcomethis problem, new multi-scale transformations such as ridgelet transform, contourlettransform, curvelet transform, and shearlet transform were introduced. Recently,these multi-scale transforms have become increasingly important in imageprocessing.

In this book, we will provide a complete introduction of multi-scale imagetransformations followed by their applications to various image processing algo-rithms including image denoising, image restoration, image enhancement, andsuper-resolution. The book is mainly divided into three parts. The readers will learnabout the basic introduction of image processing in the first part in Chaps. 1 and 2.The second part starts with Fourier transform followed by wavelet transform andnew multi-scale constructions. The third part deals with applications of themulti-scale transform in image processing.

The chapters of the present book consist of both tutorial and advanced theory.Therefore, the book is intended to be a reference for graduate students andresearchers to obtain state-of-the-art knowledge on multi-scale image processingapplications. The technique of solving problems in the transform domain is com-mon in applied mathematics as used in research and industry, but we do not devoteas much time to it as we should in the undergraduate curriculum. Also, the book isintended to be used as a reference manual for scientists who are engaged in imageprocessing research, developers of image processing hardware and software sys-tems, and practicing engineers and scientists who use image processing as a tool intheir applications.

Appendices summarize mostly used mathematical background in the book.

Seoul, South Korea Aparna VyasSoohwan YuJoonki Paik

vi Preface

Contents

Part I Introduction to Image Processing

1 Fundamentals of Digital Image Processing . . . . . . . . . . . . . . . . . . . . 31.1 Image Acquisition of Digital Camera . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Part II Multiscale Transform

2 Fourier Analysis and Fourier Transform . . . . . . . . . . . . . . . . . . . . . 152.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Frequency and Amplitude . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.4 Fourier Series of Periodic Functions . . . . . . . . . . . . . . . 192.2.5 Complex Form of Fourier Series . . . . . . . . . . . . . . . . . . 20

2.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 2D-Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 Properties of Fourier Transform . . . . . . . . . . . . . . . . . . 24

2.4 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.1 1D-Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . 272.4.2 Inverse 1D-Discrete Fourier Transform . . . . . . . . . . . . . 302.4.3 2D-Discrete Fourier Transform and 2D-Inverse

Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 312.4.4 Properties of 2D-Discrete Fourier Transform . . . . . . . . . 32

2.5 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

vii

2.6 The Discrete Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . 392.6.1 1D-Discrete Cosine Transform . . . . . . . . . . . . . . . . . . . 392.6.2 2D-Discrete Cosine Transform . . . . . . . . . . . . . . . . . . . 40

2.7 Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . 412.8 Windowed Fourier Transform or Short-Time Fourier

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.8.1 1D and 2D Short-Time Fourier Transform . . . . . . . . . . . 412.8.2 Drawback of Short-Time Fourier Transform . . . . . . . . . 42

2.9 Other Spectral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Wavelets and Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Multiresolution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.1 The Wavelet Series Expansions . . . . . . . . . . . . . . . . . . . 533.4.2 Discrete Wavelet Transform . . . . . . . . . . . . . . . . . . . . . 543.4.3 Motivation: From MRA to Discrete Wavelet

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4.4 The Quadrature Mirror Filter Conditions . . . . . . . . . . . . 57

3.5 The Fast Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.6 Why Use Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 653.7 Two-Dimensional Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.8 2D-discrete Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . 673.9 Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.9.1 1D Continuous Wavelet Transform . . . . . . . . . . . . . . . . 693.9.2 2D Continuous Wavelet Transform . . . . . . . . . . . . . . . . 69

3.10 Undecimated Wavelet Transform or Stationary WaveletTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.11 Biorthogonal Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . 703.11.1 Linear Independence and Biorthogonality . . . . . . . . . . . 703.11.2 Dual MRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.11.3 Discrete Transform for Biorthogonal Wavelets . . . . . . . . 73

3.12 Scarcity of Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . 763.13 Complex Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.14 Dual-Tree Complex Wavelet Transform . . . . . . . . . . . . . . . . . . . 793.15 Quaternion Wavelet and Quaternion Wavelet Transform . . . . . . . 83

3.15.1 2D Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 843.15.2 Quaternion Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.15.3 Quaternion Multiresolution Analysis . . . . . . . . . . . . . . . 89

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

viii Contents

4 New Multiscale Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 Ridgelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.2.1 The Continuous Ridgelet Transform . . . . . . . . . . . . . . . 944.2.2 Discrete Ridgelet Transform . . . . . . . . . . . . . . . . . . . . . 984.2.3 The Orthonormal Finite Ridgelet Transform. . . . . . . . . . 1004.2.4 The Fast Slant Stack Ridgelet Transform . . . . . . . . . . . . 1004.2.5 Local Ridgelet Transform . . . . . . . . . . . . . . . . . . . . . . . 1014.2.6 Sparse Representation by Ridgelets . . . . . . . . . . . . . . . . 101

4.3 Curvelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.3.1 The First Generation Curvelet Transform . . . . . . . . . . . . 1024.3.2 Sparse Representation by First Generation Curvelets . . . 1034.3.3 The Second-Generation Curvelet Transform . . . . . . . . . . 1044.3.4 Sparse Representation by Second Generation

Curvelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.4 Contourlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.5 Contourlet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.5.1 Multiscale Decomposition . . . . . . . . . . . . . . . . . . . . . . . 1084.5.2 Directional Decomposition . . . . . . . . . . . . . . . . . . . . . . 1094.5.3 The Discrete Contourlet Transform . . . . . . . . . . . . . . . . 110

4.6 Shearlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.7 Shearlet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.7.1 Continuous Shearlet Transform . . . . . . . . . . . . . . . . . . . 1154.7.2 Discrete Shearlet Transform . . . . . . . . . . . . . . . . . . . . . 1164.7.3 Cone-Adapted Continuous Shearlet Transform . . . . . . . . 1184.7.4 Cone-Adapted Discrete Shearlet Transform . . . . . . . . . . 1214.7.5 Compactly Supported Shearlets . . . . . . . . . . . . . . . . . . . 1234.7.6 Sparse Representation by Shearlets . . . . . . . . . . . . . . . . 125

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Part III Application of Multiscale Transforms to Image Processing

5 Image Restoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.1 Model of Image Degradation and Restoration Process . . . . . . . . . 1335.2 Image Quality Assessments Metrics . . . . . . . . . . . . . . . . . . . . . . 1345.3 Image Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.4 Noise Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.4.1 Additive Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . 1375.4.2 Multiplicative Noise Model . . . . . . . . . . . . . . . . . . . . . . 137

5.5 Types of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.5.1 Amplifier (Gaussian) Noise . . . . . . . . . . . . . . . . . . . . . . 1375.5.2 Rayleigh Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.5.3 Uniform Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Contents ix

5.5.4 Impulsive (Salt and Pepper) Noise . . . . . . . . . . . . . . . . 1395.5.5 Exponential Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.5.6 Speckle Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.6 Image Deblurring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.6.1 Gaussian Blur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.6.2 Motion Blur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.6.3 Rectangular Blur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.6.4 Defocus Blur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.7 Superresolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.8 Classification of Image Restoration Algorithms . . . . . . . . . . . . . 142

5.8.1 Spatial Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.8.2 Frequency Domain Filtering . . . . . . . . . . . . . . . . . . . . . 1465.8.3 Direct Inverse Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 1515.8.4 Constraint Least-Square Filter . . . . . . . . . . . . . . . . . . . . 1515.8.5 IBD (Iterative Blind Deconvolution) . . . . . . . . . . . . . . . 1525.8.6 NAS-RIF (Nonnegative and Support Constraints

Recursive Inverse Filtering) . . . . . . . . . . . . . . . . . . . . . 1525.8.7 Superresolution Restoration Algorithm Based on

Gradient Adaptive Interpolation . . . . . . . . . . . . . . . . . . 1525.8.8 Deconvolution Using a Sparse Prior . . . . . . . . . . . . . . . 1535.8.9 Block-Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.8.10 LPA-ICI Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.8.11 Deconvolution Using Regularized Filter (DRF) . . . . . . . 1535.8.12 Lucy-Richardson Algorithm . . . . . . . . . . . . . . . . . . . . . 1545.8.13 Neural Network Approach . . . . . . . . . . . . . . . . . . . . . . 154

5.9 Application of Multiscale Transform in Image Restoration . . . . . 1545.9.1 Image Restoration Using Wavelet Transform . . . . . . . . . 1555.9.2 Image Restoration Using Complex Wavelet

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.9.3 Image Restoration Using Quaternion Wavelet

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1725.9.4 Image Restoration Using Ridgelet Transform . . . . . . . . . 1745.9.5 Image Restoration Using Curvelet Transform . . . . . . . . . 1775.9.6 Image Restoration Using Contourlet Transform . . . . . . . 1815.9.7 Image Restoration Using Shearlet Transform . . . . . . . . . 186

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6 Image Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.2 Spatial Domain Image Enhancement Techniques . . . . . . . . . . . . 200

6.2.1 Gray Level Transformation . . . . . . . . . . . . . . . . . . . . . . 2016.2.2 Piecewise-Linear Transformation Functions . . . . . . . . . . 2026.2.3 Histogram Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 2036.2.4 Spatial Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

x Contents

6.3 Frequency Domain Image Enhancement Techniques . . . . . . . . . . 2056.3.1 Smoothing Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.3.2 Sharpening Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.3.3 Homomorphic Filtering . . . . . . . . . . . . . . . . . . . . . . . . 208

6.4 Colour Image Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2096.5 Application of Multiscale Transforms in Image

Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2106.5.1 Image Enhancement Using Fourier Transform . . . . . . . . 2126.5.2 Image Enhancement Using Wavelet Transform . . . . . . . 2146.5.3 Image Enhancement Using Complex Wavelet

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2196.5.4 Image Enhancement Using Curvelet Transform . . . . . . . 2226.5.5 Image Enhancement Using Contourlet Transform . . . . . . 2236.5.6 Image Enhancement Using Shearlet Transform . . . . . . . 225

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Appendix A: Real and Complex Number System . . . . . . . . . . . . . . . . . . . 233

Appendix B: Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Appendix C: Linear Transformation, Matrices . . . . . . . . . . . . . . . . . . . . 239

Appendix D: Inner Product Space and Orthonormal Basis . . . . . . . . . . . 241

Appendix E: Functions and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 245

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Contents xi

About the Authors

Aparna Vyas was born in Allahabad, India, in 1983. She received her B.Sc. inScience and M.Sc. in Mathematics from the University of Allahabad, Allahabad,India, in 2004 and 2006, respectively. She received her Ph.D. degree inMathematics from the University of Allahabad, Allahabad, India, in 2010. She wasAssistant Professor, Department of Mathematics, School of Basic Sciences,SHIATS, Allahabad, India, since 2006–2013. In 2014, she joined Manav RachnaUniversity, Faridabad, India, as Assistant Professor. She was a postdoctoral fellowat Soongsil University from August 2016 to September 2016. Currently, she is apostdoctoral fellow in Chung-Ang University under the BK21 Plus Project. She hasmore than 10 years of teaching and research experience. She is also a life memberof the Indian Mathematical Society and Ramanujan Mathematical Society. Herresearch interests include wavelet analysis and image processing.

Soohwan Yu was born in Incheon, Korea, in 1988. He received his B.S. inInformation and Communication Engineering from Suwon University, Korea, in2013. He received his M.S. in Image Engineering from Chung-Ang University,Korea, in 2016, where he is currently pursuing his Ph.D. in Image Engineering. Hisresearch interests include image enhancement, super-resolution, and imagerestoration.

Joonki Paik was born in Seoul, Korea, in 1960. He received his B.S. in Controland Instrumentation Engineering from Seoul National University in 1984. Hereceived his M.S. and Ph.D. degrees in Electrical Engineering and in ComputerScience from Northwestern University in 1987 and in 1990, respectively. From1990 to 1993, he worked at Samsung Electronics, where he designed image sta-bilization chipsets for consumer camcorders. Since 1993, he has been a facultymember at Chung-Ang University, Seoul, South Korea. Currently, he is Professorin the Graduate School of Advanced Imaging Science, Multimedia and Film. From1999 to 2002, he was Visiting Professor in the Department of Electrical andComputer Engineering at the University of Tennessee, Knoxville. Dr. Paik was therecipient of the Chester-Sall Award from the IEEE Consumer Electronics Society,the Academic Award from the Institute of Electronic Engineers of Korea, and the

xiii

Best Research Professor Award from Chung-Ang University. He served theConsumer Electronics Society of IEEE as a member of the editorial board. Since2005, he has been the head of the National Research Laboratory in the field ofimage processing and intelligent systems. In 2008, he worked as a full-time tech-nical consultant for the System LSI division at Samsung Electronics, where hedeveloped various computational photographic techniques including an extendeddepth-of-field (EDoF) system. From 2005 to 2007, he served as Dean of theGraduate School of Advanced Imaging Science, Multimedia and Film. From 2005to 2007, he was Director of the Seoul Future Contents Convergence (SFCC) Clusterestablished by the Seoul Research and Business Development (R&BD) Program.Dr. Paik is currently serving as a member of the Presidential Advisory Board forScientific/Technical Policy for the Korean government and as a technical consultantfor the Korean Supreme Prosecutors Office for computational forensics.

xiv About the Authors

Part IIntroduction to Image Processing

Chapter 1Fundamentals of Digital Image Processing

1.1 Image Acquisition of Digital Camera

1.1.1 Introduction

The concept of the digital image was first introduced in the transportation of thedigital image using submarine cable system in the early twenty century [3]. In addi-tion, the advance in the computational hardware and processing unit lead to thedevelopment of modern digital image processing techniques. Specifically, the digitalimage processing started in the application field of remote sensing. In 1964, the JetPropulsion Laboratory applied the digital image processing technique to improvethe visual quality of the transmitted digital image by Ranger 7 [1, 3]. In the medicalimaging, the image processing techniques were applied to develop the computer-ized tomography for medical imaging devices in early 1970s, which generates atwo-dimensional image and three-dimensional volume of the inside of the object bypassing the X-ray [3]. In addition to the remote sensing and medical imaging, thedigital image processing techniques have been widely used in various applicationfields such as consumer electronics, defense, robot vision, surveillance systems, andartificial intelligence systems.

In the modern image acquisition system, the image signal processing (ISP) chainplays an important role to obtain the high-quality digital image as shown in Fig. 1.1.The light pass the lens and color filter array (CFA). Since the imaging sensor with-out the CFA absorbs the light in all spectrum bands, we cannot obtain the colorinformation. To generate the color image, the digital camera uses the common CFAcalled Bayer pattern, which consists of two green (G), one red (R), and one blue(B) filter because the human visual system is more sensitive to the light in the greenwavelength [2]. The advanced CFA replaces the one green filter with white filter toincrease the amount of light.

In an imaging sensor such as charge coupled device (CCD) or complementarymetal oxide semiconductor (CMOS), the photon reacts to each semiconductor and

© Springer Nature Singapore Pte Ltd. 2018A. Vyas et al., Multiscale Transforms with Application to ImageProcessing, Signals and Communication Technology,https://doi.org/10.1007/978-981-10-7272-7_1

3

4 1 Fundamentals of Digital Image Processing

Fig. 1.1 The block diagram of the image signal processing chain

converts the electrical charges to the electric analog signal. The analog front end(AFE) module of ISP chain performs the sampling and quantization processes toconvert the analog signal to the digital signal. Sequentially, the AFE module alsocontrols the gain of the acquired signal to increase the signal-to-noise ratio (SNR).In low-illumination condition, since the amount of the photons is decreased to reactan imaging sensor, the digital image having low contrast is acquired with low SNR[4, 5, 7, 8]. In addition, the recent imaging devices increases the spatial resolutionof a digital image by drastically reducing the physical size of each pixel. However,the reduced pixel size results in the chrominance noise called the cross-talk becauseof the interference of the photons among the pixels and it also reduces the SNR ineach color channel.

The digital back end (DBE) module performs the digital image processing tech-niques to improve the quality of an input image. First, the DBE module performsthe demosaicing to separate the color information from the raw image data by usingthe interpolation algorithms [6]. In addition, the image enhancement techniques areperformed to improve the dynamic range of an image. The autowhite balance (AWB)performs the color constancy, which makes the digital image be acquired under theneutral light condition by correcting the chromaticity. Finally, the noise reductionshouldbeperformed to remove the amplifiednoise in the image enhancement process.Additionally, since the demosaicing and noise reduction techniques may decreasethe quality of the image by the blurring and jagging artifacts, the image restorationtechniques called super-resolution can be performed to obtain the high-resolutionimage.

1.2 Sampling

The sampling and quantization are major processes performed in the AFEmodule ofISP chain to convert a continuous image signal to a series of discrete signals. In thissection, we briefly describe the mathematical background of the sampling theorem.

1.2 Sampling 5

Fig. 1.2 The sampling operation of a continuous function using an impulse train.: a 1D continuousfunction, b impulse train with period T , and c the sampled function by the multiplication of a and b

Let x(t) be a one-dimensional (1D) continuous function, the sampling operationcan be regarded as the multiplication of x(t) and impulse train p(t) with period T ,for k = · · · ,−2,−1, 0, 1, 2, . . ., as

xs(t) = x(t)p(t) = x(t)∞∑

k=−∞δ(t − kT ). (1.2.1)

Figure1.2 shows the sampling operation of an 1D continuous function using animpulse train. As shown in Fig. 1.2, a continuous function is sampled with intervalT and the amplitude of an impulse train varies with that of the continuous functionx(t). The black dots represent the sampled values of x(t) at the location kT .

Since the impulse train is a periodic function with period T , it can be expressedas the Fourier series expansion as

p(t) =∞∑

n=−∞ane

j 2πkT t , (1.2.2)


where

an = 1

T

∫ T/2

−T/2p(t)e− j 2πkT t dt

= 1

T

∫ T/2

−T/2δ(t)e− j 2πkT t dt

= 1

T.

(1.2.3)

The Fourier transform of the impulse train is defined by using (1.2.2) as

P(u) =∫ ∞

−∞p(t)e− j2πutdt

=∫ ∞

−∞

(1

T

∞∑

k=−∞e j 2πkT t

)e− j2πutdt

= 1

T

∞∑

k=−∞

∫ ∞

−∞e− j2π(u− k

T )t dt

= 1

T

∞∑

k=−∞δ

(u − k

T

),

(1.2.4)

where P(u) is the Fourier transform of p(t). The Fourier transform of impulsetrain is an impulse train with period 1/T . In addition, since the multiplication ofFourier transformed functions in the spatial domain is the convolution in the fre-quency domain, the Fourier transform of the sampled function in (1.2.1) can beexpressed as

Xs(u) = X (u) ∗ P(u)

=∫ ∞

−∞X (τ )P(u − τ )dτ

= 1

T

∞∑

k=−∞X

(u − k

T

).

(1.2.5)

where Xs(u) represents the Fourier transform of the sampled function xs(t), and ∗the convolution operator. The Fourier transform of the sampled function xs(t) is alsoa sequence of the Fourier transform of x(t) at the location k/T with period 1/T .

Figure1.3 shows that how a continuous function is sampled in the differentsampling rate 1/T in the frequency domain. Figure1.3a is the Fourier trans-form of a continuous function x(t), which is filtered using the band-limit filter in−umax ≤ u ≤ umax . Figure1.3b shows the Fourier transform of the sampled functionwith the higher sampling rate. Since the sampled signal is completely separated, X (u)can be reconstructed from Xs(u) using the band-limit filter which used in Fig. 1.3a.

1.2 Sampling 7

Fig. 1.3 Comparative results using different sampling rate.: a the Fourier transform of band-limitedcontinuous function, b the Fourier transform of the sampled function using 1

T < 2umax , and c theFourier transform of the sampled function using 1

T > 2umax

On the other hand, if the band-limited signal is sampled at lower sampling rate, theFourier transform of the sampled function is overlapped as shown in Fig. 1.3c. Itimplies that the sampling operation is performed at the sampling rate higher thantwice the maximum frequency umax to completely reconstruct X (u). This is calledas Nyquist-Shannon sampling theorem.

1

T> 2umax . (1.2.6)

In the two-dimensional (2D) case, the sampling can be performed using an 2Dimpulse train. Given the 2D continuous function x(t, q) and impulse train p(t, q),the 2D sampled function xs(t, q) can be written as

xs(t, q) = x(t, q)p(t, q) = x(t, q)∞∑

l=−∞

∞∑

k=−∞δ(t − kT )δ(q − lQ). (1.2.7)

Since the 2D impulse train is a periodic function, it can be expressed the Fourierseries expansion as 1D impulse train in (1.2.2). The Fourier series expansion of 2Dimpulse train is defined as


p(t) =∞∑

l=−∞

∞∑

k=−∞bne

j 2πkT t e j 2πlQ q, (1.2.8)

where

bn = 1

T Q

∫ T/2

−T/2

∫ Q/2

−Q/2p(t, q)e− j 2πkT t e− j 2πlQ qdtdq

= 1

T Q

∫ T/2

−T/2

∫ Q/2

−Q/2δ(t, q)e− j 2πkT t e− j 2πlQ qdtdq

= 1

T Q.

(1.2.9)

The Fourier transform of p(t, q) is defined as

P(u, v) =∫ ∞

−∞

∫ ∞

−∞p(t, q)e− j2πut e− j2πvqdtdq

=∫ ∞

−∞

∫ ∞

−∞

(1

T Q

∞∑

l=−∞

∞∑

k=−∞e j 2πkT t e j 2πlQ q

)e− j2πut e− j2πvqdtdq

= 1

T Q

∞∑

l=−∞

∞∑

k=−∞

∫ ∞

−∞

∫ ∞

−∞e− j2π(u− k

T )t e− j2π

(v− l

Q

)qdtdq

= 1

T Q

∞∑

l=−∞

∞∑

k=−∞δ

(u − k

T

)δ

(v − l

Q

),

(1.2.10)

The Fourier transform of 2D impulse train is a periodic impulse train with period1/T and 1/Q in u and v directions. Let Xs(u, v) be the Fourier transform of 2Dsampled function xs(t, q), Xs(u, v) can be estimated using the convolution theoremin the frequency domain as

Xs(u, v) = X (u, v) ∗ P(u, v)

=∫ ∞

−∞

∫ ∞

−∞X (τu , τv)P(u − τu , v − τv)dτudτv

= 1

T Q

∫ ∞

−∞

∫ ∞

−∞X (τu , τv)

∞∑

l=−∞

∞∑

k=−∞δ

(u − τu − k

T

)δ

(v − τv − l

Q

)dτudτv

= 1

T Q

∞∑

l=−∞

∞∑

k=−∞

∫ ∞

−∞

∫ ∞

−∞X (τu , τv)δ

(u − τu − k

T

)δ

(v − τv − l

Q

)dτudτv

= 1

T Q

∞∑

l=−∞

∞∑

k=−∞X

(u − k

T, v − l

Q

).

(1.2.11)

1.2 Sampling 9

Fig. 1.4 The spectrum of the sampled function with periods 1/T and 1/Q in the frequency domain

In the same manner as the sampling in the 1D case, the Fourier transform ofthe sampling in the spatial domain results in the multiply copied version of thefrequency spectrum X (u, v) at the location k/T and l/Q. Figure1.4 shows that theperiodic frequency spectrum of the sampled function with periods 1/T and 1/Q inthe frequency domain.

In order to reconstruct the original 2D continuous signal, the Nyquist-Shannonsampling rate should be satisfied as

1

T< 2umax , (1.2.12)

and

1

Q< 2vmax . (1.2.13)

where umax and vmax respectively represent the maximum frequency of sampledspectrum in the u and v directions. Figure1.5 shows that the spectrum of the sampledfunction using the sampling rate lower than the twice of the maximum frequencyalong the u and v directions, respectively.


Fig. 1.5 The Fourier transform of the sampled function with the sampling rate than the Nyquist-Shannon sampling rate: a the spectrum of sampled function using 1

T < 2umax , and b the spectrumof sampled function using 1

Q < 2vmax

References

1. Andrews, H.C., Tescher, A.G., Kruger, R.P.: Image processing by digital computer. IEEE Spectr.9(7), 20–32 (1972)

2. Bayer, B.E.: Color imaging array. U.S. Patent No. 3,971,065 (1976)3. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 3rd edn. Prentice Hall, New Jersey

(2006)4. Ko, S., Yu, S., Kang, W., Park, C., Lee, S., Paik, J.: Artifact-free low-light video enhancement

using temporal similarity and guide map. IEEE Trans. Ind. Electron. 64(8), 6392–6401 (2017)

References 11

5. Ko, S, Yu, S., Kang, W., Park, S., Moon, B., Paik, J.: Variational framework for low-light imageenhancement using optimal transmission map and combined l1 and l2-minimization. SignalProcess.: Image Commun. 58, 99–110 (2017)

6. Malvar, H.S., He, L., Cutler, R.: High quality linear interpolation for demosaicing of bayer-patterned color images. Proc IEEE Int. Conf. Acoust. Speech Signal Process. 34(11), 2274–2282(2004)

7. Park, S., Yu, S., Moon, B., Ko, S., Paik, J.: Low-light image enhancement using variationaloptimization-based retinex model. IEEE Trans. Consum. Electron. 63(2), 178–184 (2017)

8. Park, S., Moon, B., Ko, S., Yu, S., Paik, J.: Low-light image restoration using bright channelprior-based variational retinex model. EURASIP J. Image Video Process. 2017(1), 1–11 (2017)

Part IIMultiscale Transform

Chapter 2Fourier Analysis and Fourier Transform

2.1 Overview

The origins of Fourier analysis in science can be found in Ptolemy’s decomposingcelestial orbits into cycles and epicycles and Pythagoras’ decomposing music intoconsonances. Itsmodern history beganwith the eighteenth centurywork ofBernoulli,Euler, and Gauss on what later came to be known as Fourier series. J. Fourier in1822 [Theorie analytique de la Chaleur] was the first to claim that arbitrary periodicfunctions could be expanded in a trigonometric (later called a Fourier) series, aclaim that was eventually shown to be incorrect, although not too far from the truth.It is an amusing historical sidelight that this work won a prize from the FrenchAcademy, in spite of serious concerns expressed by the judges (Laplace, Lagrange,and Legendre) regarding Fourier’s lack of rigor. Fourier was apparently a betterengineer than mathematician. Dirichlet later made rigorous the basic results forFourier series and gave precise conditions under which they applied. The rigoroustheoretical development of general Fourier transforms did not follow until about onehundred years later with the development of the Lebesgue integral.

Fourier analysis is a prototype of beautiful mathematics with many-faceted appli-cations not only in mathematics, but also in science and engineering. Since the workon heat flow of Jean Baptiste Joseph Fourier (March 21, 1768 toMay 16, 1830) in thetreatise entitled Theorie Analytique de la Chaleur, Fourier series and Fourier trans-forms have gone from triumph to triumph, permeating mathematics such as partialdifferential equations, harmonic analysis, representation theory, number theory andgeometry. Their societal impact can best be seen from spectroscopy to the effect thatatoms, molecules and hence matters can be identified by means of the frequencyspectrum of the light that they emit. Equipped with the fast Fourier transform incomputations and fulled by recent technological innovations in digital signals andimages, Fourier analysis has stood out as one of the very top achievements ofmankindcomparable with the Calculus of Sir Isaac Newton.


15

16 2 Fourier Analysis and Fourier Transform

The Fourier transform is of fundamental importance to image processing. It allowsus to perform tasks which would be impossible to perform any other way; its effi-ciency allows us to perform other tasks more quickly. The Fourier Transform pro-vides, among other things, a powerful alternative to linear spatial filtering; it is moreefficient to use the Fourier transform than a spatial filter for a large filter. The FourierTransform also allows us to isolate and process particular image frequencies, and soperform low-pass and high-pass filtering with a great degree of precision.

2.2 Fourier Series

The concept of frequency and the decomposition of waveforms into elementary“harmonic” functions first arose in the context of music and sound.

2.2.1 Periodic Functions

A periodic function is a function that repeats its value in regular intervals or periods.A function f is said to be periodic with period T (T �= 0) if f (x + T ) = f (x) forall values of x in the domain. The most important examples are the trigonometricfunctions (i.e. sine or cosine), which repeat values over the intervals of 2π.

The sine function f (x) = sin(x) has the value 0 at the origin and performs exactlyone full cycle between the origin and the point x = 2π. Hence f (x) = sin(x) is aperiodic function with period 2π, i.e.

sin(x) = sin(x + 2π) = sin(x + 4π) = · · · = sin(x + 2nπ), (2.2.1)

for all n ∈ Z. The same is true for cosine function except its value is 1 at the origini.e. cos(0) = 1, see Fig. 2.1.

2.2.2 Frequency and Amplitude

The number of oscillations of sin(x) over the distance T = 2π is one and thus thevalue of the angular frequency ω = 2π/T = 1. If f (x) = sin(3x), we obtain acompressed sine wave that oscillates three times faster than the original functionsin(x). The function sin(3x) performs five full cycles over a distance of 2π and thushas the angular frequency ω = 3 and a period T = 2π/3, see Fig. 2.2.

In general, the period T relates the angular frequency ω as

T = 2π

ω(2.2.2)

2.2 Fourier Series 17

Fig. 2.1 Sine and Cosine Graph

Fig. 2.2 sin(x) and sin(3x) Graph


for ω > 0. The relationship between the angular frequency ω and the commonfrequency f is given by

f = 1

T= ω

2πor ω = 2π f, (2.2.3)

where f is measured in cycles per length or time unit. A sine or cosine functionoscillates between the peak values +1 and −1 and its amplitude is 1. Multiplyingby a constant a ∈ R changes the peak values of the function to +a and −a and itsamplitude to a. In general, the expression

a · sin(ωx) and a · cos(ωx) (2.2.4)

denotes a sine or cosine function with amplitude a and angular velocity ω, evaluatedat position (or point in time) x .

2.2.3 Phase

Phase is the position of a point in time (an instant) on a waveform cycle. A completecycle is defined as the interval required for the waveform to return to its arbitraryinitial value. In sinusoidal functions or inwaves “phase” has two different, but closelyrelated, meanings. One is the initial angle of a sinusoidal function at its origin and issometimes called phase offset or phase difference. Another usage is the fraction ofthe wave cycle that has elapsed relative to the origin.

Shifting a sine function along the x-axis by distance ϕ,

sin(x) → sin(x − ϕ), (2.2.5)

changes the phase of the sine wave and ϕ denotes the phase angle of the resultingfunction, see Fig. 2.3. Thus, we have

sin(nx) = cos(ωx − π/2). (2.2.6)

i.e. cosine and sine functions are orthogonal in a sense and we can use this fact tocreate new sinusoidal function with arbitrary frequency, phase and amplitude. Inparticular, adding a cosine and a sine function with the identical frequency ω andarbitrary amplitude A and B, respectively create another sinusoids

A · cos(ωx) + B · sin(ωx) = C · cos(ωx − ϕ). (2.2.7)

where c = √A2 + B2 and ϕ = tan−1(B/A).


Fig. 2.3 sin(x), sin(x − π/4) and sin(x − π) Graph

2.2.4 Fourier Series of Periodic Functions

As we seen earlier, sinusoidal function of arbitrary frequency, amplitude and phasecan be described as the sum of suitably weighted cosine and sine functions. Is itpossible to write non-sinusoidal functions to sum of cosine and sine functions? Itwas Fourier [Jean Baptiste Joseph de Fourier (1768–1830)] who first extended thisidea to arbitrary functions and showed that (almost) any periodic function f (x)witha fundamental frequency ω0 can be described as a infinite sum of harmonic sinusoidsi.e.

f (x) = A0

2+

∞∑

n=1

[Ancos(ω0nx) + Bnsin(ω0nx)]. (2.2.8)

This is called Fourier series and A0, An and Bn are called Fourier coefficients of thefunction f (x), where

A0 = 1

π

∫ π

−π

f (x)dx, (2.2.9)

An = 1

π

∫ π

−π

f (x)cos(ω0nx)dx, (2.2.10)

Bn = 1

π

∫ π

−π

f (x)sin(ω0nx)dx . (2.2.11)


A Fourier series is an expression of a periodic function f (x) in terms of an infinitesumof sines and cosines. Fourier seriesmake use of the orthogonality relationships ofsine and cosine functions, since these functions form a complete orthogonal systemover [−π,π] or any interval of length 2π. The computation and study of Fourierseries is known as Harmonic analysis and is extremely useful as a way to break up anarbitrary periodic function into a set of simple terms that can be plugged in, solvedindividually and then recombined to obtain the solution of the original problem ofan approximation to it to whatever accuracy is desired or practical.

More general form of Fourier series is

f (x) = A0

2+

∞∑

n=1

[Ancos(nx) + Bnsin(nx)], (2.2.12)

where

A0 = 1

π

∫ π

−π

f (x)dx, (2.2.13)

An = 1

π

∫ π

−π

f (x)cos(nx)dx, (2.2.14)

Bn = 1

π

∫ π

−π

f (x)sin(nx)dx . (2.2.15)

Themiracle of Fourier series is that as long as f (x) is continuous (or even piecewise-continuous, with some caveats discussed in the Stewart text), such a decompositionis always possible.

2.2.5 Complex Form of Fourier Series

The Fourier series representation for a periodic function f, can be expressed moresimply using complex exponentials. Moreover, because of the unique properties ofthe exponential function, Fourier series are often easier to manipulate in complexform. The transition from the real form to the complex form of a Fourier series ismade using the following identities, called Euler identities,

eiθ = cos(θ) + isin(θ) and e−iθ = cos(θ) − isin(θ). (2.2.16)

By adding these identities, and then dividing by 2, or subtracting them, and thendividing by 2i, we have

cos(θ) = eiθ + e−iθ

2and sin(θ) = eiθ − e−iθ

2i. (2.2.17)


The complex Fourier series is obtained from (2.2.12) bywriting cos(nx) and sin(nx)in their complex exponential form and rearranging as follows:

f (x) = A0

2+

∞∑

n=1

[An

(einx + e−inx

2

)+ Bn

(einx − e−inx

2i

)]

= A0

2+

∞∑

n=1

[An − i Bn

2

]einx +

−1∑

m=−∞

[A−m + i B−m

2

]eimx

wherewe substitutedm = n in the last term on the last line. Equation clearly suggeststhe much simpler complex form of the Fourier series

f (x) =∞∑

n=−∞Cne

inx , (2.2.18)

with the coefficients given by

Cn = 1

2π

∫ π

−π

f (x)e−inxdx . (2.2.19)

Note that the Fourier coefficients Cn are complex valued. It is seen that for a real-valued function f (x), the following holds for the complex coefficients Cn

C−n = Cn, (2.2.20)

where Cn denotes the complex conjugate of Cn .

2.3 Fourier Transform

In the previous section we have seen how to expand a periodic function as a trigono-metric series. This can be thought of as a decomposition of a periodic function interms of elementary modes, each of which has a definite frequency allowed by theperiodicity. This concept can be generalized to functions periodic on any interval.

If the function has period L , then the frequencies must be integer multiples ofthe fundamental frequency k = 2π/L . The Fourier series of functions of arbitraryperiodicity is

f (x) = A0

2+

∞∑

n=1

[Ancos(2πnx/L) + Bnsin(2πnx/L)], (2.3.1)


where

A0 = 1

L

∫ L/2

−L/2f (x)dx, (2.3.2)

An = 1

L

∫ L/2

−L/2f (x)cos(2πnx/L)dx, (2.3.3)

Bn = 1

L

∫ L/2

−L/2f (x)sin(2πnx/L)dx, (2.3.4)

or in the exponential notation,

f (x) =∞∑

n=−∞Cne

i2πnx/L , (2.3.5)

where

Cn = 1

L

∫ L/2

−L/2f (x)e−i2πnx/Ldx . (2.3.6)

Fourier serieswas a powerful one and forms the backbone of theFourier transform.The Fourier transform can be viewed as an extension of the above Fourier series tonon-periodic functions and allows us to deal with non-periodic functions. A non-periodic function can be thought of as a periodic function in the limit L → ∞.Clearly, the larger L is, the less frequently the function repeats, until in the limitL → ∞ the function does not repeat at all. In the limit L → ∞ the allowedfrequencies become a continuum and the Fourier sum goes over to a Fourier integral.

Consider a function f (x) defined on the real line. If f (x) were periodic withperiod L , then f (x) can be expand by Eq. (2.3.5) as Fourier series converging to italmost everywhere within each period [−L/2, L/2]. Even if f (x) is not periodic,we can still define a function

f (x) =∞∑

n=−∞Cne

i2πnx/L , (2.3.7)

with the same Cn as above. Consider the limit in which L become very large.Define

kn = 2nπ

L,

2.3 Fourier Transform 23

then

fL(x) =∞∑

n=−∞Cne

ikn x . (2.3.8)

It is clear that for very large L the sum contains a very large number of waves withwave-vector kn and that each successive wave differs from the last by a tiny changein wave-vector (or wavelength),

�k = kn+1 − kn = 2π

L.

In the limit L → ∞ the allowed k becomes a continuous variable, the discretecoefficients, Cn , become a continuous function of k, denoted by C(k) and the sum-mation can be replaced by an integral and we have

f (x) = 1

2π

∫ ∞

−∞C(k)eikxdx, (2.3.9)

C(k) =∫ ∞

−∞f (x)e−ikxdx . (2.3.10)

The functions f and C are called a Fourier transform pair, C is called the Fouriertransform of f and f is called the inverse Fourier transform of C .

This prompts us to define the 1D-Fourier transform of the function f (x) as

f (k) =∫ ∞

−∞f (x)e−ikxdx, (2.3.11)

provided that the integral exists. Not every function f (x) has a Fourier transform.A sufficient condition is that it be square-integrable; that is, so that the followingintegral converges:

|| f ||2 =∫ ∞

−∞| f (x)|2dx . (2.3.12)

If in addition of being square-integrable, the function is continuous, then one alsohas the inversion formula

f (x) = ˇ( f ) = 1

2π

∫ ∞

−∞f (k)eikxdx . (2.3.13)


2.3.1 2D-Fourier Transform

Two-dimensional (2D) Fourier transform of the function f (x, y) is defined as

f (k, l) =∫ ∞

−∞

∫ ∞

−∞f (x, y)e−i(kx+ly)dxdy, (2.3.14)

provided that the integral exists and a the two-dimensional (2D) inverse Fouriertransform is defined by

f (x, y) = ˇ( f ) = 1

2π

∫ ∞

−∞

∫ ∞

−∞f (k, l)ei(kx+ly)dkdl. (2.3.15)

2.3.2 Properties of Fourier Transform

Let f (k) and g(k) are Fourier transform of functions f (t) and g(t), respectively.Then we have the following:

1. Linearitya f + bg(k) = a f (k) + bg(k),

here a, b are constants, i.e. if we add two functions then the Fourier transform ofthe resulting function is simply the sum of the individual Fourier transforms and ifwe multiply a function by any constant then we must multiply the Fourier transformby the same constant.

2. Shifting There are two basic shift properties of the Fourier transform:

i. Time Shifting( f (t ± t0))(k) = f (k)e±ikt0 .

ii. Frequency Shifting

( f (t)e±ik0x )(k) = f (k ± k0).

Here t0 and k0 are constants. i.e. Translating a function in one domain correspondsto a multiplication by a complex exponential function in the other domain.

3. Scaling

f (ax)(k) = 1

af (

k

a),

2.3 Fourier Transform 25

here a is constant. When a signal is expanded in time, it is compressed in frequency,and vice versa i.e. we cannot be simultaneously short in time and short in frequency.

4. Differentiation

i. Time differentiation property

f (t)(k) = ik f (k).

Differentiating a function is said to amplify the higher frequency components becauseof the additional multiplying factor k.

ii. Frequency differentiation property

t f (t)(k) = id f (k)

dk.

5. Conjugate Symmetry The Fourier transform is conjugate symmetric for timefunctions that are real-valued,

f (−k) = f (k).

From this it follows that the real part and the magnitude of the Fourier transform ofreal valued time functions are even functions of frequency and that the imaginarypart and phase are odd functions of frequency. By property of conjugate symmetry,in displaying or specifying the Fourier transform of a real-valued time function it isnecessary to display the transform only for positive values of k.

6. Reversalf (−x)(k) = f (−k), for x, k ∈ R.

7. Duality This property relates to the fact that the analysis equation and synthesisequation look almost identical except for a factor of 1

2π and the difference of a minussign in the exponential in the integral.

f (t) = 1

2π

∫ ∞

−∞f (k)eikt dk ⇐⇒ f (k) =

∫ ∞

−∞f (t)e−ikt dt

8. Convolution The convolution theorem states that convolution in time domain cor-responds to multiplication in frequency domain and vice versa:

( f ∗ g)(t)(k) = f (k )g(k).

9. Parseval’s Relation∫ ∞

−∞| f (t)|2dt = 1

2π

∫ ∞

−∞| f (k)|2dk.


2.4 Discrete Fourier Transform

We assume that vectors in CN , i.e., sequences of N complex numbers, are indexed

from 0 to N − 1 instead of {1, 2, 3, . . . , N }.we regard x as a function defined on thefinite set

ZN = {0, 1, 2, . . . , N − 1}, (2.4.1)

and we identify x with column vector

x =

⎡

⎢⎢⎢⎢⎢⎢⎣

x0x1.

.

.

xN−1

⎤

⎥⎥⎥⎥⎥⎥⎦.

This allows us to write the product of N × N matrix A by x as Ax . To save space,we usually writ such a x horizontally instead of vertically x = (x0, x1, . . . , xN−1).

In order to be consistent with the notation for functions used later in the infinitedimensional context, we write l2(ZN ) in place of C

N . So, formally,

l2(ZN ) = {x = (x0, x1, . . . , xN−1) : x j ∈ C, 0 ≤ j ≤ N − 1}. (2.4.2)

With the usual component-wise addition and scalar multiplication, l2(ZN ) is an N -dimensional vector space over C. One basis for l2(ZN ) is an standard or Euclideanbasis E = {e0, e1, . . . , eN−1}, where

e j (n) ={1, if n = j

0, otherwise.(2.4.3)

In this notation, the complex inner product on l2(ZN ) is

〈x, y〉 =N−1∑

k=0

xk yk, (2.4.4)

with the associated norm

||x || =(

N−1∑

k=0

|xk |2)1/2

, (2.4.5)

called the l2-norm.

2.4 Discrete Fourier Transform 27

2.4.1 1D-Discrete Fourier Transform

Definition 2.1 Define E0, E1, . . . , EN−1 ∈ l2(ZN ) by

Em(n) = 1√Ne2πimn/N , for 0 ≤ m, n ≤ N − 1. (2.4.6)

Clearly, the set {E0, E1, . . . , EN−1} is an orthonormal basis for l2(ZN ).We have

x =N−1∑

m=0

〈x, Em〉Em, (2.4.7)

〈x, y〉 =N−1∑

m=0

〈x, Em〉〈y, Em〉, (2.4.8)

||x ||2 =N−1∑

m=0

|〈x, Em〉|2. (2.4.9)

By definition of inner product

〈x, Em〉 =N−1∑

n=0

xn1√Ne2πimn/N = 1√

N

N−1∑

m=0

xne−2πimn/N . (2.4.10)

Definition 2.2 Suppose x = (x0, x1, . . . , xN−1) ∈ l2(ZN ). For m = 0, 1, 2, . . . ,N − 1, define

xm =N−1∑

n=0

xne−2πimn/N . (2.4.11)

Then x = (x0, x1, . . . , xN−1) ∈ l2(ZN ). The map : l2(ZN ) → l2(ZN ),which takesx to x, is called the 1D-discrete Fourier transform (DFT).

It can easily see that xm, m ∈ Z is periodic with period N :

xm+N =N−1∑

m=0

xne−2πi(m+N )n/N =

N−1∑

m=0

xne−2πimn/N e−2πi Nn/N =

N−1∑

m=0

xne−2πimn/N = xm ,

since e−2πi Nn/N = e−2πin = 1, for every n ∈ Z. Comparing the Eqs. (2.4.10) and(2.4.11), we have

xm = √N 〈x, Em〉. (2.4.12)


Remark 2.1 Equation (2.4.12) actually defines the DFT coefficients xk for any indexk and resulting xk are periodic with period N in the index. We will thus sometimesrefers to the xk on the other ranges of the index, for example −N/2 < k ≤ N/2when N is even. Actually, even if N is odd, the range −N/2 < k ≤ N/2 worksbecause k is required to be an integer.

Equation (2.4.12) leads to the following reformulation of formulae (2.4.7), (2.4.8)and (2.4.9). Let x = (x0, x1, . . . , xN−1) and y = (y0, y1, . . . , yN−1) ∈ l2(ZN ). Then

(i) Fourier Inversion Formula:

xn = 1

N

N−1∑

m=0

xme−2πimn/N , for n = 0, 1, 2, . . . , N − 1. (2.4.13)

(ii) Parseval’s Relation:

〈x, y〉 = 1

N

N−1∑

m=0

xm ym = 1

N〈xm, ym〉. (2.4.14)

(ii) Plancherel Theorem:

||x ||2 = 1

N

N−1∑

m=0

|xm |2 = 1

N||xm ||2. (2.4.15)

The DFT can be represented by matrix, since Eq. (2.4.11) shows that the maptaking x to x is a linear transformation. Define

wN = e−2πi/N .

Then we havee−2πimn/N = wmn

N and e2πimn/N = w−mnN ,

and

xm =N−1∑

n=0

xnwmnN . (2.4.16)

Definition 2.3 LetWN be thematrix [wmn]0≤m,n≤N−1, such thatwmn = wmnN .Hence

WN =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 1 1 . . 11 wN w2

N · · wN−1N

1 w2N w4

N · · w2(N−1)N· · · · · ·

· · · · · ·1 w

(N−1)N w

2(N−1)N · · w(N−1)(N−1)

N

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

.


Regarding x, x ∈ l2(ZN ) as column vectors (as Eq.2.4.11), the mth componentof WNx is

N−1∑

n=0

wmnxn =N−1∑

n=0

xnwmnN = xm 0 ≤ m ≤ N − 1.

Hence

x = WNx . (2.4.17)

Now, we only compute a simple example in order to demonstrate the definitions.We could use Eq. (2.4.12) for this, but it is easier to use Eq. (2.4.17). The values ofmatrices W2 and W4 are as follows:

W2 =[1 11 −1

]and W4 =

⎡

⎢⎢⎣

1 1 1 11 −i −1 i1 −1 1 −11 i −1 −i

⎤

⎥⎥⎦ .

Example 2.1 Let x = (1, 0,−3, 4) ∈ l2(Z4). Find x .

Solution

x = W4x =

⎡

⎢⎢⎣

1 1 1 11 −i −1 i1 −1 1 −11 i −1 −i

⎤

⎥⎥⎦

⎡

⎢⎢⎣

10

−34

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

24 + 4i−6

4 − 4i

⎤

⎥⎥⎦ .

The matrix WN has a lot of structure. This structure can even be exploited todevelop an algorithm called the fast Fourier transform that provides a very efficientmethod for computing DFT’s without actually doing the full matrix multiplication.

Definition 2.4 (Convolution) For x, y ∈ l2(ZN ), the convolution x ∗ y ∈ l2(ZN ) isthe vector with components

(x ∗ y)(m) =N−1∑

m=0

x(m − n)y(n), (2.4.18)

for all m.

Suppose x, y ∈ l2(ZN ). Then for each m,

(x ∗ y)m = xm ym . (2.4.19)


2.4.2 Inverse 1D-Discrete Fourier Transform

To interpret the Fourier inversion formula (2.4.13), wemake the following definition:

Definition 2.5 Let m = 0, 1, 2, . . . , N − 1. Define Fm ∈ l2(ZN ) by

Fm(n) = 1

Ne2πimn/N , for 0 ≤ n ≤ N − 1. (2.4.20)

Then F = {F0, F1, . . . , FN−1} is called the Fourier basis for l2(ZN ).

Form Eq. (2.4.6) we have

Fm = 1√NEm . (2.4.21)

Since Em is orthonormal basis for l2(ZN ), F is an orthogonal basis for l2(ZN ).Withthis notation, Eq. (2.4.13) becomes

x =N−1∑

m=0

xm Fm . (2.4.22)

The Fourier inversion formula (2.4.13) shows that the linear transformation: l2(ZN ) → l2(ZN ) is a one-onemap. Therefore is invertible. Hence, Eq. (2.4.13)gives us a formula for the inverse of discrete Fourier transform and it is denoted by .

Definition 2.6 Let y = (y0, y1, . . . , yN−1) ∈ l2(ZN ). Define

yn = 1

N

N−1∑

m=0

yne2πimn/N For n = 0, 1, 2, . . . , N − 1. (2.4.23)

Then y = (y0, y1, . . . , yN−1) ∈ l2(ZN ). The mapˇ : l2(ZN ) → l2(ZN ), which takesy to y, is called the 1D-inverse discrete Fourier transform (IDFT).

We can easily see that ym,m ∈ Z is also periodic function with period N . Fourierinversion formula states that for x ∈ l2(ZN ),

ˇ(xn) = xn or (xn) = xn, for n = 0, 1, 2, . . . , N − 1.

Since the DFT is an invertible linear transformation, the matrix WN is invertible andwe must have x = W−1

N x . Substituting x = y and equivalently x = y in equations,we have

y = W−1N y. (2.4.24)


In the notation of formula (2.4.16), formula (2.4.21) becomes

y =N−1∑

n=0

yn1

Nw−mn

N =N−1∑

n=0

yn1

Nwmn

N .

This shows that the (n,m) entry of W−1N is wmn

NN , which is 1/N times of the complex

conjugate of the (n,m) entry of WN . If we denote by WN the matrix whose entriesare the complex conjugates of the entries of WN , we have

W−1N = 1

NWN .

We have

W−12 = 1

2

[1 11 −1

]and W−1

4 = 1

4

⎡

⎢⎢⎣

1 1 1 11 i −1 −i1 −1 1 −11 −i −1 i

⎤

⎥⎥⎦ .

Example 2.2 Let y = (2, 4 + 4i,−6, 4 − 4i) ∈ l2(Z4). Find y.

Solution

W−14 = 1

4

⎡

⎢⎢⎣

1 1 1 11 i −1 −i1 −1 1 −11 −i −1 i

⎤

⎥⎥⎦

⎡

⎢⎢⎣

24 + 4i−6

4 − 4i

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

10

−34

⎤

⎥⎥⎦ .

2.4.3 2D-Discrete Fourier Transform and 2D-InverseDiscrete Fourier Transform

The definition of the two-dimensional (2D) discrete Fourier transform is very similarto that for one dimension. The forward and inverse transforms for an M × N matrix,where for notational convenience we assume that the m indices are from 0 to M − 1and the n indices are from 0 to N − 1 are:

x(r,s) =M−1∑

m=0

N−1∑

n=0

x(m.n)e−2πi( mr

M + nsN ), (2.4.25)

and

x(r,s) = 1

MN

M−1∑

m=0

N−1∑

n=0

x(m,n)e2πi( mr

M + nsN ). (2.4.26)


2.4.4 Properties of 2D-Discrete Fourier Transform

All the properties of the one-dimensional DFT transfer into two dimensions. Butthere are some further properties not previously mentioned, which are of particularuse for image processing.

1. Similarity First notice that the forward and inverse transforms are very similar,with the exception of the scale factor 1/MN in the inverse transform, and the negativesign in the exponent of the forward transform. This means that the same algorithm,only very slightly adjusted, can be used for both the forward an inverse transforms.

2. The DFT as a spatial filter Note that the values

e±2πi( mrM + ns

N ) (2.4.27)

are independent of the values x or x . This means that they can be calculated inadvance, and only then put into the formulas above. It also means that every valueof x(r,s) is obtained by multiplying every value of x(r,s) by a fixed value, and addingup all the results. But this is precisely what a linear spatial filter does: it multipliesall elements under a mask with fixed values, and adds them all up. Thus we canconsider the DFT as a linear spatial filter which is as big as the image. To deal withthe problem of edges, we assume that the image is tiled in all directions, so that themask always has image values to use.

3. Separability Notice that the discrete Fourier transform filter elements can beexpressed as products:

e2πi(mrM + ns

N ) = e2πi(mrM )e2πi(

nsN ). (2.4.28)

The first product valuee2πi(

mrM ) (2.4.29)

depends only on m and r, and is independent of n and s. Conversely, the secondproduct value

e2πi(nsN ) (2.4.30)

depends only on n and s, and is independent of m and r . This means that we canbreak down our formulas above to simpler formulas that work on single rows orcolumns:

xm =M−1∑

r=0

xr e−2πimr/M , (2.4.31)

xr = ˇ(xm)r = 1

M

M−1∑

m=0

xme2πimr/M . (2.4.32)


If we replacem and r with n and s we obtain the corresponding formulas for the DFTof matrix columns. These formulas define the one-dimensional DFT of a vector, orsimply the DFT.

The 2D DFT can be calculated by using this property of separability; to obtainthe 2D DFT of a matrix, we first calculate the DFT of all the rows, and then calculatethe DFT of all the columns of the result. Since a product is independent of the order,we can equally well calculate a 2D DFT by calculating the DFT of all the columnsfirst, then calculating the DFT of all the rows of the result.

4. Linearity An important property of the DFT is its linearity. the DFT of a sum isequal to the sum of the individual DFT’s, and the same goes for scalar multiplication:

x + y = x + y

andkx = kx,

where k is a scalar, and x and y are matrices.This property is of great use in dealing with image degradation such as noise

which can be modelled as a sum:

d = f + η,

where f is the original image, η is the noise, and d is the degraded image. Since

d = f + η

we may be able to remove or reduce η by modifying the transform. As we shall see,some noise appears on the DFT in a way which makes it particularly easy to remove.

5. Convolution Theorem This result provides one of the most powerful advantagesof using the DFT. Suppose we wish to convolve an image M with a spatial filter S.Our method has been place S over each pixel of M in turn, calculate the product ofall corresponding gray values of M and elements of S, and add the results. The resultis called the digital convolution of M and S, is denoted by

M ∗ S.

This method of convolution can be very slow, especially if S is large.

The DC coefficient The value x(0,0) of the DFT is called the DC coefficient. If weput r = s = 0 in the Eq. (3.2.25), we have

x(0,0) =M−1∑

m=0

N−1∑

n=0

x(m,n)e−2πi( m0

M + n0N ) =

M−1∑

m=0

N−1∑

n=0

x(m,n).

http://dx.doi.org/10.1007/978-981-10-7272-7_3


That is, this term is equal to the sum of all terms in the original matrix.

6. Shifting For purposes of display, it is convenient to have the DC coefficient inthe centre of the matrix. This will happen if all elements x(m,n) in the matrix aremultiplied by (−1)m+n before the transform.

7. Conjugate Symmetry An analysis of the Discrete Fourier transform definitionleads to a symmetry property, i.e., if wemake the substitutionsm = −m and n = −nin Eq. (2.4.25), then

x(m,n) = x(−m+pM,−n+qN ),

for any integers p and q. This means that half of the transform is a mirror image ofthe conjugate of the other half. We can think of the top and bottom halves, or the leftand right halves, being mirror images of the conjugates of each other.

8. Displaying Transforms Having obtained the Fourier transform x of an image x ,we would like to see what it looks like. As the elements x are complex numbers, wecan’t view them directly, but we can view their magnitude |x |, since these will benumbers of type double, generally with large range. The display of the magnitude ofa Discrete Fourier transform is called the spectrum of the transform.

2.5 Fast Fourier Transform

The fast Fourier transform (FFT) is a class of algorithms for computing the DFTand IDFT efficiently and it is one of the most influential algorithms of the twentiethcentury. Cooley and Tukey [6] in 1965 published the modern version of the FFTalgorithms, but most of the ideas appeared earlier. Given the matrixWN , it is not sur-prising that shortcuts can be found for computing thematrix-vector productWN x . Allmodern software uses the FFT algorithm for computing discrete Fourier transformsand the details are usually transparent to the user. There are many of points of viewon the FFT as a computer scientist would classify the FFT as a classic divide andconquer algorithm, a mathematician might view the FFT as a natural consequenceof the structure of certain finite groups and other practitioners might simply view itas a efficient technique for organizing the sums involved in Eq. (2.4.11).

From Eq. (2.4.17), we have x = WNx, where WN is the matrix in Definition 2.3.Clearly, direct computation of x takes N 2 complex multiplications. More precisely,we could also count the number of additions. Though, we get a good idea of thespeed of computation by just considering the number of complex multiplicationsrequired because multiplication is much slower on a computer than addition. Here,complex multiplication, we mean the multiplication of two complex numbers. Thiswould appear to require four real multiplications, but by a trick, it requires only threereal multiplications. In signal and image processing, the vectors under considerationcan be very large. Computation of the DFTs of these vectors in real time by directmeans may be beyond the capacity of ones computational hardware. So a fast algo-

2.5 Fast Fourier Transform 35

rithm is needed known as FFT. Our intention is merely to demonstrate a very briefintroduction to the idea behind one form of the FFT algorithm.

Now, We begin with the simplest version of the FFT, in which the length N of thevector is assumed to be even.

Theorem 2.1 Let M ∈ N, with N = 2M and let x ∈ l2(ZN ). Define u, v ∈ l2(ZM)

by

uk = x2k f or k = 0, 1, 2, . . . ,M − 1, or uk = (x0, x2, x4, . . . xN−4, xN−2

),

and

vk = x2k+1 f or k = 0, 1, 2, . . . ,M − 1 or vk = (x1, x3, x5, . . . xN−3, xN−1

).

Let x denote the DFT of x defined on N points, that is, x = WNx . Let u and v denotethe DFTs of u and v respectively,defined on M = N/2 points, that is, u = WMu andv = WMv. Then for m = 0, 1, 2, . . . ,M − 1,

x(m) = u(m) + e−2πim/N v(m). (2.5.1)

Also, for m = M,M + 1,M + 2, . . . , N − 1, let l = m − M. Note that thecorresponding values of l are l = 0, 1, 2, . . . ,M − 1. Then

x(m) = x(l + M) = u(l) − e−2πil/N v(l). (2.5.2)

Proof By definition, for m = 0, 1, 2, . . . , N − 1,

xm =N−1∑

n=0

xne−2πimn/N ,

The sum over n = 0, 1, 2, . . . , N − 1 can be broken up into the sum over theeven values n = 2k, k = 0, 1, 2, . . . ,M − 1, plus the sum over the odd valuesn = 2k + 1, k = 0, 1, 2, . . . ,M − 1 :

xm =M−1∑

k=0

x2ke−2πi2 km/N +

M−1∑

k=0

x2k+1e−2πi(2k+1)m/N ,

=M−1∑

k=0

uke−2πikm/(N/2) + e−2πim/N

M−1∑

k=0

vke−2πikm/(N/2),

=M−1∑

k=0

uke−2πikm/M + e−2πim/N

M−1∑

k=0

vke−2πikm/M .


In the case m = 0, 1, 2, . . . ,M − 1, the last expression is u(m)+ e−2πim/N v(m), sowe have Eq. (2.5.1). Now suppose m = M,M + 1,M + 2, . . . , N − 1. By writingm = l + M as in the statement of the theorem and substituting this for m above, weget

xm =M−1∑

k=0

uke−2πik(l+m)/M + e−2πi(l+m)/N

M−1∑

k=0

vke−2πik(l+m)/M ,

=M−1∑

k=0

uke−2πikl/M − e−2πil/N

M−1∑

k=0

vke−2πikl/M ,

since the exponentials e−2πikl/M are periodic with period M , and e−2πiM/N =e−πi = −1 for N = 2M. Hence Eq. (2.5.2) proves. �

Notice that the same values are used in Eqs. (2.5.1) and (2.5.2) and to applyEqs. (2.5.1) and (2.5.2), we first compute u and v. Each can be computed directlywithM2 complex multiplications since each of these is a vector of length M = N/2.Thencompute the products e−2πim/N v(m) for m = 0, 1, 2, . . . ,M − 1, this requires anadditional M multiplications. Rest is done using only additions and subtractionsof these quantities, which we do not count. Hence, the total number of complexmultiplications required to compute x by Eqs. (2.5.1) and (2.5.2) is at most

2M2 + M = 2

(N

2

)2

+ N

2= 1

2(N 2 + N ).

For N large, this is essentially N 2/2,whereas the number of complexmultiplicationsrequired to compute x directly is N 2. Thus, Theorem2.1 already cuts the computationtime nearly in half.

If N is divisible by 4 instead of just 2, we can proceed further. Similarly, ifN is divisible by 8, we can carry this one step further, and so on. Since u and v

have even order, we can then apply the same method to reduce the time required tocompute them. A more general way to describe this is to define #N , for any positiveinteger N , to be the least number of complex multiplications required to computethe DFT of a vector of length N . If N = 2M , then Eqs. (2.5.1) and (2.5.2) reducethe computation of x to the computation of two DFTs of size M , plus M additionalcomplex multiplications. Hence

#N ≤ 2#M + M. (2.5.3)

The most favorable case is when N is a power of 2. He we have the following:

Theorem 2.2 Let N = 2n for some n ∈ N. Then

2.5 Fast Fourier Transform 37

#N ≤ 1

2Nlog2N .

Proof For n = 1, a vector of length 21 is of the form x = (a, b). Then x = (a +b, a−b).Notice that this computation does not require any complex multiplications,so #2 = 0 < 1 = (2log22)/2. The result holds for n = 1. By induction, suppose itholds for n = k − 1. Then for n = k, we have by Eq. (2.5.3) and

#2k ≤ 2#2k−1 + 2k−1 ≤ 21

22k−1(k − 1) + 2k−1 = k2k−1 = 1

2k2k = 1

2N log2N .

Hence result holds for n = k. Thus the result true for all n. �

For a vector of size 262, 144 = 218, the FFT reduces the number of complexmultiplications needed to compute the DFT from 6.87 × 1010 to 2, 359, 296, thusmaking the computation more than 29, 000 times faster. Hence, if it takes 8 hours todo this via DFT directly, then it would take about 1 second to do it via the FFT. As Nincreases, this ratio becomes more extreme to the point that some computations thatcan be done by the FFT in a reasonable length of time could not be done directly inan entire lifetime. The FFT is usually implemented without explicit recursion. TheFFT is not limited to N that are the powers of 2. What if N is not even? If N is prime,the method of the FFT does not apply. However, an efficient FFT algorithm can bederived most easily when N is “highly Composite” i.e. factors completely into smallintegers. In general, if N is composite, say N = pq, a generalization of Theorem2.1can be applied.

Theorem 2.3 Let p, q ∈ N, and N = pq. Let x ∈ l2(ZN ). Define w0, w1, . . . ,

wp−1 ∈ l2(Zq) by

wl(k) = xkp+l f or k = 0, 1, 2, . . . , q − 1.

For b = 0, 1, 2, . . . , q − 1, define vb ∈ l2(Zp) by

vb(l) = e−2πibl/N wl(b) f or l = 0, 1, 2, . . . , p − 1.

Then for a = 0, 1, 2, . . . , p − 1 and b = 0, 1, 2, . . . , q − 1,

x(aq + b) = vb(a). (2.5.4)

Note that by the division algorithm, every m = 0, 1, 2, . . . , N − 1 is of the formaq + b for some a ∈ {0, 1, 2, . . . , p− 1} and b ∈ {0, 1, 2, . . . , q − 1}, so Eq. (2.5.4)gives the full DFT of x .

Proof We can write each n = 0, 1, . . . , N − 1 uniquely in the form kp + l for somek ∈ {0, 1, 2, . . . , q − 1} and l ∈ {0, 1, 2, . . . , p − 1}. Hence


xaq+b =N−1∑

n=0

xne−2πi(aq+b)n/N =

p−1∑

l=0

q−1∑

k=0

xkp+l e−2πi(aq+b)(kp+l)/(pq).

Note that

e−2πi(aq+b)(kp+l)/(pq) = e−2πiake−2πial/pe−2πibk/qe−2πibl/(pq).

Since e−2πiak = 1 and pq = N , using the definition of wl(k) we obtain

xaq+b =p−1∑

l=0

e−2πial/pe−2πibl/Nq−1∑

k=0

e−2πibk/q

=p−1∑

l=0

e−2πial/pe−2πibl/N wl(b)

=p−1∑

l=0

e−2πial/pvb(l) = vb(a).

�This proof shows the basic principle behind the FFT. In computing xaq+b, the

same quantities vb(l), 0 ≤ l ≤ p − 1, arise for each value of a. The FFT algorithmrecognizes this and computes these values only once. We first compute the vectorswl , for l = 0, 1, ..., p − 1. Each of these is a vector of length q, so computingeach wl requires #q complex multiplications. So this step requires a total of p#qcomplex multiplications. The next step is to multiply each wl(b) by e e−2πibl/N toobtain the vectors vb(l). This requires a total of pq complex multiplications, onefor each of the q values of b and p values of l. Finally we compute the vectors vbfor b = 0, 1, ..., q − 1. Each vb is a vector of length p, so each of the q vectors vbrequires #p complex multiplications, for a total of q#p multiplications. Adding up,we have an estimate for the number of multiplications required to compute a DFTof size N = pq, namely

#pq ≤ p#q + q#p + pq. (2.5.5)

This estimate can be used inductively to make various estimates on the time requiredto compute the FFT. The advantage of using the FFT is greater the more compositeN is. There are many variations on the FFT algorithm, sometimes leading to slightadvantages over the basic one given here. But the main point is that the DFT ofa vector of length N = 2n can be computed with at most n2n−1 = (N/2)log2Ncomplex multiplications as opposed to N 2 = 22n if done directly.

Since xn = 1N xN−n and x ∗ y = (x y), the FFT algorithm can be used to compute

the IDFT and convolutions quickly also. In computing IDFT, at most (N/2)log2Nsteps requires if N is a power of 2. We do not count division by N because integerdivision is relatively fast. If x, y ∈ l2(ZN ), for N a power of 2, it takes at most

2.6 The Discrete Cosine Transform 39

N log2N multiplications to compute x and y, and N multiplications to compute x y,and at most (N/2)log2N multiplications to take the IDFT of x y. Thus overall it takesno more than N + (3N/2)log2N multiplications to compute z ∗ w.

2.6 The Discrete Cosine Transform

The Fourier transform and the DFT are designed for processing complex valuedsignals, and they always produce a complex-valued spectrum even in the case wherethe original signal was strictly real-valued. The reason is that neither the real nor theimaginary part of the Fourier spectrum alone is sufficient to represent (i.e., recon-struct) the signal completely. In other words, the corresponding cosine (for the realpart) or sine functions (for the imaginary part) alone do not constitute a complete setof basis functions.

Further, we know that a real-valued signal has a symmetric Fourier spectrum, soonly one half of the spectral coefficients need to be computed without losing anysignal information.

There are several spectral transformations that have properties similar to the DFTbut do not work with complex function values. The discrete cosine transform (DCT)is well known example that is particularly interesting in our context because it is fre-quently used for image and video compression. The DCT uses only cosine functionsof various wave numbers as basis functions and operators on real-valued signals andspectral coefficients. Similarly, there is also a discrete sine transform (DST) basedon a system of sine functions.

2.6.1 1D-Discrete Cosine Transform

In the one-dimensional case, the discrete cosine transform (DCT) for a signal g(n)of length N is defined as

G(m) =√

2

N

N−1∑

n=0

g(n) · cmcos(

πm(2n + 1)

2N

), (2.6.1)

for 0 ≤ m < N , and the inverse discrete cosine transform (IDCT) is

g(n) =√

2

N

N−1∑

m=0

G(m) · cmcos(

πm(2n + 1)

2N

), (2.6.2)

for 0 ≤ n < N , respectively, with


cm ={

1√2, ifm = 0

1, otherwise.(2.6.3)

Note that the index variables (n,m) are used differently in the forward transform andthe inverse transform, so the two transform are, in contrast to theDFT, not symmetric.One may ask why it is possible that the DCT can work without any sine functions,while they are essential in the DFT. The trick is to divide all frequencies in half suchthat they are spaced more densely and thus the frequency resolution in the spectrumis doubled. Comparing the cosine parts of the DFT basis functions (Eq.2.4.11) andthose of the DCT (Eq.2.6.1), we have

DFT: CNn (m) = cos

(2π mn

N

)

DCT: DNn (m) = cos

(π m(2n+1)

2N

)= cos

(π 2m(n+0.5)

2N

),

one can only see that the period of the DCT basis functions (2N/m) is double theperiod of DFT functions (M/m) and DCT functions are also phase-shifted by 0.5units. of course, much more efficient (fast) algorithms exist. Moreover, the DCT canalso be computed in O(M log2M) time using FFT. The DCT is often used for imagecompression, in particular JPEG compression, where the size of the transformed subimages is fixed at 8 × 8 and the processing can highly be optimized.

2.6.2 2D-Discrete Cosine Transform

The two-dimensional formof theDCTfollows immediately from theone-dimensionaldefinition, resulting in 2D forward transform

G(m1,m2) =√

2

N1N2

N1−1∑

n1=0

N2−1∑

n2=0

g(n1, n2)

· cm1cos

(πm1(2n1 + 1)

2N1

)· cm2cos

(πm2(2n2 + 1)

2N2

), (2.6.4)

for 0 ≤ m1 < N1, 0 ≤ m2 < N2, and the 2D-inverse DCT is

g(n1, n2) =√

2

N1N2

N1−1∑

n1=0

N2−1∑

n2=0

G(m1,m2)

· cm1cos

(πm1(2n1 + 1)

2N1

)· cm2cos

(πm2(2n2 + 1)

2N2

), (2.6.5)

for 0 ≤ m1 < N1, 0 ≤ m2 < N2.

2.7 Heisenberg Uncertainty Principle 41

2.7 Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle was originally stated in physics, and claimsthat it is impossible to know both the position and momentum of a particle simul-taneously. However, it has an analog basis in signal processing. In terms of signals,the Heisenberg uncertainty principle is given by the rule that it is impossible to knowboth the frequency and time at which they occur. The time and frequency domains arecomplimentary. If one is local, the other is global. Formally, the uncertainty principleis expressed as

(�t)2(�ω)2 ≥ 1

4. (2.7.1)

In the case of an impulse signal, which assumes a constant value for a brief periodof time, the frequency spectrum is finite; whereas in the case of a step signal whichextends over infinite time, its frequency spectrum is a single vertical line. This factshows that we can always localize a signal in time or in frequency but not bothsimultaneously. If a signal has a short duration, its band of frequency is wide andvice-versa.

2.8 Windowed Fourier Transform or Short-Time FourierTransform

The short-time Fourier transform (STFT) is a modified version of Fourier transform.The Fourier transform separates the input signal into a sum of sinusoids of differentfrequencies and also identities their respective amplitudes. Thus, the Fourier trans-form gives the frequency-amplitude representation of an input signal. The Fouriertransform is not an effective tool to analyses non-stationary signals. In STFT, thenon-stationary signal is divided into small portions, which are assumed to be station-ary. This is done using a window function of a chosen width, which is shifted andmultiplied with the signal to obtain small stationary signals.

2.8.1 1D and 2D Short-Time Fourier Transform

The short-time Fourier transform maps a signal into two-dimensional function oftime and frequency. The STFT of a one-dimensional signal f (t) is represented byX (τ ,ω) where

X (τ ,ω) =∫ ∞

−∞f (t)g∗(t − τ )e−iωt dt, (2.8.1)


Fig. 2.4 The time-frequency tiling of STFT

Here, g∗ denotes the conjugate of g, f (t) represents the input signal, gτ ,ω(t) =g(t − τ )eiωt is a temporal window with finite support and X (τ ,ω) is the time fre-quency atom. Also, the non-stationary signal f (t) is assumed to be approximatelystationary in the span of the temporal window gτ ,ω(t).

In the case of a 2D signal f (x, y), the space-frequency atom or STFT is given by

X (τ1, τ2,ω1,ω2) =∫ ∞

−∞

∫ ∞

−∞f (x, y)g∗(x − τ1, x − τ2)e

−i(ω1x+ω2 y)dxdy, (2.8.2)

where τ1, τ2 represents the spatial position of the two-dimensional windowgτ1,τ2,ω1,ω2(x, y) and ω1,ω2 represents the spatial frequency parameters. The per-formance of STFT for specific application depends on the choice of the window.Different types of windows that can be used in STFT are Hamming, Hanning, Gaus-sian and Kaiser windows. The time-frequency tiling of STFT is given in Fig. 2.4.

2.8.2 Drawback of Short-Time Fourier Transform

The main drawback of STFT is that once a particular size time window is chosen, thewindow remains the same for all frequencies. To analyze the signal effectively, amoreflexible approach is needed where the window size can vary in order to determinemore accurately either the time or frequency information of the signal. This problemis known as resolution problem.

2.9 Other Spectral Transforms

Apparently, the Fourier transform is not the only way to represent a given signal infrequency space; in fact, numerous similar transforms exist. Some of these, such asthe discrete cosine transform, also use sinusoidal basis functions, while others, such

2.9 Other Spectral Transforms 43

as the Hadamard transform (also known as the Walsh transform), build on binary0/1-functions. All of these transforms are of global nature; i.e., the value of anyspectral coefficient is equally influenced by all signal values, independent of thespatial position in the signal. Thus a peak in the spectrum could be caused by ahigh-amplitude event of local extent as well as by a widespread, continuous waveof low amplitude. Global transforms are therefore of limited use for the purpose ofdetecting or analyzing local events because they are incapable of capturing the spatialposition and extent of events in a signal. A solution to this problem is to use a set oflocal, spatially limited basis functions (wavelets) in place of the global, spatially fixedbasis functions. The correspondingwavelet transform, ofwhich several versions havebeen proposed, allows the simultaneous localization of repetitive signal componentsin both signal space and frequency space.

References

1. Bachmann, G., Narici, L., Beckenstein, E.: Fourier and Wavelet Analysis. Springer, New York(1999)

2. Bracewell, R.N.: The Fourier Transform and its Applications. Mcgraw-Hill International Edi-tors (2000)

3. Brigham, E.O.: The Fast Fourier Transform: An Introduction to its Theory and Applications.Prentice Hall, New Jersey (1973)

4. Broughton, S.A., Bryan, K.: Discrete Fourier Analysis and Wavelets: Applications to Signaland Image Processing. Wiley, Inc., Hoboken, New Jersey (2009)

5. Chu, E.: Discrete and Continuous Fourier Transforms: Analysis, Applications and Fast Algo-rithms. CRC Press, Boca Raton (2008)

6. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex fourier series.Math. Comput. 19, 297–301 (1965)

7. Folland, G.B.: Fourier Analysis and Its Applications. The Wadsworth and Brooks/Cole Math-ematics Series (1992)

8. Frazier, M.W.: An Introduction to Wavelets through Linear Algebra. Springer, Berlin (1999)9. Rao, K.R., Yip, P.: Discrete Cosine Transform: Algorithms, Advantages, Applications. Aca-

demic, New York (1990)10. Rjasanow, S., Steinbach, O.: Fast Fourier Transform and its Applications, Prentice Hall, New

Jersey (1988)11. Walker, J.S.: Fourier analysis and wavelet analysis. Notices AMS. 44(6), 658–670 (1997)

Chapter 3Wavelets and Wavelet Transform

3.1 Overview

Wavelet transforms are themost powerful and themost widely used tool in the field ofimage processing. Wavelet transform has received considerable attention in the fieldof image processing due to its flexibility in representing non-stationary image signalsand its ability in adapting to human visual characteristics. Wavelet transform is anefficient tool to represent an image. The wavelet transform allows multiresolutionanalysis of an image. The aim of the transform is to extract relevant informationfrom an image. A wavelet transform divides a signal into a number of segments,each corresponding to a different frequency band.

Fourier transform is a powerful tool that has been available to signal analysisfor many years. It gives information regarding the frequency content of a signal.However, the problemwith using Fourier transform is that frequency analysis cannotoffer both good frequency and time resolution at the same time. A Fourier transformdoes not give information about the time at which a particular frequency has occurredin the signal. Hence, a Fourier transform is not an effective tool to analyses a non-stationary signal. To overcome this problem, windowed Fourier transform, or short-time Fourier transform, was introduced. Even though a short-time Fourier transformhas the ability to provide time information, but multiresolution is not possible withshort-time Fourier transforms. Wavelet is the answer to the multiresolution problem.A wavelet has the important property of not having a fixed-width sampling window.Thewavelet transformcanbebroadly classified into (i) continuouswavelet transform,and (ii) discrete wavelet transform. For long signals, continuous wavelet transformcan be time consuming since it needs to integrate over all times. To overcome the timecomplexity, discrete wavelet transform was introduced. Discrete wavelet transformscan be implemented through subband coding. TheDWT is useful in image processingbecause it can simultaneously localize signals in time and scale, whereas the DFTor DCT can localize in the frequency domain.


45

46 3 Wavelets and Wavelet Transform

The first literature that relates to the wavelet transform is Haar wavelet. It was pro-posed by the mathematician Alfrd Haar in 1909. However, the concept of the waveletdid not exist at that time. Until 1981, the concept was proposed by the geophysicistJean Morlet. Afterward, Morlet and the physicist Alex Grossman invented the termwavelet in 1984. Before 1985, Haar wavelet was the only orthogonal wavelet peopleknow. A lot of researchers even thought that there was no orthogonal wavelet exceptHaar wavelet. Fortunately, the mathematician Yves Meyer constructed the secondorthogonal wavelet calledMeyer wavelet in 1985. As more and more scholars joinedin this field, the 1st international conference was held in France in 1987. In 1989,Stephane Mallat and Meyer proposed the concept of multiresolution [21–25]. In thesame year, Ingrid Daubechies found a systematical method to construct the compactsupport orthogonal wavelet. In 1989, Mallat proposed the fast wavelet transform.With the appearance of this fast algorithm, the wavelet transform had numerousapplications in the signal and image processing field.

3.2 Wavelets

The notion of wavelets came into being because the Fourier analysis which dependson oscillating building blocks is poorly suited to signals that change suddenly. Awavelet is crudely a function which together with its dilates and their translatesdetermine all functions of our need.

Definition 3.1 A function ψ ∈ L2(R) is called a wavelet if it posses the followingproperties:1. It is square integrable, or equivalently, has finite energy:

∫ ∞

−∞|ψ(x)|2dx < ∞, (3.2.1)

2. The function integrates to zero, or equivalently, its Fourier transform denoted asψ(ξ) is zero at the origin: ∫ ∞

−∞|ψ(x)|dx = 0, (3.2.2)

3. The Fourier transform of ψ(x) must satisfy the admissibility condition given by

Cψ =∫ ∞

−∞|ψ(ξ)|2

|ξ| dξ < ∞, (3.2.3)

where ψ(ξ) is the Fourier transform of ψ(x).

Equation (3.2.1) implies that most of the energy in ψ(x) is confined to a finiteinterval, or ψ(x) has good space localisation. Ideally, the function is exactly zero

3.2 Wavelets 47

outside the finite interval and this implies that the function is compactly supportedfunction. Equation (3.2.2) suggests that the function is either oscillatory or has awavyappearance. Equation (3.2.3) is useful in formulating the inverse Fourier transform.From Eq. (3.2.3), it is obvious that ψ(ξ) must have a sufficient decay in frequency.This means that the Fourier transform of a wavelet is localised, that is, a waveletmostly contains frequencies from a certain frequency band. Since the Fourier trans-form is zero at the origin, and the spectrum decays at high frequencies, a wavelethas a band pass characteristics. Thus a wavelet is a ‘small wave’ that exhibits goodtime-frequency localisation.

We generate a doubly-indexed family of wavelets from ψ by dilating and trans-lating,

ψa,b(x) = a−1/2ψ

(x − b

a

), (3.2.4)

where a, b ∈ R, a �= 0 (we use negative as well as positive a at this point). Thenormalization has been chosen so that ||ψa,b|| = ||ψ|| for all a, b ∈ R. We willassume that ||ψ|| = 1. The discretization of the dilation parameter, we choose a =a− j0 , where j ∈ Z, and assume a0 > 1, and discretize b by taking only the integer(positive and negative) multiples of one fixed b0, where b0(> 0) is appropriatelychosen so that the ψ(x − kb0) cover the whole line. For different values of j , thewidth of a j/2

0 ψ(a j0 x) is a

j0 times of the width of ψ(x), so that the choice b = kb0a

− j0

will ensure that the discretized wavelets at level j cover the line in the same waythat the ψ(x − kb0) do. Thus we choose a = a− j

0 , b = kb0a− j0 , where j, k ∈ Z and

a0 > 1, b0 > 0, are fixed. Hence discrete form of wavelet is

ψ j,k(x) = a j/20 ψ

(x − kb0a

− j0

a− j0

)= a j/2

0 ψ(a j0 x − kb0). (3.2.5)

For a0 = 2, b0 = 1, we have

ψ j,k(x) = 2 j/2ψ(2 j x − k). (3.2.6)

Alternatively, A function ψ ∈ L2(R) is called an orthonormal wavelet if the system{ψ j,k} j,k∈Z forms an orthonormal basis for L2(R). This is also known as dyadicwavelet.

The Haar wavelet (1910) is the oldest wavelet which has limited application as itis not continuous. To suit for approximating data with sharp discontinuities, waveletsthat could automatically adapt to different components of a signal were targeted forinvestigation in early 1980s. Daubechies [9] brought a big breakthrough in 1988 andher work immediately stimulated a rapid development in the theory and applicationsof wavelet analysis. The Haar wavelet is not continuous, while the Shannon waveletas well as Meyer wavelets are smooth. Meyer wavelets can be so chosen that theirFourier transforms are also smooth.


Definition 3.2 An orthonormal wavelet ψ ∈ L2(R) is called compactly supportedwavelet if the support of the wavelet ψ is compact.

Definition 3.3 An orthonormal wavelet ψ ∈ L2(R) is said to be a band-limitedwavelet if its Fourier transform has compact support.

The Haar wavelet is compactly supported while the Shannon wavelet, the Journéwavelet andMeyer wavelets are examples of band-limited wavelets. For more detailsabout Haar, Shannon and Meyers wavelets, one can read, Hernandez and Weiss [13]and for Daubechies wavelets, Daubechies [8].

3.3 Multiresolution Analysis

The concept of Multiresolution analysis (MRA) was first introduced by Mallat [21,22] and Meyer [25]. It is a general framework that makes constructing orthonormalwavelets basis for the space L2(R). The MRA based compactly supported orthonor-mal wavelet systems were constructed by Daubechies [8]. Multiresolution analysis(MRA) is a very well-known and unique mathematical theory that incorporates andunifies various image processing techniques. The main purpose of this analysis is toobtain different approximations of a function f (x) at different resolution. Further-more, the MRA structure grants fast implementation of wavelet decomposition andreconstruction which makes wavelets a very practical tool for image processing andanalysis. Multiresolution analysis is a family of closed subspaces of L2(R) satisfyingcertain properties:

Definition 3.4 A pair ({Vj}j∈Z ,ϕ) consisting of a family

{Vj}j∈Z of closed sub-

spaces of L2(R) together with a function ϕ ∈ V0 is called aMultiresolution analysis(MRA) if it satisfies the following conditions:

(a) Vj ⊂ Vj+1, for all j ∈ Z,(b) f ∈ V0 ⇐⇒ f (2 j ·) ∈ Vj , for all j ∈ Z,(c) ∩ j∈ZVj = {0},(d) ∪ j∈ZVj = L2(R),(e) {ϕ(· − k) : k ∈ Z}, denoted by ϕ0,k , is an orthonormal basis for V0.

The function ϕ is called a scaling function for the given MRA. We note that anorthonormal basis for Vj , j ∈ Z is given by translates of normalized dilations{2 j/2ϕ(2 j · −k), k ∈ Z}, denoted by ϕ j,k , of ϕ. For each j ∈ Z, define the approx-imation operator Pj on functions f (x) ∈ L2(R) by,

Pj f (x) =∑

〈 f,ϕ j,k〉ϕ j,k, (3.3.1)

and for each j ∈ Z, the detail operator Q j on functions f (x) ∈ L2(R) is definedby,

3.3 Multiresolution Analysis 49

Q j f (x) = Pj+1 f (x) − Pj f (x). (3.3.2)

A multiresolution analysis gives(1) an orthogonal direct sum decomposition of L2 (R), and(2) a wavelet ψ called an MRA Wavelet.LetW0 be the orthogonal complement of V0 in V1; that is, V1 = V0 ⊕W0. Then if

we dilate the elements of W0 by 2 j , we obtain a closed subspace Wj of Vj in Vj+1,such that

Vj+1 = Vj ⊕ Wj , for each j ∈ Z. (3.3.3)

Conditions (c) and (d) of the Definition 3.4 of an MRA provide

L2 (R) = ⊕∞j=−∞Wj . (3.3.4)

Furthermore, the Wj spaces inherit the scaling property (Definition 3.4(b)) from theVj :

f ∈ W0 ⇐⇒ f (2 j ·) ∈ Wj . (3.3.5)

Equation (3.3.5) ensures that if {ψ0,k; k ∈ Z} is an orthonormal basis for W0, then{ψ j,k; k ∈ Z} will likewise be an orthonormal basis forWj , for any j ∈ Z. Our mainaim is to finding ψ ∈ W0 such that the ψ(· − k) constitute an orthonormal basis forW0. To construct this ψ let us write some interesting properties of ϕ and W0.

1. Since ϕ ∈ V0 ⊂ V1 and ϕ1,k is an orthonormal basis in V1, we have

ϕ =∑k∈Z

h(k)ϕ1,k, (3.3.6)

with h(k) = 〈ϕ,ϕ1,k〉 and∑k∈Z |h(k)|2 = 1, i.e. h(k) ∈ l2(Z). Hence,

ϕ(x) = √2∑k∈Z

h(k)ϕ(2x − k), (3.3.7)

is known as the scaling relation or the refinement equation. The sequence {h(k)}k∈Z iscalled the scaling sequence or scaling filter associated with ϕ(x). By taking Fouriertransform both side of equation (3.3.7), we have

ϕ(ξ) = 1√2

∑k∈Z

h(k)e−ikξ/2ϕ(ξ/2), (3.3.8)

where convergence in either sum holds in L2-sense. Equation (3.3.8) can be rewrittenas

ϕ(ξ) = m0(ξ/2)ϕ(ξ/2), (3.3.9)


where

m0(ξ) = 1√2

∑k∈Z

h(k)e−ikξ. (3.3.10)

Equality in Eq. (3.3.9) holds point-wise almost everywhere. Equation (3.3.10) showsthat m0 is a 2π-periodic function in L2([0, 2π]).

2. The orthonormality of the ϕ(x − k) leads to special properties for m0. We have

δk,0 =∫

ϕ(x)ϕ(x − k)dx =∫

|ϕ(ξ)|2eikξdξ =∫ 2π

0

∑l∈Z

|ϕ(ξ + 2πl)|2eikξdξ,

implying ∑l∈Z

|ϕ(ξ + 2πl)|2 = (2π)−1 a.e. (3.3.11)

Substituting equation (3.3.9) in Eq. (3.3.11) and put η = ξ/2 leads to

∑l∈Z

|m0(η + πl)|2|ϕ(η + πl)|2 = (2π)−1,

by splitting the sum into even and odd l, using the periodicity of m0 and applying(3.3.11) once more gives

|m0(ξ)|2 + |m0(ξ + π)|2 = 1 a.e. (3.3.12)

3. Characterization of W0: f ∈ W0 is equivalent to f ∈ V1 and f ⊥ V0. Since,f ∈ V1, we have

f =∑k∈Z

f (k)ϕ1,k,

with f (k) = 〈 f,ϕ1,k〉. This implies

f (ξ) = 1√2

∑k∈Z

f (k)e−ikξ/2ϕ(ξ/2) = m f (ξ/2)ϕ(ξ/2), (3.3.13)

where

m f (ξ) = 1√2

∑k∈Z

f (k)e−ikξ; (3.3.14)

clearly m f is a 2π-periodic function in L2([0, 2π]), convergence in (3.3.14) holdspointwise a.e.. The constraint f ⊥ V0 implies f ⊥ ϕ0,k for all k, i.e.,


0 =∫

f (ξ)ϕ(ξ)eikξdξ =∫ (∑

l

f (ξ + 2πl)ϕ(ξ + 2πl)

)eikξdξ,

hence ∑l

f (ξ + 2πl)ϕ(ξ + 2πl) = 0, (3.3.15)

where the series in (3.3.15) converges absolutely in L1([π,π]). Substituting (3.3.9)and (3.3.13), regrouping the sums for odd and even l (which we allow to do, becauseof the absolute convergence), using (3.3.11) leads to

m f (ξ)m0(ξ) + m f (ξ + π)m0(ξ + π) = 0 a.e.. (3.3.16)

Since m0(ξ) and m0(ξ + π) cannot vanish together on a set of nonzero measure(because of (3.3.12)), this implies the existence of 2π-periodic function λ(ξ) so that

m f (ξ) = λ(ξ)m0(ξ + π) a.e., (3.3.17)

andλ(ξ) + λ(ξ + π) = 0 a.e.. (3.3.18)

This last equation can be reorganize as

λ(ξ) = eiξν(2ξ), (3.3.19)

where ν is 2π-periodic. Substituting equations (3.3.19) and (3.3.17) into Eq. (3.3.13)gives

f (ξ) = eiξ/2 m0(ξ/2 + π)ν(ξ)ϕ(ξ/2), (3.3.20)

where ν is 2π-periodic.4. The general form of Eq. (3.3.20) for the Fourier transform of f ∈ W0 suggeststhat we take

ψ(ξ) = eiξ/2 m0(ξ/2 + π)ϕ(ξ/2), (3.3.21)

as a candidate of our wavelet. Disregarding convergence questions, Eq. (3.3.20) canbe rewritten as

f (ξ) =(∑

k∈Zν(k)e−ikξ

)ψ(ξ),

orf (x) =

∑k∈Z

ν(k)ψ(x − k),


so that the ψ(· − k) are a good candidate for a basis of W0. Next, we need to verifythat the ψ0,k are indeed an orthonormal basis for W0. The properties of m0 and ϕensure that Eq. (3.3.21) defines indeed an L2-function i.e. ψ ∈ V1 and ψ ⊥ V0, sothat ψ ∈ W0. Orthonormality of the ψ0,k is easy to check:

∫ψ(x)ψ(x − k)dx =

∫|ψ(ξ)|2eikξdξ =

∫ 2π

0eikξ

∑l∈Z

|ψ(ξ + 2πl)|2dξ.

Now

∑l∈Z |ψ(ξ + 2πl)|2 =

∑l∈Z |m0(ξ/2 + πl + π)|2|ϕ(ξ/2 + πl)|2

= |m0(ξ/2 + π)|2∑n∈Z

|ϕ(ξ/2 + 2πn)|2

+ |m0(ξ/2)|2∑n∈Z

|ϕ(ξ/2 + π + 2πn)|2

= (2π)−1[|m0(ξ/2 + π)|2 + |m0(ξ/2)|2] a.e. (by Eq. (3.3.11))

= (2π)−1 a.e. (by Eq. (3.3.12)).

Hence∫

ψ(x)ψ(x−k)dx = δk,0. In order to check that theψ0,k are indeed a basisfor all of W0, it then suffices to check that any f ∈ W0 can be written as

f =∑n∈Z

γ(n)ψ0,n,

with∑

n∈Z |γ(n)|2 < ∞, orf (ξ) = γ(ξ)ψ(ξ), (3.3.22)

with γ is 2π-periodic and L2([0, 2π]).Let us go back to Eq. (3.3.20), we have f (ξ) =ν(ξ)ψ(ξ), with

∫ 2π0 |ν(ξ)|2 = ∫ π

0 |λ(ξ)|2. by Eq. (3.3.14)∫ 2π

0|m f (ξ)|2dξ =

∫ 2π

0|λ(ξ)|2|m0(ξ + π)|2dξ

=∫ π

0|λ(ξ)|2[|m0(ξ)|2

+ |m0(ξ + π)|2]dξ (use Eq. (3.3.18))

=∫ π

0|λ(ξ)|2 (use Eq. (3.3.12)).


Hence∫ 2π0 |ν(ξ)|2dξ = 2π|| f ||2 < ∞, and f is of the form (3.3.22) with square

integrable 2π-periodic γ. We have thus proved the following theorem:

Theorem 3.1 Let {Vj } j∈Z be a sequence of closed subspaces of L2(R) satisfyingconditions (a)-(e) of Definition (3.4). Then there exists an associated orthonormalwavelet basis {ψ j.k; j, k ∈ Z} for L2(R) such that

Pj+1 f = Pj f +∑k∈Z

〈 f,ψ j,k〉ψ j,k f or all f ∈ L2(R), (3.3.23)

where Pj is the orthogonal projection onto Vj . One possibility for the constructionof the wavelet ψ is

ψ(ξ) = eiξ/2 m0(ξ/2 + π)ϕ(ξ/2),

(with m0 as defined by (3.3.10) and (3.3.6)), or equivalently

ψ(x) =∑k∈Z

g(k)ϕ1,k(x) = √2∑k∈Z

g(k)ϕ(2x − k), (3.3.24)

where g(k) = (−1)k h(−k + 1), is called wavelet filter.

3.4 Wavelet Transform

The wavelet transform (WT) provides a time-frequency representation of the signal.Thewavelet transformwas developed to overcome the shortcomings of the short-timeFourier transform, which can be used to analyse non-stationary signals. The maindrawback of STFT is that it gives a constant resolution at all frequencies, while thewavelet transform uses a multiresolution technique by which different frequenciesare analysed with different resolutions. The wavelet transform is generally termedmathematical microscope in which big wavelets give an approximate image of thesignal, while the smaller wavelets zoom in on the small details. The basic idea ofthe wavelet transform is to represent the signal to be analysed as a superpositionof wavelets. Historically, the continuous wavelet transform came first. It is com-pletely different from the discrete wavelet transform. It is popular among physicists,whereas the discrete wavelet transform ismore common in numerical analysis, signalprocessing and image processing.

3.4.1 The Wavelet Series Expansions

Likewise Fourier series expansion, a wavelet series is a representation of squareintegrable real or complex valued function by a certain orthonormal series generated


by a wavelet. A function f (x) ∈ L2(R) can be represented by a scaling functionexpansion in subspace VJ0 and some other number of wavelet expansions in subspaceWJ0 ,WJ0+1, . . .. Thus

f (x) =∑k∈Z

c j0(k)ϕ j0,k(x) +∞∑j= j0

∑k∈Z

d j (k)ψ j,k(x), (3.4.1)

where j0 is arbitrary starting scale and the c j0(k) and d j (k), defined by,

c j0(k) = 〈 f (x),ϕ j0,k(x)〉 =∫

f (x)ϕ j0,k(x)dx, (3.4.2)

d j (k) = 〈 f (x),ψ j,k(x)〉 =∫

f (x)ψ j,k(x)dx, (3.4.3)

are called approximation coefficient or scaling coefficient and detail coefficient orwavelet coefficient, respectively.

3.4.2 Discrete Wavelet Transform

Thewavelet series expansionmaps a function of a continuous variable into a sequenceof coefficients. If the sequence being expanded is discrete, the resulting coefficientsare called the discrete wavelet transform (DWT). The discrete wavelet transformcoefficients of function f (n) are defined as

Wϕ( j0, k) = 1√M

M−1∑n=0

f (n)ϕ j0,k(n), (3.4.4)

Wψ( j, k) = 1√M

M−1∑n=0

f (n)ψ j,k(n), j ≥ j0, (3.4.5)

and Wϕ( j0, k) and Wψ( j, k) are called approximation coefficient and detail coeffi-cient, respectively. The DFT coefficients enable us to reconstruct the discrete signalf (x) in l2(Z) by

f (n) = 1√M

∑k

Wϕ( j0, k)ϕ j0,k(n) + 1√M

∞∑j= j0

∑k

Wψ( j, k)ψ j,k(n). (3.4.6)

Here f (n),ϕ j0,k(n) and ψ j,k(n) are discrete functions defined in [0, M − 1].Generally, we take j0 = 0 and select M = 2J , so that the summations in

3.4 Wavelet Transform 55

Eqs. (3.4.4)–(3.4.6) are performedovern = 0, 1, 2, . . . , M−1, j = 0, 1, 2, . . . , J−1 and k = 0, 1, 2, . . . , 2 j − 1.

There is another way to define DWT which will be described in next subsection.It is necessary to mention here that we are only providing the details of constructionof discrete wavelet transform fromMRA. See Daubechies [7], Walnut [38] for moredetails and proof of the theorems provided in this subsection.

3.4.3 Motivation: From MRA to Discrete Wavelet Transform

Let {Vj } j∈Z be an MRA with scaling function ϕ(x). Then the refinement equationof ϕ(x) is

ϕ(x) =∑n

h(n)ϕ1,n(x) = √2∑

h(n)ϕ(2x − n),

and the corresponding wavelet ψ(x) is defined by

ψ(x) =∑n

g(n)ϕ1,n(x) = √2∑

g(n)ϕ(2x − n),

where g(n) = (−1)nh(1 − n). For any j, k ∈ Z,

ϕ j,k(x) = 2 j/2ϕ(2 j x − k)

= 2 j/2√2∑n

h(n)ϕ(2(2 j x − k) − n) (using Eq. (3.3.7))

= 2( j+1)/2∑

nh(n)ϕ(2 j+1x − 2k − n)

= 2( j+1)/2∑

nh(n − 2k)ϕ(2 j+1x − n)

=∑n

h(n − 2k)ϕ j+1,n(x). (3.4.7)

Similarly,ψ j,k(x) =

∑n

g(n − 2k)ϕ j+1,n(x). (3.4.8)

For k ∈ Z, definec0(k) = 〈 f,ϕ0,k〉,

and for j ∈ N and k ∈ Z, define c j (k) and d j (k) by


c j (k) = 〈 f,ϕ− j,k〉 and d j (k) = 〈 f,ψ− j,k〉.

Then, by using Eq. (3.4.7)

c j+1(k) = 〈 f,φ− j−1,k〉= 〈 f,

∑nh(n − 2k)φ− j,n〉

=∑

nh(n − 2k)〈 f,φ− j,n〉

=∑n

c j (n)h(n − 2k) (3.4.9)

and by Eq. (3.4.8), we have

d j+1(k) =∑n

c j (n)g(n − 2k). (3.4.10)

In order to see that the calculation of c j+1(k) and d j+1(k) is completely reversible,recall that by Eq. (3.3.1)

P− j f (x) =∑n

〈 f,ϕ− j,n〉ϕ− j,n(x) =∑n

c j (n)ϕ− j,n(x), (3.4.11)

andQ− j f (x) =

∑n

〈 f,ψ− j,n〉ψ− j,n(x) =∑n

d j (n)ψ− j,n(x). (3.4.12)

Also, by Eq. (3.3.1), we have

P− j f (x) = P− j−1 f (x) + Q− j−1 f (x).

Using Eqs. (3.4.11) and (3.4.12), we have

∑nc j (n)ϕ− j,n(x) =

∑nc j+1(n)ϕ− j−1,n(x) +

∑nd j+1(n)ψ− j−1,n(x)

=∑

nc j+1(n)

∑kh(k − 2n)ϕ− j,k(x)

+∑

nd j+1(n)

∑kg(k − 2n)ϕ− j,k(x)

=∑

k

(∑nc j+1(n)h(k − 2n) +

∑nd j+1(n)g(k − 2n)

)ϕ− j,k(x).

By matching the coefficients, we conclude that

c j (k) =∑n

c j+1(n)h(k − 2n) +∑n

d j+1(n)g(k − 2n). (3.4.13)


We summarize these results in the following theorem:

Theorem 3.2 Let {Vj } be an MRA with associated scaling function ϕ(x), scalingfilter h(k), wavelet function ψ(x) and wavelet filter g(k). Given a function f (x) inL2(R), define for k ∈ Z,

c0(k) = 〈 f,ϕ0,k〉,

and for j ∈ N and k ∈ Z, define c j (k) and d j (k) by


Thenc j+1(k) =

∑n

c j (n)h(n − 2k) d j+1(k) =∑n

c j (n)g(n − 2k)

andc j (k) =

∑n

c j+1(n)h(k − 2n) +∑n

d j+1(n)g(k − 2n).

The above theorem suggests that the key object in calculating 〈 f,ϕ〉 and 〈 f,ψ〉 isthe scaling filter h(k) and not the scaling functionϕ(x). It also suggests that as long asEq. (3.4.13) holds, Eqs. (3.4.9) and (3.4.10) define an exactly invertible transform forsignals. The question is: What conditions must the scaling filter h(k) satisfy in orderfor the transform defined by Eqs. (3.4.9) and (3.4.10) to be invertible by Eq. (3.4.13)?These properties will be referred as quadrature mirror filter (QMF) conditions andwill be used to define the discrete wavelet transform.

3.4.4 The Quadrature Mirror Filter Conditions

In this subsection, we will provide the QMF conditions, reformulate them in thelanguage of certain filtering operations on signals called the approximation and detailoperators, and finally give a very simple characterization of the QMF conditions thatwill be used in the design of wavelet filter and scaling filter.

Theorem 3.3 Let {Vj } be an MRA with scaling filter h(k) and the wavelet filterg(k). Then

(a)∑

n h(n) = √2,

(b)∑

n g(n) = 0,(c)∑

k h(k)h(k − 2n) =∑k g(k)g(k − 2n) = δ(n),

(d)∑

k g(k)h(k − 2n) = 0, f or all n ∈ Z, and(e)∑

k h(m − 2k)h(n − 2k) +∑k g(m − 2k)g(n − 2k) = δ(n − m).


Proof (a)

∫R

ϕ(x)dx =∫R

∑n

h(n)21/2ϕ(2x − n)dx =∑n

h(n)2−1/2∫R

ϕ(x)dx .

Since we know that∫R

ϕ(x)dx �= 0, we can cancel the nonzero factor∫R

ϕ(x)dxfrom both sides. it follows that

∑n

h(n) = √2. (3.4.14)

(b) From Eq. (3.2.2),∫R

ψ(x)dx = 0, so that

0 =∫R

ψ(x)dx =∫R

∑n

g(n)21/2ϕ(2x − n)dx =∑n

g(n)2−1/2∫R

ϕ(x)dx .

Hence, ∑n

g(n) = 0. (3.4.15)

This is also equivalent to the statement that

∑n

h(2n) =∑n

h(2n + 1) (3.4.16)

(c) Since {ϕ0,n(x)} and {ϕ1,n(x)} are the orthonormal systems on R,

∫R

ϕ(x)ϕ(x − n) =∫R

∑k

h(k)ϕ1,k(x)∑m

h(m)ϕ1,m(x − n)dx

=∫R

∑kh(k)ϕ1,k(x)

∑m

h(m)ϕ1,m+2n(x)dx

=∑

k

∑mh(k)h(m − 2n)

∫R

ϕ1,k(x)ϕ1,m(x)dx

=∑

kh(k)h(k − 2n).

Hence, ∑k

h(k)h(k − 2n) = δ(n). (3.4.17)

Since {ψ0,n(x)}n∈Z is also an orthonormal systems onR, the same argument gives

∑k

g(k)g(k − 2n) = δ(n). (3.4.18)


(d) Since 〈ψ0,n,ϕ0,m(x)〉 = 0 for all m, n ∈ Z, we have

∑k

g(k)h(k − 2n) = 0, (3.4.19)

for all n ∈ Z.

(e) Since for any signal c0(n),

c0(n) =∑k

c1(k)h(n − 2k) +∑k

d1(k)g(n − 2k),

wherec1(k) =

∑m

c0(m)h(m − 2k)

andd1(k) =

∑m

c0(m)g(m − 2k),

it follows that

c0(n) =∑k

∑m

c0(m)h(m − 2k)h(n − 2k) +∑k

∑m

c0(m)g(m − 2k)g(n − 2k),

=∑m

c0(m)

(∑k

h(m − 2k)h(n − 2k) +∑k

g(m − 2k)g(n − 2k)

).

Hence we must have

∑k

h(m − 2k)h(n − 2k) +∑k

g(m − 2k)g(n − 2k) = δ(n − m). (3.4.20)

�

Remark 3.1 Condition (a) is referred to as a normalization condition. The value√2

arises from the fact that we have chosen to write the two-scale dilation equation asϕ(x) =∑n h(n)21/2ϕ(2x − n). In some of the literature on wavelets and especiallyon two-scale dilation equations, the equation is written ϕ(x) =∑n h(n)ϕ(2x − n).

This leads to the normalization,∑

n h(n) = 2. The choice of normalization is just aconvention and has no real impact on any of the results that follow.

Remark 3.2 Conditions (c) and (d) are referred to as orthogonality conditions sincethey are immediate consequences of the orthogonality of the scaling functions at agiven scale, the orthogonality of the wavelet functions at a given scale, and the factthat the wavelet functions are orthogonal to all scaling functions at a given scale.


Remark 3.3 Condition (e) is referred to as the perfect reconstruction (PR) conditionsince it follows from the reconstruction formula for orthonormal wavelet bases.

Definition 3.5 Let c(n) be a signal.(a) Given m ∈ Z, the shift operator τm is defined by

τmc(n) = c(n − m). (3.4.21)

(b) the downsampling operator ↓ is defined by

(↑ c)(n) = c(2n), (3.4.22)

i.e. (↓ c)(n) is formed by removing every odd term in c(n).

(c) the upsampling operator ↑ is defined by

(↑ c)(n) ={c(n2

), n is even

0, i f n is odd(3.4.23)

i.e. (↑ c)(n) is formed by inserting a zero between adjacent entries of c(n).

Definition 3.6 Given a signal c(n) and a filter h(k), define g(k) by g(k) =(−1)kh(1 − k). Then the approximation operator H and detail operator G corre-sponding to h(k) are defined by

(Hc)(k) =∑n

c(n)h(n − 2k), (Gc)(k) =∑n

c(n)g(n − 2k). (3.4.24)

The approximation adjoint H∗ and detail adjoint G∗ are defined by

(H∗c)(k) =∑n

c(n)h(k − 2n), (G∗c)(k) =∑n

c(n)g(k − 2n). (3.4.25)

Remark 3.4 The operators H and G can be thought of as convolution with the filtersh(n) = h(−n) and g(n) = g(−n) followed by downsampling. That is,

(Hc)(n) =↓ (c ∗ h)(n), (Gc)(n) =↓ (c ∗ g)(n). (3.4.26)

Remark 3.5 H∗ and G∗ can be thought of as upsampling followed by convolutionwith the filters h and g. That is,

(H∗c)(n) = (↑ c) ∗ h(n) and (G∗c)(n) = (↑ c) ∗ g(n). (3.4.27)

Remark 3.6 The operators H∗ and G∗ are the formals adjoints of H and G. That is,for all signals c(n) and d(n),


〈Hc, d〉 =∑k

(Hc)(k)d(k) =∑k

c(k)(H∗d)(k) = 〈c, H∗d〉 (3.4.28)

and

〈Gc, d〉 =∑k

(Gc)(k)d(k) =∑k

c(k)(G∗d)(k) = 〈c,G∗d〉 (3.4.29)

Now, we can reformulate the conditions of Theorem 3.3(c)–(e) as follows:

Theorem 3.4 Givena scaling filter h(k)and thewavelet filter g(k)definedby g(k) =(−1)kh(1 − k). Then

(a)∑

n h(n) = √2,

(b)∑

n g(n) = 0,(c)∑

k h(k)h(k − 2n) = ∑k g(k)g(k − 2n) = δ(n) if and only if HH∗ =

GG∗ = I, where I is the identity operator on sequences,(d)∑

k g(k)h(k − 2n) = 0, f or all n ∈ Z, if and only if HG∗ = GH∗ =0, and

(e)∑

k h(m − 2k)h(n − 2k) +∑k g(m − 2k)g(n − 2k) = δ(n − m) if and onlyif H∗H + G∗G = I.

Next, we will show that all of the conditions in Theorem 3.3 can be written as asingle condition (Theorem 3.4) on the auxiliary function m0(ξ) = 1√

2

∑n h(n)e−inξ

plus the normalization condition m0(0) = 1. These two conditions will be referredto as the quadrature mirror filter (QMF) conditions.

Theorem 3.5 Given a filter h(k), define g(k) by g(k) = (−1)kg(1 − k), mo(ξ) byEq. (3.3.10), mψ(ξ) or m1(ξ) by Eq. (3.3.14), and the operators H, G, H∗ and G∗by Eqs. (3.4.24) and (3.4.25). Then the following are equivalent:

(a) |m0(ξ)|2 + |m0(ξ + π)|2 = 1.(b)∑

n h(n)h(n − 2k) = δ(k).(c) H∗H + G∗G = I.(d) HH∗ = GG∗ = I.

Definition 3.7 Given a filter h(k) and mo(ξ). Then h(k) is a QMF provided that:(a) m0(0) = 1 and(b) |m0(

ξ2 )|2 + |m0(

ξ2 + π)|2 = 1 for all ξ ∈ R.

Theorem 3.6 Suppose that h(k) is aQMFanddefineg(k)byg(k) = (−1)kg(1 − k).Then the following holds:

(a)∑

n h(n) = √2,

(b)∑

n g(n) = 0,(c)∑

n h(2n) =∑n h(2n + 1),(c)∑

k h(k)h(k − 2n) =∑k g(k)g(k − 2n) = δ(n),

(d)∑

k g(k)h(k − 2n) = 0, f or all n ∈ Z, and(e)∑

k h(m − 2k)h(n − 2k) +∑k g(m − 2k)g(n − 2k) = δ(n − m).


Definition 3.8 Let h(k) be a QMF, define g(k) by g(k) = (−1)kg(1 − k) and letH,G, H∗ and G∗ be given by Eqs. (3.4.24) and (3.4.25). Fix J ∈ N, the discretewavelet transform (DWT) of a signal c0(n), is the collection of sequences

{d j (k) : 1 ≤ j ≤ J ; k ∈ Z} ∪ {cJ (k) : k ∈ Z}, (3.4.30)

wherecJ+1(n) = (HcJ )(n) and dJ+1(n) = (GcJ )(n). (3.4.31)

The inverse discrete wavelet transform (IDWT) is defined by

cJ (n) = (H∗cJ+1)(n) + (G∗dJ+1)(n). (3.4.32)

If J = ∞, then the DWT of c0 is the collection of sequences

{d j (k) : j ∈ N; k ∈ Z}. (3.4.33)

3.5 The Fast Wavelet Transform

The fast wavelet transform (FWT) is a mathematical algorithm designed to turn awaveform or signal in the time domain into a sequence of coefficients based onan orthogonal basis of small finite waves, or wavelets. The transform can be eas-ily extended to multidimensional signals, such as images, where the time domainis replaced with the space domain. It has as theoretical foundation the device of afinitely generated, orthogonal multiresolution analysis (MRA). Transform codingis a widely used method of compressing image information. In a transform-basedcompression system two-dimensional (2D) images are transformed from the spa-tial domain to the frequency domain. An effective transform will concentrate usefulinformation into a few of the low-frequency transform coefficients.

We first consider the refinement equation (3.3.4) of multiresolution analysis

ϕ(x) = √2∑k∈Z

hϕ(k)ϕ(2x − k). (3.5.1)

By a scaling of x by 2 j , translation of x by n, and making m = 2n + k, we wouldget

ϕ(2 j x − k) = √2∑

k∈Z hϕ(k)ϕ(2(2 j x − n) − k)

= √2∑m∈Z

hϕ(m − 2n)ϕ(2( j+1)x − m), (3.5.2)

3.5 The Fast Wavelet Transform 63

and analogously,

ψ(2 j x − k) = √2∑m∈Z

hψ(m − 2n)ϕ(2( j+1)x − m), (3.5.3)

where hϕ(k) = (−1)khψ(1−k). A property that involves the convolution of a scalingfunction and a wavelet coefficient can be derived by the following steps:We begin byconsidering the definition of the discrete wavelet transform as shown in Eqs. (3.4.4)and (3.4.5). By substituting equation (3.2.6) in Eq. (3.4.5), we get

Wψ( j, k) = 1√M

M−1∑x=0

f (x)2 j/2ψ(2 j x − k), (3.5.4)

using Eq. (3.5.3), we have

Wψ( j, k) = 1√M

M−1∑x=0

f (x)2 j/2

[√2∑m∈Z

hψ(m − 2n)ϕ(2( j+1)x − m)

]. (3.5.5)

Rearrange the summation part of the equation, we have

Wψ( j, k) =∑m∈Z

hψ(m − 2n)

[1√M

M−1∑x=0

f (x)2( j+1)/2ϕ(2( j+1)x − m)

], (3.5.6)

where the bracketed quantity is identical to Eq. (3.5.4) with j0 = j + 1. Therefore,

Wψ( j, k) =∑m∈Z

hψ(m − 2n)Wϕ( j + 1,m), (3.5.7)

and similarly the DWT approximation coefficient at scale j + 1 can be expressed as

Wϕ( j, k) =∑m∈Z

hϕ(m − 2n)Wϕ( j + 1,m). (3.5.8)

Equations (3.5.7) and (3.5.8) demonstrate that both the approximation coefficientWϕ( j, k) and the detail coefficientWψ( j, k) can be obtained by convolvingWϕ( j, k),approximation coefficients at the scale j + 1, with the time-reversed scaling andwavelet vectors, hϕ(−n) and hψ(−n) followed by the subsequent sub-sampling. TheEqs. (3.5.7) and (3.5.8) can then expressed in the following convolution formats:

Wψ( j, k) = hψ(−n) ∗ Wϕ( j + 1, n), for n = 2k, k ≥ 0, (3.5.9)

Wϕ( j, k) = hϕ(−n) ∗ Wϕ( j + 1, n), for n = 2k, k ≥ 0. (3.5.10)


Fig. 3.1 An FWT analysis filter bank

Fig. 3.2 Two scale FWT analysis filter bank

For the commonly used discrete signal, say, a digital image, the original data canbe viewed as approximation coefficients with order J . That is, f [n] = W [J ; n] byEqs. (3.4.4) and (3.4.5), next level of approximation and detail can be obtained. Thisalgorithm is “fast” because one can find the coefficients level by level rather thandirectly using Eqs. (3.4.4) and (3.4.5) to find the coefficients. For a sequence of lengthM = 2J , the number of mathematical operations involved is on the order of O(M).

That is, the number ofmultiplications and additions is linearwith respect to the lengthof the input sequence since the number of multiplications and additions involved inthe convolutions performed by the FWT analysis bank (see Fig. 3.1) is proportionalto the length of the sequences being convolved. Hence, the FWT compares favorablywith the FFT algorithm, which requires on the order of O(M log2M) operations.

We simply note that the filter bank in Fig. 3.1 can be iterated to create multi-stage structures for computing DWT coefficients at two or more successive scales.Figure3.2 shows the two scale FWT analysis filter bank.

As one might expect, a fast inverse transform for the reconstruction of from theresults of the forward transform can be formulated, called the inverse fast wavelettransform. It uses the scaling andwavelet vectors employed in the forward transform,together with the level j − 1 approximation and detail coefficients, to generate the jlevel approximation coefficients (see Fig. 3.3). As with the forward FWT, the inversefilter bank can also be iterated as shown in Fig. 3.4.

3.6 Why Use Wavelet Transforms 65

Fig. 3.3 The inverse FWT synthesis filter bank

Fig. 3.4 Two scale inverse FWT synthesis filter bank

3.6 Why Use Wavelet Transforms

Wavelets and wavelet transforms (including the DWT) have many advantages overrival multiscale analysis techniques, such as the fast Fourier transform (FFT). Thekey advantages are listed below:

Structure Extraction wavelets can be used to analyze the structure (shape) of afunction, that is the coefficients d j,k tell you how much of the corresponding waveletψ j,k makes up the function.

Localization If the function f (x) has a discontinuity at x∗ then only the waveletsψ j,k(x) which overlap the discontinuity will be affected and the associated waveletcoefficients, d j,k will show this.

Efficiency Wavelet transforms are usually much faster than other methods (or atleast as good as). For example the discrete wavelet transform is O(n) whereas theFFT is O(nlogn).

Sparsity If the right wavelet function is chosen then the coefficients computed canbe very sparse. Thismeans that the key information from the function can be summedup concisely. This is obviously advantageous when it comes to storing details abouta function or sequence, as it will take up less space. When images are looked at thisis especially of use due to the fact the image files tend to be quite large.


3.7 Two-Dimensional Wavelets

Two-dimensional (2D) wavelets are a natural extension from the single dimensioncase. As a concept they can be applied to many 2D situations, such as 2D functionalspaces. However they really come into there own when images are considered. Ina world where digital images are processed by computers ever second, methodsfor condensing the information carried in an image are needed. Also with so manydifferent images in circulation via the internet methods are needed for computationalanalysis of the content of these images. 2D wavelets provide ways to tackle both ofthese problems.

Before 2Dwavelets are introduced, it will be helpful to know about outer productsor tensor products. Outer products work in a similar way to inner products, howeverwhilst inner products take two vectors and combine to form a scalar, outer productswork the other way, extrapolating a matrix from two vectors. In the case of thestandard outer product this is done in a natural way.

Definition 3.9 Consider two vectors a and b of length n and m respectively. Thenwe can define the standard outer product, ⊗, as follows:

a ⊗ b = abT = C, (3.7.1)

where C is a matrix with elements determined by:

Ci, j = ai × b j . (3.7.2)

This concept of outer products can then be used to define 2D discrete wavelets. Thisis done by taking the outer products of one-dimensional scaling function and waveletfunction as follows:

Definition 3.10 For 1D wavelets ψ and scaling function ϕ, the 2D scaling functionand wavelet functions are defined by the matrices given by:

�(x, y) = ϕ(x)ϕ(y), (3.7.3)

and�H (x, y) = ϕ(x)ψ(y), (3.7.4)

�V (x, y) = ψ(x)ϕ(y), (3.7.5)

�D(x, y) = ψ(x)ψ(y), (3.7.6)

where �H measures the horizontal variations(horizontal edges), �V correspondsto the vertical variations (vertical edges) and �D detects the variations along thediagonal directions.

3.7 Two-Dimensional Wavelets 67

For each j,m, n ∈ Z, define

� j,m,n(x, y) = 2 j�(2 j x − m, 2 j y − n) = ϕ j,m(x)ϕ j,n(y), (3.7.7)

�Hj,m,n(x, y) = 2 j�H (2 j x − m, 2 j y − n) = ϕ j,m(x)ψ j,n(y), (3.7.8)

�Vj,m,n(x, y) = 2 j�V (2 j x − m, 2 j y − n) = ψ j,m(x)ϕ j,n(y), (3.7.9)

�Dj,m,n(x, y) = 2 j�D(2 j x − m, 2 j y − n) = ψ j,m(x)ψ j,n(y). (3.7.10)

Theorem 3.7 (i) The collection

{� ij,m,n(x, y)}i=H,V,D, j,m,n∈Z (3.7.11)

form an orthonormal basis on R2.

(ii) For each J ∈ Z, the collection

{�J,m,n(x, y)}m,n∈Z ∪ {� iJ,m,n(x, y)}i=H,V,D, j≥J,m,n∈Z (3.7.12)

form an orthonormal basis on R2.

The proof of the above theorem is beyond the scope of this book.

3.8 2D-discrete Wavelet Transform

Given separable 2D scaling and wavelet functions, extension of the 1D DWT to twodimensions is straightforward. The two-dimensional (2D) discrete wavelet transformof function f (x, y) of size M × N is defined as

Wϕ( j0,m, n) = 1√MN

M−1∑x=0

N−1∑y=0

f (x, y)ϕ j0,m,n(x, y), (3.8.1)

Wiψ( j,m, n) = 1√

MN

M−1∑x=0

N−1∑y=0

f (x, y)ψij,m,n(x, y), i = {H, V, D}, (3.8.2)

whereϕ j,m,n(x, y) = 2 jϕ(2 j x − m, 2 j y − n), (3.8.3)

ψij,m,n(x, y) = 2 jψ(2 j x − m, 2 j y − n), i = {H, V, D}. (3.8.4)


Fig. 3.5 The 2D FWT analysis filter bank

Fig. 3.6 The 2D inverse FWT synthesis filter bank

As in one dimensional case, j0 is an arbitrary starting scale and the Wϕ( j0,m, n)

coefficients define an approximation of f (x, y) at scale j0. The Wψ( j,m, n) coeffi-cients add horizontal, vertical, and diagonal details for scales j ≥ j0. Given the Wϕ

and Wiψ , f (x, y) is obtained via the two-dimensional (2D) inverse discrete wavelet

transform

f (x, y) = 1√MN

∑m

∑n

Wϕ( j0,m, n)ϕ j0,m,n(x, y) + 1√MN

∑i=H,V,D

∞∑j= j0

∑m

∑n

W iψ( j,m, n)ψi

j,m,n(x, y). (3.8.5)

Likewise the 1D discrete wavelet transform, the 2D DWT can be implementedusing digital filters and downsamplers. With separable two-dimensional scaling andwavelet functions, we simply take the 1D FWT of the rows of f (x, y), followed bythe 1D FWT of the resulting columns (see Figs. 3.5 and 3.6).

3.9 Continuous Wavelet Transform 69

3.9 Continuous Wavelet Transform

3.9.1 1D Continuous Wavelet Transform

The continuous wavelet transform (CWT) of one-dimensional signal f (t) is givenby

Wψ(a, b) = 1√|a|∫ ∞

−∞f (t)ψ∗

(t − b

a

)dt. (3.9.1)

The continuous wavelet transform is a function of two variables a and b, where ais the scaling parameter and b is the shifting parameter. Here ψ(t) is the motherwavelet or the basic function. The scale parameter a gives the frequency informationin the wavelet transform. The translating parameter or shifting parameter b gives thetime information in the wavelet transform. It indicates the location of the windowas it is shifted through the signal. A low scale corresponds to wavelets of smallerwidth, which gives the detailed information in the signal. A high scale correspondsto wavelets of larger width which gives the global view of the signal.

The inverse continuous wavelet transform (ICWT) of one-dimensional signal isgiven by

f (t) = 1

Cψ

∫ ∞

0

∫ ∞

−∞Wψ(a, b)ψa,b(t)

dbda

a2, (3.9.2)

where ψa,b(t) = 1√|a|ψ(t−ba

).

The difference between the wavelet and windowed Fourier transform or STFTlies in the shapes of the analyzing functions gτ ,ω and ψa,b. The functions gτ ,ω , allconsist of the same envelope function g, translated to the proper time location, and“filled in” with higher frequency oscillations. All the gτ ,ω, regardless of the valueof ω, have the same width. In contrast, the ψa,b have time-widths adapted to theirfrequency: high frequency ψa,b are very narrow, while low frequency ψa,b are muchbroader. As a result, the wavelet transform is better able than the windowed Fouriertransform or STFT to “zoom in” on very short-lived high frequency phenomena,such as transients in signals (or singularities in functions or integral kernels).

3.9.2 2D Continuous Wavelet Transform

The two-dimensional (2D) continuous wavelet transform (CWT) of signal f (x, y)is given by

Wψ(a, b) = 1√|a|∫ ∞

−∞f (x, y)ψ∗

(x − m

a,y − n

a

)dxdy, (3.9.3)


wherem, n are shifting parameter or translation parameter and a is the scaling param-eter.

3.10 Undecimated Wavelet Transform or StationaryWavelet Transform

We know that the classical DWT suffers a drawback that the DWT is not a time-invariant transform. This means that, even with periodic signal extension, the DWTof a translated version of a signal x is not, in general, the translated version of theDWT of x .

How to restore the translation invariance, which is a desirable property lost bythe classical DWT? The idea is to average some slightly different DWT, called ε-decimated DWT, to define the Stationary wavelet transform (SWT) [19].

ε-decimated DWT: There exist a lot of slightly different ways to handle the dis-crete wavelet transform. Let us recall that the DWT basic computational step is aconvolution followedby adecimation.Thedecimation retains even indexed elements.

But the decimation could be carried out by choosing odd indexed elements insteadof even indexed elements. This choice concerns every step of the decomposition pro-cess, so at every level we chose odd or even. If we perform all the different possibledecompositions of the original signal, we have 2J different decompositions, for agiven maximum level J .

Let us denote by ε j = 1 or 0 the choice of odd or even indexed elements at step j .Every decomposition is labeled by a sequence of 0′s and 1′s : ε = ε1, ε2, · · ·, εJ . Thistransform is called the ε-decimated DWT or stationary wavelet transform (SWT).

3.11 Biorthogonal Wavelet Transform

A biorthogonal wavelet is a wavelet where the associatedwavelet transform is invert-ible but not necessarily orthogonal. Designing biorthogonal wavelets allows moredegrees of freedom than orthogonal wavelets. One additional degree of freedom isthe possibility to construct symmetric wavelet functions.

3.11.1 Linear Independence and Biorthogonality

The notion of the linear independence of vectors is an important concept inthe theory of finite-dimensional vector spaces. Specifically, a collection of vec-tors {x1, x2, . . . , xn} in R

n is linearly independent if any collection of scalars

3.11 Biorthogonal Wavelet Transform 71

{a1, a2, . . . , an} such that

a1x1 + a2x2 + · · · + anxn = 0 (3.11.1)

must satisfy a1 = a2 = · · · = an = 0. If in addition m = n that is, if the number ofvectors in the set matches the dimension of the space, then {x1, x2, . . . , xn} is calleda basis for R

n . This means that any vector x ∈ Rn has a unique representation as

x = b1x1 + b2x2 + · · · + bnxn, (3.11.2)

where b′i s are real scalars. For computing b′

i s, there exists a unique collection ofvectors {x1, x2, . . . , xn} called the dual basis that is biorthogonal to the collection{x1, x2, . . . , xn}. That is

〈xi , x j 〉 = δ(i − j). (3.11.3)

Hence, b′i s are given by bi = 〈xi , x〉. In generalizing the notion of a basis to the

infinite-dimensional setting, we retain the notion of linear independence.

Definition 3.11 A collection of functions {gn}n∈N ∈ L2(R) is linearly independentif given any l2-sequence of coefficients a(n) such that

∞∑n=1

a(n)gn(x) = 0 (3.11.4)

in L2(R), then a(n) = 0 for all n ∈ N.

Definition 3.12 A collection of functions {gn}n∈N ∈ L2(R) is biorthogonal to acollection {gn}n∈N ∈ L2(R), if

〈gn, gm〉 =∫R

gn(x )gm(x) = δ(n − m). (3.11.5)

It is often difficult to verify directly whether a given collection of functions islinearly independent. The next lemma gives a sufficient condition for linear indepen-dence of collection of functions:

Lemma 3.1 Let {gn(x)} be a collection of functions in L2(R), suppose that there isa collection {gn(x)}n∈N ∈ L2(R) biorthogonal to {gn(x)}. Then {gn(x)} is linearlyindependent.

Proof Let {a(n)}n∈N be an l2-sequence, and satisfy

∞∑n=1

a(n)gn(x) = 0

in L2(R). Then for each m ∈ N,


0 = 〈0, gm〉 =⟨ ∞∑n=1

a(n)gn(x), gm(x)

⟩=

∞∑n=1

a(n)〈gn(x), gm(x)〉 = a(m)

by biorthogonality. Therefore {gn(x)} is linearly independent. �

3.11.2 Dual MRA

Definition 3.13 A pair of MRA’s {Vj } j∈Z with scaling function ϕ(x) and {V j } j∈Zwith scaling function ϕ(x) are dual to each other if ϕ(x − n) is biorthogonal toϕ(x − n).

Since theremay bemore than one function ϕ(x) such thatϕ(x−n) is biorthogonalto ϕ(x − n), there may be more than one MRA {V j } j∈Z dual to {Vj } j∈Z.

Definition 3.14 Let ϕ(x) and ϕ(x) be scaling function for dual MRA. For eachj ∈ Z, define an approximation operators PJ , PJ , and the detail operators QJ andQ J on L2(R) functions f (x) by

PJ f (x) =∑k

〈 f, ϕ j,k〉ϕ j,k(x), (3.11.6)

PJ f (x) =∑k

〈 f,ϕ j,k〉ϕ j,k(x), (3.11.7)

QJ f (x) = PJ+1 f (x) − PJ f (x), (3.11.8)

Q J f (x) = PJ+1 f (x) − PJ f (x). (3.11.9)

Definition 3.15 Let ϕ(x) and ϕ(x) be scaling function for dual MRA and let h(n)

and h(n) be the scaling filters corresponding to ϕ(x) and ϕ(x). Define the filtersg(n) and g(n) by

g(n) = (−1)nh(1 − n) and g(n) = (−1)nh(1 − n). (3.11.10)

Define the wavelet ψ(x) and the dual wavelet ψ(x) by

ψ(x) =∑n

g(n)21/2ϕ(2x − n) and ψ(x) =∑n

g(n)21/2ϕ(2x − n). (3.11.11)

The following illustrate some basic properties of thewavelet and its dual. Letψ(x)and ψ(x) be the wavelet and dual wavelet corresponding to the MRA’s {Vj } j∈Z withscaling function ϕ(x) and {V j } j∈Z with scaling function ϕ(x). Then the followingholds:


(a) ψ(x) ∈ V1 and ψ(x) ∈ V1.

(b) {ψ0,n(x)} is biorthogonal to {ψ0,n(x)}.(c) {ψ0,n(x)} is a orthogonal basis for span{ψ0,n(x)} and {ψ0,n(x)} is a orthogonal

basis for span{ψ0,n(x)}.(d) For all n,m ∈ Z, 〈ψ0,n, ϕ0,n〉 = 〈ψ0,n,ϕ0,n〉 = 0.(e) for any f (x) ∈ C0

c (R),

Q0 f (x) ∈ span{ψ0,n(x)} and Q0 f (x) ∈ span{ψ0,n(x)}

It can be easily seen that the collections {ψ j,k(x)} j,k∈Z and {ψ j,k(x)} j,k∈Z definedby Eq. (3.11.11) are orthogonal basis on R.

3.11.3 Discrete Transform for Biorthogonal Wavelets

As with orthogonal wavelets, there is a very simple and fast discrete version of thebiorthogonal wavelet expansion. Recall that by Definition 3.14, for any j ∈ Z,

P− j f (x) =∑n

〈 f,ϕ− j,n〉ϕ− j,n(x) =∑n

c j (n)ϕ− j,n(x), (3.11.12)

andQ− j f (x) =

∑n

〈 f,ψ− j,n〉ψ− j,n(x) =∑n

d j (n)ψ− j,n(x). (3.11.13)

Also by definition 3.14,

P− j f (x) = P− j−1 f (x) + Q− j−1 f (x).

Writing out in terms of equations (3.11.12) and (3.11.13), we have

∑n

c j (n)ϕ− j,n(x) =∑n

c j+1(n)ϕ− j−1,n(x) +∑n

d j+1(n)ψ− j−1,n(x)

=∑n

c j+1(n)∑k

h(k − 2n)ϕ− j,k(x) +∑n

d j+1(n)∑k

g(k − 2n)ϕ− j,k(x)

=∑k

(∑n

c j+1(n)h(k − 2n) +∑n

d j+1(n)g(k − 2n)

)ϕ− j,k(x).

By matching the coefficients, we conclude that

c j (k) =∑n

c j+1(n)h(k − 2n) +∑n

d j+1(n)g(k − 2n). (3.11.14)


Now, we summarize these results in the following theorem:

Theorem 3.8 Letϕ(x) and ϕ(x) be scaling function for dualMRA’s and let h(n) andh(n) be the scaling filters corresponding to ϕ(x) and ϕ(x). Define the wavelet filtersg(n) and g(n) by Eq. (3.11.10) and the wavelets ψ(x) and ψ(x) by Eq. (3.11.11).Given a function f (x) ∈ L2(R), define for k ∈ Z,

c0(k) = 〈 f,ϕ0,k〉,

and for every j ∈ N and k ∈ Z, define c j (k) and d j (k) by


Then

c j+1(k) =∑n

c j (n)h(n − 2k) d j+1(k) =∑n

c j (n)g(n − 2k) (3.11.15)

andc j (k) =

∑n

c j+1(n)h(k − 2n) +∑n

d j+1(n)g(k − 2n). (3.11.16)

The operations inEq. (3.11.15) are precisely the approximation operator and detailoperator corresponding to the filters h(n) and g(n). Equation (3.11.16) involves theapproximation adjoint and detail adjoint corresponding to the filters h(n) and g(n).This leads to the following definition:

Definition 3.16 Given a signal c(n) and apair of filtersh(k) and h(k), defineg(k) and

g(k) by g(k) = (−1)k h(1 − k) and g(k) = (−1)kh(1 − k). Define the correspondingapproximation operators H and H and detail operators G and G on signal c(n) by

(Hc)(k) =∑n

c(n)h(n − 2k), (Gc)(k) =∑n

c(n)g(n − 2k), (3.11.17)

(Hc)(k) =∑n

c(n)h(n − 2k), (Gc)(k) =∑n

c(n)g(n − 2k),

and the approximation adjoints H∗, H∗ and detail adjoints G∗, G∗ are defined by

(H∗c)(k) =∑n

c(n)h(k − 2n), (G∗c)(k) =∑n

c(n)g(k − 2n), (3.11.18)

(H∗c)(k) =∑n

c(n)h(k − 2n), (G∗c)(k) =∑n

c(n)g(k − 2n).


Theorem 3.9 Keeping the same notation as Theorem 3.8, we have

c j+1 = Hcj , d j+1 = Gcj , (3.11.19)

andc j = H∗c j+1 + G∗d j+1. (3.11.20)

Next, we will define the analogue of the QMF conditions in the biorthogonal case.suppose that ϕ(x) and ϕ(x) are scaling functions for dual GMRA’s, with scalingfilters h(n) and h(n) and g(n) and g(n) are corresponding wavelet filters. Then wecan prove the following analogue of Theorem 3.3.

Theorem 3.10 With h(n), h(n), g(n) and g(n) are defined as above. Then

(a)∑

n h(n)h(n − 2k) =∑n g(n)g(n − 2k) = δ(k),

(b)∑

n g(n)h(n − 2k) =∑n g(n)h(n − 2k) = 0, f or all k ∈ Z, and(c)∑

k h(m − 2k )h(n − 2k) +∑k g(m − 2k )g(n − 2k) = δ(m − n).

We also have the following analogue of Theorem 3.4.

Theorem 3.11 With h(n), h(n), g(n) and g(n) are defined as above, define

m0(ξ) = 1√2

∑n

h(n)e−inξ, m0(ξ) = 1√2

∑n

h(n)e−inξ, (3.11.21)

m1(ξ), and m1(ξ), by

m1(ξ) = e−i(ξ+π)m0(ξ + π), m1(ξ) = e−i(ξ+π)m0(ξ + π). (3.11.22)

Define the operators H, H ,G, G, H∗, H∗,G∗ and G∗ by Eqs. (3.11.17) and(3.11.18). Then the following are equivalent:

(a) m0(ξ)m0(ξ) + m0(ξ + π)m0(ξ + π) = 1.(b)∑

n h(n)h(n − 2k) = δ(k).(c) H∗H + G∗G = I.(d) H H∗ = GG∗ = I.

This leads to the following definition.

Definition 3.17 Given a filter h(k) and h(k), define mo(ξ) and m0(ξ) by

m0(ξ) = 1√2

∑n

h(n)e−inξ, m0(ξ) = 1√2

∑n

h(n)e−inξ .

Then h(k) and h(k) form a QMF provided that:(a) m0(0) = m0(0) = 1 and(b) m0(

ξ2 )m0(ξ) + m0(

ξ2 + π)m0(ξ + π) = 1, for all ξ ∈ R.


conditions (a) and (b) are called the biorthogonal QMF conditions.

Theorem 3.12 Given h(n) and h(n) are a QMF pair. Then:(a)∑

n h(n) =∑n h(n) = √2.

(b)∑

n g(n) =∑n g(n) = 0.

(c)∑

n h(n)h(n − 2k) =∑n g(n)g(n − 2k) = δ(k).

(d)∑

n g(n)h(n − 2k) =∑n g(n)h(n − 2k) = 0, f or all k ∈ Z, and(e)∑

k h(m − 2k )h(n − 2k) +∑k g(m − 2k )g(n − 2k) = δ(m − n).

3.12 Scarcity of Wavelet Transform

Why have wavelets and multiscale analysis proved so useful in such a wide rangeof applications? The primary reason is because they provide an extremely efficientrepresentation for many types of signals that appear often in practice but are not wellmatchedby theFourier basis,which is ideallymeant for periodic signals. In particular,wavelets provide an optimal representation for many signals containing singularities.The wavelet representation is optimally sparse for such signals, requiring an order ofmagnitude fewer coefficients than the Fourier basis to approximate within the sameerror. The key to the sparsity is that since wavelets oscillate locally, only waveletsoverlapping a singularity have large wavelet coefficients; all other coefficients aresmall.

The sparsity of the wavelet coefficients of many real-world signals enables near-optimal signal processing based on simple thresholding (keep the large coefficientsand kill the small ones), the core of a host of powerful image compression, denoising,approximation, deterministic and statistical signal and image algorithms.

In spite of its efficient computational algorithm and sparse representation, thewavelet transform suffers from four fundamental, intertwined shortcomings.

1. Oscillations

Since wavelets are bandpass functions, the wavelet coefficients tend to oscillate posi-tive and negative around singularities. This considerably complicates wavelet-basedprocessing, making singularity extraction and signal modeling, in particular, verychallenging. Moreover, since an oscillating function passes often through zero, wesee that the conventional wisdom that singularities yield large wavelet coefficientsis overstated. Indeed, it is quite possible for a wavelet overlapping a singularity tohave a small or even zero wavelet coefficient.

2. Shift Variance

A small shift of the signal greatly perturbs the wavelet coefficient oscillation patternaround singularities. Shift variance also complicates wavelet-domain processing.Algorithms must be made capable of coping with the wide range of possible wavelet

3.12 Scarcity of Wavelet Transform 77

coefficient patterns caused by shifted singularities. To better understand waveletcoefficient oscillations and shift variance, consider a piecewise smooth signal x(t−t0)like the step function u(t) = 0, t < 0 and u(t) = 1, t ≥ 0 analyzed by awavelet basishaving a sufficient number of vanishing moments. Its wavelet coefficients consist ofsamples of the step response of the wavelet

d j (n) ≈ 2−3 j/2 �∫ 2 j t0−n

−∞ψ(t)dt,

where � is the height of the jump. Since ψ(t) is a bandpass function that oscillatesaround zero, so does its step response d j (n) as a function of n. Moreover, the factor2 j in the upper limit ( j ≥ 0) amplifies the sensitivity of d j (n) to the time shift t0,leading to strong shift variance.

3. Aliasing

The wide spacing of the wavelet coefficient samples, or equivalently, the fact that thewavelet coefficients are computed via iterated discrete-time downsampling opera-tions interspersed with non ideal low-pass and high-pass filters, results in substantialaliasing. The inverse DWT cancels this aliasing, but only if the wavelet and scalingcoefficients are not changed. Any wavelet coefficient processing upsets the delicatebalance between the forward and inverse transforms, leading to artifacts in the recon-structed signal.

4. Lack OF Directionality

Finally, while Fourier sinusoids in higher dimensions correspond to highly direc-tional plane waves, the standard tensor product construction of multidimensionalsignal (MD) wavelets produces a checkerboard pattern that is simultaneously ori-ented along several directions. This lack of directional selectivity greatly complicatesmodeling and processing of geometric image features like ridges and edges.

Fortunately, there is a simple solution, called complex wavelet transform (CWT),to these four DWT shortcoming. The key is to note that the Fourier transform doesnot suffer from these problems.

(i) Themagnitude of the Fourier transform does not oscillate positive and negativebut rather provides a smooth positive envelope in the Fourier domain.

(ii) The magnitude of the Fourier transform is perfectly shift invariant, with asimple linear phase offset encoding the shift.

(iii) The Fourier coefficients are not aliased and do not rely on a complicatedaliasing cancellation property to reconstruct the signal;

(iv) The sinusoids of the MD Fourier basis are highly directional plane waves.

What is the difference between FT and DWT?Unlike the DWT, which is based onreal-valued oscillating wavelets, the Fourier transform is based on complex-valuedoscillating sinusoids


eiξt = cos(ξt) + isin(ξt) with i = √−1.

The oscillating cosine and sine components, respectively, form a Hilbert transformpair, i.e., they are 90◦ out of phase with each other. Together they constitute ananalytic signal eiξt that is supported on only one-half of the frequency axis.

3.13 Complex Wavelet Transform

Inspired by the Fourier representation, define a complex wavelet transform (CWT) assimilar to the DWT but with a complex-valued scaling function and complex-valuedwavelet

ψc(x) = ψre(x) + iψim(x), (3.13.1)

where ψre(x) is real and even and iψim(x) is imaginary and odd. Moreover, if ψre(x)and iψim(x) form a Hilbert transform pair, then ψc(x) is an analytic signal andsupported on only one-half of the frequency axis. projecting the signal onto ψc, j,n ,we obtain the complex wavelet coefficient

dc, j (k) = dre, j (k) + i dim, j (k)

with magnitude

|dc, j (k)| =√

[dre, j (k)]2 + [dim, j (k)]2

and phase

∠dc, j (k) = arctan

(dim, j (k)

dre, j (k)

)

when |dc, j (k)| > 0. As with the Fourier transform, complex wavelets can beused to analyze and represent both real-valued signals (resulting in symmetries inthe coefficients) and complex-valued signals. In either case, the CWT enables newcoherent multiscale signal processing algorithms that exploit the complex magnitudeand phase. In particular, a large magnitude indicates the presence of a singularitywhile the phase indicates its position within the support of the wavelet.

The theory and practice of discrete complex wavelets can be divided into twoclass. The first approach seeks a ψc(x) that forms an orthonormal or biorthogonalbasis [1, 4, 10, 20, 32, 37]. This strong constraint prevents the resulting CWTfrom overcoming most of the four DWT shortcomings discussed above. The secondapproach seeks a redundant representation, with bothψre(x) andψim(x) individuallyforming orthonormal or biorthogonal bases. The resulting CWT is a 2× redundanttight frame [7] in 1D, with the power to overcome the four shortcomings.

In this subsection, we will focus on a particularly second approach proposed byKingsbury in [15, 16], redundant type of CWT, is also called the dual-tree CWT

3.13 Complex Wavelet Transform 79

approach, which is based on two filter bank trees and thus two bases. Any CWTbased on wavelets of compact support cannot exactly possess the Hilbert transformand analytic signal properties, and hence any such CWTwill not perfectly overcomethe four DWT shortcomings. The key challenge in dual-tree wavelet design is thejoint design of its two filter banks to yield a complex wavelet and scaling functionthat are as close as possible to analytic.

As a result, the dual-tree CWT comes very close to mirroring the attractive prop-erties of the Fourier transform, including a smooth, non-oscillating magnitude; anearly shift-invariant magnitude with a simple near-linear phase encoding of signalshifts; substantially reduced aliasing and directional wavelets in higher dimensions.The only cost for all of this is a moderate redundancy: 2× redundancy in 1D. This ismuch less than the log2N× redundancy of a perfectly shift-invariant DWT, which,moreover, will not offer the desirablemagnitude and phase interpretation of theCWTnor the good directional properties in higher dimensions.

3.14 Dual-Tree Complex Wavelet Transform

The discrete complex wavelet transform (DCWT) is a form of discrete wavelet trans-form, which generates complex coefficients by using a dual tree of wavelet filtersto obtain their real and imaginary parts. What makes the complex wavelet basisexceptionally useful for denoising purposes is that it provides a high degree of shift-invariance and better directionality compared to the real DWT. The DWT suffersfrom the following two problems:

Lack of shift invariance - this results from the downsampling operation at eachlevel. When the input signal is shifted slightly, the amplitude of the wavelet coeffi-cients varies so much.

Lack of directional selectivity - as the DWTfilters are real and separable the DWTcannot distinguish between the opposing diagonal directions.

These problems hinder the use of wavelets in other areas of image processing. Thefirst problem can be avoided if the filter outputs from each level are not downsampledbut this increases the computational costs significantly and the resulting undecimatedwavelet transform still cannot distinguish between opposing diagonals since thetransform is still separable. To distinguish opposing diagonals with separable filtersthe filter frequency responses are required to be asymmetric for positive and negativefrequencies. A good way to achieve this is to use complex wavelet filters which canbe made to suppress negative frequency components.

One effective approach for implementing an analyticwavelet transform, first intro-duced by Kingsbury in 1998 [17], is called the dual-tree CWT (DTCWT). Like theidea of positive and negative post-filtering of real subband signals, the idea behindthe dual-tree approach is quite simple. The dual-tree CWT employs two real DWTs.The first DWT gives the real part of the transform while the second DWT givesthe imaginary part. The analysis and synthesis filter banks used to implement thedual-tree CWT and its inverse are illustrated in given Figs. 3.7 and 3.8.


Fig. 3.7 The analysis filter bank for the Dual-Tree CWT

Fig. 3.8 The synthesis filter bank for the Dual-Tree CWT

The two real wavelet transforms use two different sets of filters, with each satisfy-ing the perfect reconstruction conditions. The two sets of filters are jointly designed sothat the overall transform is approximately analytic. Let h0(n), h1(n) denote the low-pass, high-pass filter pair respectively for the upper filter bank, and let g0(n), g1(n)

denote the low-pass, high-pass filter pair respectively for the lower filter bank.Wewilldenote the two real wavelets associated with each of the two real wavelet transformsas ψh(x) and ψg(x). In addition to satisfying the perfect reconstruction (PR) condi-tions, the filters are designed so that the complex wavelet ψ(x) = ψh(x) + iψg(x)is approximately analytic. Equivalently, they are designed so that ψg(x) is approxi-mately the Hilbert transform of ψh(x) [denoted ψg(x) ≈ H{ψh(x)}].

Note that the filters are themselves real and no complex arithmetic is required forthe implementation of the dual-tree CWT. Also note that the dual-tree CWT is nota critically sampled transform and it is two times expansive in 1D because the totaloutput data rate is exactly twice the input data rate.

The inverse of the dual-treeCWT is as simple as the forward transform. To inversethe transform, the real part and the imaginary part are each inverted, the inverse ofeach of the two real DWTs are used to obtain two real signals. These two real signalsare then averaged to obtain the final output. Note that the original signal x(n) can be

3.14 Dual-Tree Complex Wavelet Transform 81

recovered from either the real part or the imaginary part alone, however, such inversedual-tree CWTs do not capture all the advantages an analytic wavelet transformoffers.

If the two real DWTs are represented by the square matrices Fh and Fg, then thedual-tree CWT can be represented by the rectangular matrix

F =[Fh

Fg

]. (3.14.1)

If the vector x represents a real signal, then xh = Fhx represents the real partand xg = Fgx represents the imaginary part of the dual-tree CWT. The complexcoefficients are given by xh + i xg . A (left) inverse of F is then given by

F−1 = 1

2

[F−1h F−1

g

], (3.14.2)

as we can easily see that

F−1 · F = 1

2

[F−1h F−1

g

] ·[Fh

Fg

]= 1

2[I + I ] = I.

We can just as well share the factor of one half between the forward and inversetransforms, to obtain

F = 1√2

[Fh

Fg

]F−1 = 1√

2

[F−1h F−1

g

]. (3.14.3)

If the two real DWTs are orthonormal transforms, then the transpose of Fh is itsinverse Ft

h · Fh = I and similarly for Fg. The dual-tree wavelet transform definedin (3.14.3) keeps the real and imaginary parts of the complex wavelet coefficientsseparate. However, the complex coefficients can be explicitly computed using thefollowing form:

Fc = 1

2

[I i II − i I

]·[Fh

Fg

]. (3.14.4)

F−1c = 1

2

[F−1h F−1

g

] ·[

I I−i I i I

]. (3.14.5)

Note that the complex sum and difference matrix in (3.14.4) is unitary. Therefore,if the two real DWTs are orthonormal transforms then the dual-tree CWT satisfiesF∗c · Fc = I, where ∗ denotes conjugate transpose. If

[uv

]= Fc · x .


When x is real, we have v = u∗. When the input signal x is complex, then v �= u∗,so both u and v need to be computed.

When the DTCWT is applied to a real signal, the output of the upper and lowerfilter banks will be the real and imaginary parts of the complex coefficients and theycan be stored separately as represented by Eq. (3.14.3). However, if the DTCWT isapplied to a complex signal, then the output of both the upper and lower filter bankswill be complex and it is no longer correct to label them as the real and imaginaryparts.

TheDTCWT is also very easy to implement because there is no data flow betweenthe two real DWTs. They can be easily implemented using existing software andhardware. Moreover, the transform id naturally parallelized for efficient hardwareimplementation. In addition, the use of the DTCWT can be informed by the existingtheory and practice of real wavelet transforms because the DTCWT is implementedusing two real wavelet transforms.

The DTCWT has the following properties:1. nearly shift invariance.2. good selectivity and directionality in 2D (or higher dimension) with Gabor-like

filters.3. perfect reconstruction (PR) using short linear phase filters,4. limited redundancy, independent of the number of scales, for example, a redun-

dancy factor of only 2n for n-dimensional signals,5. efficient order N computation: only 2n times the simple real DWT for n-

dimensional signal.ThemultidimensionalDTCWTis non-separable but is based on a computationally

efficient, separable filter bank. A 2D dual-tree complex wavelet can be defined asψ(x, y) = ψ(x)ψ(y) associated with the row-column implementation of the wavelettransform, where ψ(x) and ψ(y) are two complex wavelets, ψ(x) = ψh(x)+ iψg(x)and ψ(y) = ψh(y) + iψg(y), ψh(·) and ψg(·) are real wavelet transforms of upperfilter bank and lower filter bank, respectively. Then we obtain the following for theexpression:

ψ(x, y) = [ψh(x) + iψg(x)][ψh(y) + iψg(y)]

= [ψh(x)ψh(y) − ψg(x)ψg(y)] + i[ψg(x)ψh(y) + ψh(x)ψg(y)]. (3.14.6)

A 2D DTCWT is oriented and approximately analytic at the cost of being fourtimes expansive. It also possesses the full shift-invariant properties of the constituent1D transforms. The real parts of six oriented complex wavelets of DTCWT can bedefined as follows:

ϕk(x, y) = 1√2(ψ1,k(x, y) − ψ2,k(x, y))

ϕk+3(x, y) = 1√2(ψ1,k(x, y) − ψ2,k(x, y)) (3.14.7)

3.14 Dual-Tree Complex Wavelet Transform 83

where k = 1, 2, 3 and

ψ1,1(x, y) = φh(x)ψh(y) ψ2,1(x, y) = φg(x)ψg(y)

ψ1,2(x, y) = ψh(x)φh(y) ψ2,2(x, y) = ψg(x)φg(y)

ψ1,3(x, y) = ψh(x)ψh(y) ψ2,3(x, y) = ψg(x)ψg(y).

(3.14.8)

The imaginary parts of six oriented complex wavelets of DTCWT can be defined asfollows:

ξk(x, y) = 1√2(ψ3,k(x, y) − ψ4,k(x, y))

ξk+3(x, y) = 1√2(ψ3,k(x, y) − ψ4,k(x, y))

(3.14.9)

where k = 1, 2, 3 and

ψ3,1(x, y) = φg(x)ψh(y) ψ4,1(x, y) = φh(x)ψg(y)

ψ3,2(x, y) = ψg(x)φh(y) ψ4,2(x, y) = ψh(x)φg(y)

ψ3,3(x, y) = ψg(x)ψh(y) ψ4,3(x, y) = ψh(x)ψg(y).

(3.14.10)

In Eqs. (3.14.8) and (3.14.10), h(·) and g(·) are the low-pass functions of upperfilter bank and lower filter bank, respectively along the first dimension. h(·) and g(·)are the high-pass functions of upper filter bank and lower filter bank, respectivelyalong the second dimension. A 2D DTCWT produces three sub-bands in each ofspectral quadrants 1 and 2, giving six sub-bands of complex coefficients at eachlevel, which are strongly oriented at angles of 15◦, 45◦, 75◦.

3.15 Quaternion Wavelet and Quaternion WaveletTransform

As a mathematical tool, wavelet transform is a major breakthrough over the Fouriertransform since it has good time-frequency localization and multiple resolution anal-ysis features. Wavelet analysis theory has become one of the most useful tools insignal analysis, image processing, pattern recognition andmany other fields. In imageprocessing, the basic idea of the wavelet transform is to decompose an image intomultiple resolutions. Specifically, the original image is decomposed into differentspace and frequency sub-images, and coefficients of the sub-image are then pro-cessed. A small shift in the signal significantly changes the distribution of the realdiscrete wavelet transform. Although the dual-tree complex wavelet transform canovercame the problem, it cannot avoid the phase ambiguity of two-dimensional(2D)


images features. As an improved alternative, the quaternion wavelet transform isa new multiscale analysis tool for image processing. It is based on the 2D Hilberttransform theory, which is shift invariant, and can overcome the drawbacks of realand complex wavelet transforms.

Quaternion wavelet transform [5, 6, 42] is established based on the quaternionalgebra, quaternion Fourier transform andHilbert transform.A quaternion, or quater-nion analytic signal, has a real part and three imaginary parts. Given a 2D real signal,its real DWT is the real part of the quaternion wavelet and its three Hilbert trans-forms become three imaginary part of the quaternion wavelet. It can be understood asthe improved real wavelet and complex wavelet’s promotion, which are shift invari-ant with abundant phase information and limited redundancy. Since the quaternionwavelet retains the traditional wavelet’s time-frequency localization ability, it is easyto design filters Hilbert transform pair of the dual tree structure.

3.15.1 2D Hilbert Transform

Definition 3.18 Let f (x, y) ∈ L2(R2). Then Hilbert transform of f (x, y), denotedby fHx (x, y), fHy (x, y) and fHxy (x, y) along x-axis, y-axis and x, y-axis respec-tively, are defined by

fHx (x, y) = 1

π

∫R

f (ξ1, y)

x − ξ1dξ1, (3.15.1)

fHy (x, y) = 1

π

∫R

f (x, ξ2)

y − ξ2dξ2, (3.15.2)

and

fHxy (x, y) = 1

π2

∫R

f (ξ1, ξ2)

(x − ξ1)(y − ξ2)dξ1dξ2 (3.15.3)

and the corresponding frequency domain are

fHx (u, v) = − j sgn(u) f (u, v), (3.15.4)

fHy (u, v) = − j sgn(v) f (u, v) (3.15.5)

andfHxy (u, v) = −sgn(u)sgn(v) f (u, v). (3.15.6)

3.15 Quaternion Wavelet and Quaternion Wavelet Transform 85

3.15.2 Quaternion Algebra

The quaternion is proposed by W.R. Hamilton in 1843. The quaternion algebra overR, denoted byH, is an associative non-commutative four-dimensional algebra. Everyelement of H is a linear combination of a scalar and three imaginary units i, j, andk with real coefficients

H = {q : q = q0 + iq1 + jq2 + kq3, q0, q1, q2, q3 ∈ R}, (3.15.7)

where i, j and k obey Hamiltons multiplication rules

i j = − j i = k, jk = −k j = i, ki = −ik = j, i2 = j2 = k2 = i jk = −1.(3.15.8)

The quaternion conjugate of a quaternion q is given by

q = q0 − iq1 − jq2 − kq3 q0, q1, q2, q3 ∈ R. (3.15.9)

The quaternion conjugation (3.15.9) is a linear anti-involution

q = q, p + q = p + q, pq = q p, f or all p, q ∈ H. (3.15.10)

The multiplication of a quaternion q and its conjugate can be expressed as

qq = q20 + q2

1 + q22 + q2

3 . (3.15.11)

This leads to the modulus |q| of a quaternion q defined as

|q| = √qq =√q20 + q2

1 + q22 + q2

3 . (3.15.12)

Using (3.15.9) and (3.15.12), we can defined the inverse of q ∈ H\{0} as

q−1 = q

|q|2 , (3.15.13)

which shows thatH is a normed division algebra. Furthermore, we get |q−1| = |q|−1.In addition, quaternion q can also be expressed as:

q = |q|eiφeiθeiξ, (3.15.14)

where |q| is the modulus of q and (φ, θ, ξ) are the three phase angles which isuniquely definedwithin the range (φ, θ, ξ) ∈ [−π,π]×[−π/2,π/2]×[−π/4,π/4].

The quaternion module L2(R2, H) is defined as


L2(R2, H) = { f : R2 → H, || f ||L2(R2,H) < ∞}. (3.15.15)

For all quaternion functions f, g ∈ R2 → H, the inner product is defined as follows:

( f, g)L2(R2,H) =∫R2

f (x)g(x)d2x, f or x ∈ R2. (3.15.16)

If f = g almost everywhere, we obtain the associated norm

|| f ||L2(R2,H) = ( f, g)1/2 =(∫

R2| f (x)|2d2x

)1/2

, x ∈ R2. (3.15.17)

With the usual addition and scalar multiplication of functions together with the innerproduct L2(R2, H) becomes a Hilbert space.

Definition 3.19 If f (x, y) is a real two-dimensional signals, then quaternion ana-lytic signal can be defined as f q(x, y) = f (x, y) + i fHx (x, y) + j fHy (x, y) +k fHxy (x, y), where fHx (x, y), fHy (x, y), and fHxy (x, y) are the Hilbert transform off (x, y) along the x-axis, the y-axis and along the xy-axis respectively.

This section is based on the study of quaternion wavelet theory, which proved andpresented the correlative properties and concepts of scale basis and wavelet basis ofquaternion wavelet.

Lemma 3.2 Let {ϕ(x − k)}k∈Z be the space of standard orthogonal basis for V ⊂L2(R), then the system {ϕg(x − k)}k∈Z, where ϕg(x) = H(φ(x)), is the standard

orthogonal basis of space V = HV ⊂ L2(R).

Proof According toMallat, we know that standard orthogonal basis in the frequencydomain is equivalent to

∑k∈Z |φ(ξ+2kπ)|2 = 1,by theproperty ofHilbert transform,

we have∑

k∈Z |φg(ξ + 2kπ)|2 = 1. �

According to the lemma, it is not difficult to prove the following.

Theorem 3.13 Let {Vj } j∈Z be a one dimensional orthogonal multiresolution analy-sis (MRA), and the corresponding scale and wavelet functions are ϕh(x) and ψh(x),respectively, then {ϕg(x − l)ϕh(x − k)}l,k∈Z or {ϕh(x − l)φg(x − k)}l,k∈Z are thestandard orthogonal basis of the space V0 ⊗ V0 and the space V0 ⊗ V0, whereV0 = HV0. Hx (φh(x)φh(y)) = φg(x)φh(y), Hy(φh(x)φh(y)) = φh(x)φg(y)and Hxy(φh(x)φh(y)) = ϕg(x)ϕg(y) respectively denote Hilbert transforms of thefunction (ϕh(x)ϕh(y)) along x-axis, y-axis and xy-axis directions.

By above Theorem and the analysis, we have the following.Let {Vj } j∈Z be one dimensional orthogonal MRA and ϕh(x) and ψh(x) are the

corresponding scale and wavelet functions respectively. Then {�qj,k,m(x, y)}k,m∈Z is

the orthogonal basis of the quaternion wavelet scale space in L2(R2, H), where


�q(x, y) = ϕh(x)ϕh(y)+iϕg(x)ϕh(y)+ jϕh(x)ϕg(y)+kϕg(x)ϕg(y), (3.15.18)

�qj,k,m(x, y) = ϕh, j,k(x)ϕh, j,m(y) + iϕg, j,k(x)ϕh, j,m(y) + jϕh, j,k(x)ϕg, j,m(y)

+ kϕg, j,k(x)ϕg, j,m(y), (3.15.19)

and ϕh, j,k(x) = 2− j/2φh(2− j x − k), j, k, m ∈ Z. �q(x, y) is called a quaternionwavelet scale function in L2(R2, H) and {�q

j,k,m(x, y)}k,m∈Z is called the discretequaternion wavelet scale function in L2(R2, H).

Further, Let {Vj } j∈Z beonedimensional orthogonalMRAandϕh(x) andψh(x) arethe corresponding scale and wavelet functions respectively. Then {�q,1(x, y), �q,2

(x, y), �q,3(x, y)} are the quaternion wavelet basis functions in L2(R2, H) and{�q,1

j,k,m, �q,2j,k,m, �

q,3j,k,m} arediscrete quaternionwavelet basis functions in L2(R2, H),

where

�q,1(x, y) = ϕh(x)ψh(y) + iϕg(x)ψh(y) + jϕh(x)ψg(y)

+ kϕg(x)ψg(y), (3.15.20)

�q,2(x, y) = ψh(x)ϕh(y) + iψg(x)ϕh(y) + jψh(x)ϕg(y)

+ kψg(x)ϕg(y), (3.15.21)

�q,2(x, y) = ψh(x)ψh(y) + iψg(x)ψh(y) + jψh(x)ψg(y)

+ kψg(x)ψg(y), (3.15.22)

the shift and expand form of {�q,1(x, y), �q,2(x, y) and �q,3(x, y)} are�

q,1j,k,m(x, y) = ϕh, j,k(x)ψh, j,m(y) + iϕg, j,k(x)ψh, j,m(y) + jϕh, j,k (x)ψg, j,m(y) + kϕg, j,k(x)ψg, j,m(y),

(3.15.23)

�q,2j,k,m(x, y) = ψh, j,k(x)ϕh, j,m(y) + iψg, j,k(x)ϕh, j,m(y) + jψh, j,k (x)ϕg, j,m(y) + kψg, j,k(x)ϕg, j,m(y),

(3.15.24)

�q,3j,k,m(x, y) = ψh, j,k (x)ψh, j,m(y) + iψg, j,k(x)ψh, j,m(y) + jψh, j,k (x)ψg, j,m(y) + kψg, j,k(x)ψg, j,m(y),

(3.15.25)and ψh, j,k(x) = 2− j/2ψh(2− j x − k), j, k, m ∈ Z.

Definition 3.20 For all f (x, y) ∈ L2(R2, H), define

aqj,k,m =(f (x, y), �

qj,k,m(x, y)

), (3.15.26)

dq,ij,k,m =

(f (x, y), �

q,ij,k,m(x, y)

), (i = 1, 2, 3, j, k, m ∈ Z). (3.15.27)


Fig. 3.9 Decomposition filter bank for QWT

dq,ij,k,m (i = 1, 2, 3), is called the discrete quaternion wavelet transform (DQWT) off (x, y).

The above discussion shows that quaternion wavelet transform by using four realdiscrete wavelet transforms, the first real discrete wavelet corresponding quaternionwavelet real part, the other real discrete wavelets are formed by the first real discretewavelet transform by Hilbert transform, corresponding to the three imaginary partsof quaternion wavelet, respectively. The quaternion scale function �q(x, y) and thequaternion wavelet basis functions �q,1(x, y), �q,2(x, y), �q,3(x, y) correspond-ing real component are taken out to form a matrix:

G =

⎛⎜⎜⎝

ϕh(x)ϕh(y) ϕh(x)ψh(y) ψh(x)ϕh(y) ψh(x)ψh(y)ϕg(x)ϕh(y) ϕg(x)ψh(y) ψg(x)ϕh(y) ψg(x)ψh(y)ϕh(x)ϕg(y) ϕh(x)ψg(y) ψh(x)ϕg(y) ψh(x)ψg(y)ϕg(x)ϕg(y) ϕg(x)ψg(y) ψg(x)ϕg(y) ψg(x)ψg(y)

⎞⎟⎟⎠ .

Then each row of the matrix G corresponds to the one real wavelet of quaternionwavelet, the first column corresponding to quaternion wavelet scale function, theother columns are quaternion wavelets three wavelet functions corresponding tohorizontal, vertical and diagonal three subbands. In the space L2(R2), there are fourstandard orthogonal real wavelet bases, by wavelet frame and the concept of 2D realwavelet, we know that quaternion wavelet base in L2(R2) form a tight frame withframe bound 4.


Fig. 3.10 Reconstruction structure of filter bank

Let h0 and h1 are low-pass and high-pass filter of real wavelet, g0 and g1 are low-pass and high-pass filter, corresponding to Hilbert transform of h0 and h1, respec-tively and let, h0, h1, g0 and g1 are synthesis filter. Figures3.9 and 3.10 show thedecomposition and reconstruction filter bank for quaternion wavelet transform.

In order to calculate the coefficients of QWT, the quaternionwavelet filters systemis similar to dual-tree complex wavelet, quaternion wavelet filters and coefficientsis quaternion, and it is realized by using the dual-tree algorithm, using an analyticquaternionwavelet bases in order to satisfy theHilbert transform.Quaternionwaveletfilters are dual-tree filters and each filters subtree part comprises 2 analysis filtersand 2 synthesis filters, respectively.

3.15.3 Quaternion Multiresolution Analysis

For a two dimensional image function f (x, y), a quaternionmultiresolution analysis(QMRA) can be expressed as


f (x, y) = Aqn f (x, y) +

n∑j=1

[Dq,1

j f (x, y) + Dq,2j f (x, y) + Dq,3

j f (x, y)].

(3.15.28)We can characterize each approximation function Aq

j f (x, y) and the difference com-

ponents Dq,ij f (x, y) for i = 1, 2, 3, by means of a 2D scaling function �q(x, y)

and its associated wavelet function �q,i (x, y) as follows:

Aqj f (x, y) =

∑k∈Z

∑m∈Z

a j,k,m�qj,k,m(x, y), (3.15.29)

Dq,ij f (x, y) =

∑k∈Z

∑m∈Z

dij,k,m�

q,ij,k,m(x, y), (3.15.30)

whereaqj,k,m =

(f (x, y), �

qj,k,m(x, y)

), (3.15.31)

dq,ij,k,m =

(f (x, y),�q,i

j,k,m(x, y))

, (i = 1, 2, 3, j, k, m ∈ Z). (3.15.32)

References

1. Ates, H.F., Orchard, M.T.: A nonlinear image representation in wavelet domain using complexsignals with single quadrant spectrum. Proc. Asilomar Conf. Signals Syst. Comput. 2, 1966–1970 (2003)

2. Antoine, J.P., Carrette, P., Murenzi, R., Piette, B.: Image analysis with two-dimensional con-tinuous wavelet transform. Signal Process. 31, 241–272 (1993)

3. Bachmann, G., Narici, L., Beckenstein, E.: Fourier and Wavelet Analysis. Springer, New York(1999)

4. Belzer, B., Lina, J.M., Villasenor, J.: Complex, linear-phase filters for efficient image coding.IEEE Trans. Signal Process. 43(10), 2425–2427 (1995)

5. Corrochano, E.B.: Multiresolution image analysis using the quaternion wavelet transform.Numer. Algorithms 39, 35–55 (2005)

6. Corrochano, E.B.: The theory and the use of the quaternionwavelet transform. J.Math. ImagingVis. 24(1), 19–35 (2006)

7. Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEETrans. Inform. Theory 36(5), 961–1005 (1990)

8. Daubechies, I.: Ten Lectures on Wavelets. Capital City Press (1992)9. Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl.

Math. 41(7), 909–996 (1988)10. Fernandes, F., Wakin, M., Baraniuk, R. G.: Non-redundant, linear-phase, semi-orthogonal,

directional complex wavelets. In: Proceedings of the IEEE International Conference on Acous-tics Speech Signal Processing (ICASSP), Montreal, vol. 2, pp. 953–956 (2004)

11. Frazier,M.W.:An Introduction toWavelets throughLinearAlgebra. Springer, NewYork (1999)12. Gripenberg, G.: A necessary and sufficient condition for the existence of a father wavelet.

Studia Math. 114(3), 207–226 (1995)13. Hernández, E., Weiss G.: A First Course on Wavelets. CRC Press, Boca Raton (1996)

References 91

14. Kingsbury, N.G., Magarey, J.F.A.: Wavelet transforms in image processing. In: Proceedings1st European Conference Signal Analysis Prediction, pp. 23–34 (1997)

15. Kingsbury, N.G.: Complex wavelets for shift invariant analysis and filtering of signals. Appl.Comput. Harmon. Anal. 10(3), 234–253 (2001)

16. Kingsbury, N.G.: Image processing with complex wavelets. Philos. Trans. R. Soc. Lond. A.Math. Phys. Sci. 357(1760), 2543–2560 (1999)

17. Kingsbury,N.G.: The dual-tree complexwavelet transform: a new technique for shift invarianceand directional filters. In: Proceedings of the 8th IEEE DSP Workshop, Utah (1998)

18. Kingsbury, N.G., Magarey, J.F.A.: Wavelet transforms in image processing. In: Proceedings ofthe First European Conference Signal Analysis Prediction, pp. 23–34 (1997)

19. Lang, M., Guo, H., Odegard, J.E., Burrus, C.S., Wells Jr., R.O.: Noise reduction using anundecimated discrete wavelet transform. IEEE Signal Process. Lett. 3(1), 10–12 (1996)

20. Lina, J.M., Mayrand, M.: Complex daubechies wavelets. Appl. Comput. Harmon. Anal. 2(3),219–229 (1995)

21. Mallat, S.G.: Multiresolution approximation and wavelet orthonormal bases of L2(R). Trans.Am. Math. Sot. 315(1), 69–87 (1989)

22. Mallat, S.G.:ATheory forMultiresolutionSignalDecomposition:TheWaveletRepresentation,vol. 11, pp. 674–693 (1989)

23. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego, CA (1998)24. Meyer, Y.: Wavelets and Operators, vol. 37. Cambridge University Press, Cambridge (1992)25. Meyer, Y.: Wavelets Algorithms and Applications. SIAM, Philadelphia (1993)26. Papadakis, M.: On the dimension function of orthonormal wavelets. Proc. Am. Math. Soc.

128(7), 2043–2049 (2000)27. Romber, G.J., Wakin, M., Choi, H., Baraniuk, R.G.: A geometric hidden markov tree wavelet

model. In: Proceedings of the Wavelet Applications Signal Image Processing X (SPIE 5207),San Diego, pp. 80–86 (2003)

28. Romberg, J.K., Wakin, M., Choi, H., Kingsbury, N.G., Baraniuk, R.G.: A HiddenMarkov TreeModel for the Complex Wavelet Transform. Technical Report, Rice ECE (2002)

29. Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)30. Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1974)31. Selesnick, I.W., Baraniuk, R.G., Kingsbury, N.G.: The dual-tree complex wavelet transform.

IEEE Signal Process. Mag. 22(6), 123–151 (2005)32. Spaendonck, R., Van, B.T., Baraniuk, R., Vetterli M.: Orthogonal Hilbert transform filter banks

and wavelets. In: Proceedings of the IEEE International Conference Acoustic, Speech, SignalProcessing (ICASSP), vol. 6, pp. 505–508 (2003)

33. The Wutam Consortium: Basic Properties of Wavelets. J. Fourier Anal. Appl. 4, 575–594(1998)

34. Tian, L.S., Zhang, X.P., Peng, Y.N.: From the wavelet series to the discrete wavelet transform.IEEE Trans. Signal Process. 44(1), 129–133 (1996)

35. Vetterli, M., Kovacevic, J.: Wavelets and Subband Coding. Prentice-Hall, Englewood Cliffs,NJ (1995)

36. Wakin, M.: Image Compression using Multiscale Geometric Edge Models. Master thesis, RiceUniversity, Houston, TX (2002)

37. Wakin, M., Orchard, M., Baraniuk, R.G., Chandrasekaran, V.: Phase and magnitude perceptualsensitivities in nonredundant complexwavelet representations. In: Proceedings of the AsilomarConference on Signals, Systems, Computers, vol. 2, pp. 1413–1417 (2003)

38. Walnut, D.F.: An Introduction to Wavelet Analysis. Birkhäuser, Boston (2002)39. Wang, Z., Simoncelli, E.P.: Local Phase Coherence and the Perception of Blur. In: Thrun, S.,

Saul, L., Schlkopf, B. (eds.) Adv. Neural Information Processing Systems, vol. 16. MIT Press,Cambridge, MA (2004)

40. Wang, X.: The Study ofWavelets from the Properties of their Fourier Transforms, Ph.D. Thesis,Washington University, (1995)

41. Wojtaszczyk, P.: AMathematical Introduction toWavelets. Cambridge University Press, Cam-bridge (1997)


42. Yin, M., Liu, W., Shui, J., Wu, J.: Quaternion Wavelet Analysis and Application in ImageDenoising.Mathematical problems inEngineering, vol. 2012.Hindwai publishingCorporation,Cairo (2012)

43. Young, R.K.: Wavelet Theory and Its Applications. Kluwer Academic Publishers, Dordrecht(1993)

Chapter 4New Multiscale Constructions

4.1 Overview

Although the wavelet transform has been proven to be powerful in many signaland image processing applications such as compression, noise removal, image edgeenhancement, and feature extraction; wavelets are not optimal in capturing the two-dimensional singularities found in images. Therefore, several transforms have beenproposed for image signals that have incorporated directionality and multiresolutionand hence, could more efficiently capture edges in natural images. Multiscale meth-ods have become very popular, especially with the development of the wavelets inthe last decade. Despite the success of the classical wavelet viewpoint, it was arguedby Candes and Donoho [13, 14] that the traditional wavelets present some stronglimitations that question their effectiveness in higher-dimension than 1.Wavelets relyon a dictionary of roughly isotropic elements occurring at all scales and locations,do not describe well highly anisotropic elements, and contain only a fixed numberof directional elements, independent of scale. Despite the fact that wavelets havehad a wide impact in image processing, they fail to efficiently represent objects withhighly anisotropic elements such as lines or curvilinear structures (e.g. edges). Thereason is that wavelets are non-geometrical and do not exploit the regularity of theedge curve.

Following this reasoning, new constructions have been proposed such as theridgelets by Candes and Donoho [13], Candes [15] and the curvelets by Candesand Donoho [11, 14], Donoho and Duncan [26], Strack et al. [59]. The Ridgeletand the Curvelet transforms were developed by Candes and Donoho [13, 14] as ananswer to the weakness of the separable wavelet transform in sparsely representingwhat appears to be simple building atoms in an image, that is lines, curves and edges.Curvelets and ridgelets take the form of basis elements which exhibit high directionalsensitivity and are highly anisotropic [14, 15, 26, 59]. These, very recent geomet-ric image representations, are built upon ideas of multiscale analysis and geometry.They have had an important success in a wide range of image processing applications


93

94 4 New Multiscale Constructions

including denoising [41, 55, 59], deconvolution [33, 60], contrast enhancement [60],texture analysis [1, 62], detection [43], watermarking [68], component separation[61], inpainting [32, 34] or blind source separation [5, 6]. Curvelets have also provenuseful in diverse fields beyond the traditional image processing application: for exam-ple seismic imaging [30, 41, 42], astronomical imaging [49, 57, 60], scientific com-puting and analysis of partial differential equations [9, 10]. Another reason for thesuccess of ridgelets and curvelets is the availability of fast transform algorithmswhich are available in non-commercial software packages following the philosophyof reproducible research (BeamLab [4], The curvelet toolbox [66]).

Ridgelets and curvelets are specialmembers of the family ofmultiscale orientation-selective transforms, which has recently led to a flurry of research activity in the fieldof computational and applied harmonic analysis.Many other constructions belongingto this family have been investigated in the literature, and go by the name contourlets[20], directionlets [67], bandlets [53] and shearlets [48] etc.

In this chapter, we shall mainly focus on ridgelet transform, curvelet transform,contourlet transform and shearlet transform.

4.2 Ridgelet Transform

4.2.1 The Continuous Ridgelet Transform

The two-dimensional continuous ridgelet transform inR2 is defined by Candes [17].Let ψ : R → R be a smooth univariate function with sufficient decay and satisfyingthe admissibility condition ∫ |ψ(ξ)|2

|ξ|2 dξ ≤ ∞, (4.2.1)

which holds if ψ has a vanishing mean∫

ψ(x)dx = 0. We will assume a specialnormalization for ψ so that

∫ ∞0 |ψ(ξ)|2|ξ|−2dξ = 1.

For each scale a > 0, each position b ∈ R and each orientation θ ∈ [0, 2π), wedefine the bivariate ridgelet ψa,b,θ : R2 → R by

ψa,b,θ(x) = ψa,b,θ(x1, x2) = a−1/2 · ψ((x1cosθ + x2sinθ − b)/a) (4.2.2)

A ridgelet is constant along lines x1cosθ + x2sinθ = constant. Transverse to theseridges it is a wavelet.

Figure4.1 depicts few examples of ridgelets. The second to fourth panels areobtained after simple geometric manipulations of the ridgelet (left panel), namelyrotation, rescaling, and shifting.

4.2 Ridgelet Transform 95

Fig. 4.1 Few Examples of Ridgelets

Given an integrable bivariate function f (x), we define its ridgelet coefficients by

� f (a, b, θ) := 〈 f,ψa,b,θ〉 =∫R2

f (x)ψa,b,θ(x)dx . (4.2.3)

We have the exact reconstruction formula

f (x) =∫ 2π

0

∫ ∞

−∞

∫ ∞

0� f (a, b, θ)ψa,b,θ(x)

da

a3db

dθ

4π, (4.2.4)

valid almost everywhere for functions which are both integrable and square inte-grable. This formula is stable and so we have a Parseval’s relation:

∫| f (x)|2dx =

∫ 2π

0

∫ ∞

−∞

∫ ∞

0|� f (a, b, θ)|2 da

a3db

dθ

4π. (4.2.5)

This approach generalizes to any dimension. Given a ψ obeying∫ |ψ(ξ)|2ξ−n

dξ = 1, define ψa,b,θ(x) = ψ((θ′x −b)/a)/√a and � f (a, b, θ) = 〈 f,ψa,b,θ〉. Then

there is an n-dimensional reconstruction formula

f = cn

∫ ∫ ∫� f (a, b, θ)ψa,b,θ(x)

da

an+1dbdθ, (4.2.6)

with dθ the uniform measure on the sphere and a Parseval’s relation

|| f ||2L2(Rn) =∫ ∫ ∫

|� f (a, b, θ)|2 da

an+1dbdθ. (4.2.7)

The CRT appears similar to the 2D CWT except that the point parameters (k1, k2)are replaced by the line parameters (b, θ). In brief, those 2D transform are relatedby:

Wavelets:−→ ψscale, point posi tion

Ridgelets:−→ ψscale, line posi tion


The consequence of this is: as wavelet analysis is very effective at representingobjects with isolated point singularities and ridgelet analysis can be very effectiveat representing objects with singularities along lines. In fact, one can loosely viewridgelets as a way of concatenating 1D wavelets along lines. Hence the motivationfor using ridgelets in image processing tasks is very appealing as singularities areoften joined together along edges or contours in images.

In the Radon domain, ridgelet analysis may be constructed as wavelet analysis.The rationale behind this is that the Radon transform translates singularities alonglines into point singularities, for which the wavelet transform is known to provide asparse representation. The Radon transform of an object f is the collection of lineintegrals indexed by (θ, t) ∈ [0, 2π) × R given by

R f (θ, t) =∫R2

f (x1, x2)δ(x1cosθ + x2sinθ − t)dx1dx2, (4.2.8)

where δ is the Dirac distribution. Then the ridgelet transform is precisely the appli-cation of a 1-dimensional wavelet transform to the slices of the Radon transformwhere the angular variable θ is constant and t is varying. Thus, the basic strategy forcalculating the continuous ridgelet transform is first to compute the Radon transformR f (θ, t) and second, to apply a one-dimensional wavelet transform to the slicesR f (, θ).

Several digital ridgelet transforms (DRTs) have been proposed, and we willdescribe some of them in this section, based on different implementations of theRadon transform.

1. The RectoPolar Ridgelet Transform

A fast implementation of the Radon transform based on the projection-slice-theoremcan be proposed in the Fourier domain. First, the 2D FFT of the given image iscomputed and then the resulting function in the frequency domain is to be used toevaluate the frequency values in a polar grid of rays passing through the origin andspread uniformly in angle. This conversion, from Cartesian to Polar grid, could beobtained by interpolation, and this process is known by theGridding in tomography.Given the polar grid samples with the number of rays corresponds to the number ofprojections and the number of samples on each ray corresponds to the number of shiftsper such angle. The Radon projections are obtained by applying one-dimensionalinverse Fourier transform for each ray.

Due to the sensitivity to the interpolation involved, the above described processis known to be inaccurate. This implies that for a better accuracy, the first 2D FFTemployed should be done with high-redundancy. An alternative solution for theFourier-basedRadon transform exists, where the polar grid is replacedwith a pseudo-polar one.

Concentric circles of linearly growing radius in the polar grid are replaced byconcentric squares of linearly growing sides. The rays are spread uniformly not inangle but in slope. These two changes give a grid vaguely resembling the polarone, but for this grid a direct FFT can be implemented with no interpolation. When


applying 1D FFT for the rays, we get a variant of the Radon transform, where theprojection angles are not spaced uniformly. For the pseudo-polar FFT to be stable,it was shown that it should contain at least twice as many samples, compared tothe original image we started with. A by-product of this construction is the fact thatthe transform is organized as a 2D array with rows containing the projections as afunction of the angle. Thus, processing the Radon transform in one axis is easilyimplemented (See Strack et al. [59] for more details).

2. One-Dimensional Wavelet Transform

To complete the ridgelet transform, we must take a one-dimensional wavelet trans-formalong the radial variable inRadon space.Wenowdiscuss the choice of the digital1D-WT. As we know that compactly supported wavelets can lead to many visual arti-facts when used in conjunction with nonlinear processing, such as hard-thresholdingof individual wavelet coefficients, particularly for decimated wavelet schemes usedat critical sampling. Also, because of the lack of localization of such compactly sup-ported wavelets in the frequency domain, fluctuations in coarse-scale wavelet coef-ficients can introduce fine-scale fluctuations. A frequency-domain approach must betaken, where the discrete Fourier transform is reconstructed from the inverse Radontransform. These considerations lead to use band-limited wavelet, whose support iscompact in the Fourier domain rather than the time-domain. A specific over-completewavelet transform has been used by Strack et al. [58] and Strack et al. [63]. Thewavelet transform algorithm is based on a scaling function ϕ such that ϕ vanishesoutside of the interval [ξc, ξc].We define the Fourier transform of the scaling functionas a re-normalized B3-spline

ϕ(ξ) = 3

2B3(4ξ), (4.2.9)

and ψ as the difference between two consecutive resolutions

ψ(2ξ) = ϕ(ξ) − ϕ(2ξ) (4.2.10)

Because ψ is compactly supported, the sampling theorem shows than one can easilybuild a pyramid of n + n/2+ · · · + 1 = 2n elements (see Strack et al. [63] for moredetails). This 1D-WT transform possesses the following useful properties:

(i) The wavelet coefficients are directly calculated in the Fourier space. Thisallows avoiding the computation of the one-dimensional inverse Fourier transformalong each radial line in the context of the ridgelet transform.

(ii) Each sub-band is sampled above the Nyquist rate, hence, avoiding aliasingphenomenon typically encountered by critically sampled orthogonal wavelet trans-forms.

(iii) The reconstruction is trivial. The wavelet coefficients simply need to be co-added to reconstruct the input signal at any given point.

This implementation would be useful to the practitioner whose focuses ondata analysis, for which it is well-known that over-completeness through almost


translation-invariance can provide substantial advantages. The discrete ridgelet trans-form (DRT) of an image of size n × n is an image of size 2n × 2n, introducing aredundancy factor equal to 4.

We note that, because this transform is made of a chain of steps, each one of whichis invertible, the whole transform is invertible, and so has the exact reconstructionproperty. For the same reason, the reconstruction is stable under perturbations of thecoefficients. This discrete transform is also computationally attractive.

4.2.2 Discrete Ridgelet Transform

For applications point of view, it is important that one obtains a discrete representationusing ridgelets. Typical discrete representations include expansions in orthonormalbasis. Here we describe an expansion only in two dimensions by frames. For higherdimensions, see Candes [8].

The formula for the continuous ridgelet transform (CRT) using Fourier domainis defined by

� f (a, b, θ) := 1

2π

∫ψa,b,θ(ξ) f (ξ)dξ, (4.2.11)

where ψa,b,θ(ξ) is interpreted as a distribution supported on the radial line in thefrequency plane. Letting ξ(λ, θ) = (λ · cos(θ),λ · sin(θ)) we can write

� f (a, b, θ) := 1

2π

∫ ∞

−∞a1/2ψ(aλ)e−iλb f (ξ(λ, θ))dλ. (4.2.12)

This says that the CRT is obtainable by integrating the weighted Fourier transformωa,b f (ξ) along a radial line in the frequency domain, with weight ωa,b given by

a1/2ψ(a|ξ|) times a complex exponential in e−iλb. Or, we can see that the functionof b, with a and θ considered fixed, ρa,θ(b) = � f (a, b, θ), satisfies

ρa,θ(b) = {ρa,θ(λ)},

andρa,θ(λ) = a1/2ψ(aλ) f (ξ(λ, θ)) − ∞ < λ < ∞

is the restriction of ωa,0 f (ξ) to the radial line.Hence the CRT at a certain scale a and angle θ can be obtained by first taking 2D

Fourier transform for obtaining f (ξ), take radial windowing for obtaining ωa,0 f (ξ)and finally take 1D inverse Fourier transform along radial lines for obtaining ρa,θ(b),for all b ∈ R.

Now we are providing a method for sampling (a j , b j,k, θ j,l) so that we obtainframe bounds, i.e. so we have equivalence


∑j,k,l

|� f (a j , bk, θ j,l)|2 �∫ ∫ ∫

|� f (a, b, θ)|2da/a3dbdθ. (4.2.13)

To simplify exposition, assume ψ(λ) = 1{1≤|ξ|≤2}, a j = a02 j and b j,k = 2πk2− j .

The ridgelet coefficients may be written as by using Eq. (4.2.12)

� f (a j , b j,k, θ) = 1

2π2− j/2

∫2 j≤|λ|≤2 j+1

e−iλ2π2− jf (ξ(λ, θ))dλ, (4.2.14)

and hence, the Plancherel theorem gives

∑k

� f (a j , b j,k, θ) = 1√2π

∫2 j≤|λ|≤2 j+1

|ω2 j ,0|2| f (ξ(λ, θ))|2dλ. (4.2.15)

Discretizing the angular variable θ amounts to performing a sampling of suchsegment-integrals from which the integral of | f (ξ)|2 over the whole frequencydomain needs to be inferred. This is not possible without support constraints on f ,as functions f can always be constructed with f (x) having slow decay as |x | → ∞so that f will vanish on a collection of disjoint segments without being identicallyzero. However, under a support restriction, so that f is supported inside the unit disk(or any other compact set), the integrals over the segments can provide sufficientinformation to infer

∫ | f (ξ)|2dξ.

Indeed, under a support constraint, the Fourier transform f (ξ) is a band-limitedfunction, and over ‘cells’ of appropriate size can only display very banal behavior.If we sample once per cell, we will capture sufficiently much of the behavior of thisobject that wewill be in a position to infer the size of the function from those samples.The solution found by Candes is to sample with increasing angular resolution atincreasingly fine scales, something like the following:

θ j,l = 2πl2− j .

This strategy gives the equivalence in Eq. (4.2.13). It then follows that the collec-tion

{2 j/2ψ(2 j (x1cos(θ j,l) + x2sin(θ j,l) − 2πk2− j ))}( j≥ j0,l,k)

is a frame for the unit disk. For any f supported in the disk with finite L2 norm,

∑j,k,l

|〈ψa j ,b j,k ,θ j,l , f 〉|2 � || f ||2.

The existence of frame bounds implies, by soft analysis, that there are dual ridgeletsψ j,k,l so that

f =∑j,k,l

〈 f, ψ j,k,l〉ψ j,k,l


and

f =∑j,k,l

〈 f,ψ j,k,l〉ψ j,k,l

with equality in an L2-sense, and so that

∑j,k,l

|〈 f, ψ j,k,l〉|2 �∑j,k,l

|〈 f,ψ j,k,l〉|2 � || f ||2L2 .

4.2.3 The Orthonormal Finite Ridgelet Transform

The orthonormal finite ridgelet transform (OFRT) has been proposed by Do andVetterli [24] for image compression and filtering. This transform is based on the finiteRadon transform (seeMatus andFlusser [51]) and a1Dorthogonalwavelet transform.OFRT is not redundant and reversible. It would have been a great alternative to thepreviously described ridgelet transform if the OFRT were not based on an awkwarddefinition of a line. In fact, a line in the OFRT is defined algebraically rather thatgeometrically, and so the points on a line can be arbitrarily and randomly spread outin the spatial domain.

Because of this specific definition of a line, the thresholding of the OFRT coef-ficients produces strong artifacts. Finally, the OFRT presents another limitation: theimage size must be a prime number. This last point is however not too restrictive,because we generally use a spatial partitioning when denoising the data, and a primenumber block size can be used. The OFRT is interesting from the conceptual pointof view, but still requires work before it can be used for real applications such asdenoising.

4.2.4 The Fast Slant Stack Ridgelet Transform

The Fast Slant Stack (FSS) (see Averbuch et al. [2]) is a Radon transform of dataon a Cartesian grid, which is algebraically exact and geometrically more accurateand faithful than the previously described methods. The back-projection of a pointin Radon space is exactly a ridge function in the spatial domain.

The transformation of an n × n image is a 2n × 2n image. n line integrals withangle between

[− π4 , π

4

]are calculated from the zero padded image on the y-axis, and

n line integrals with angle between[

π4 , 3π

4

]are computed by zero padding the image

on the x-axis. For a given angle inside[−π

4 , π4

], 2n line integrals are calculated by

first shearing the zero-padded image, and then integrating the pixel values along allhorizontal lines (resp. vertical lines for angles in

[π4 , 3π

4

]. The shearing is performed

one column at a time (resp. one line at a time) by using the 1D FFT. A DRT based on


the FSS transform has been proposed in (Donoho and Flesia, [27]). The connectionbetween the FSS and the Linogram has been investigated by Averbuch et al. [2]. AFSS algorithm is also proposed by Averbuch et al. [2], based on the 2D Fast Pseudo-polar Fourier transformwhich evaluates the 2D Fourier transform on a non-Cartesiangrid, operating in O(n2log n) flops.

4.2.5 Local Ridgelet Transform

The ridgelet transform is optimal for finding global lines of the size of the image. Todetect line segments, a partitioning must be introduced by Candes [15]. The imagecan be decomposed into overlapping blocks of side-length b pixels in such a waythat the overlap between two vertically adjacent blocks is a rectangular array of sizeb by b/2; we use overlap to avoid blocking artifacts. For an n by n image, we count2n/b such blocks in each direction, and thus the redundancy factor grows by a factorof 4.

The partitioning introduces redundancy, as a pixel belongs to 4 neighboringblocks. We present two competing strategies to perform the analysis and synthe-sis:

1. The block values are weighted by a spatial window w (analysis) in such away that the co-addition of all blocks reproduce exactly the original pixel value(synthesis).

2. The block values are those of the image pixel values (analysis) but are weightedwhen the image is reconstructed (synthesis).

4.2.6 Sparse Representation by Ridgelets

The continuous ridgelet transform provides sparse representation of both smoothfunctions (in the Sobolev space W 2

2 ) and of perfectly straight lines (Candes [16],Donoho [28]). As we know that there are also various DRTs, i.e. expansions withcountable discrete collection of generating elements, which correspond either toframes or orthonormal bases. It has been shown for these schemes that the DRTachieves near optimal M-term approximation, that is the non-linear approximationof f using the M highest ridgelet coefficients in magnitude, to smooth images withdiscontinuities along straight lines [13, 28]. In short, ridgelets provide sparse pre-sentation for piecewise smooth images away from global straight edges.


4.3 Curvelets

The ridgelet transform is optimal at representing straight-line singularities. Thistransformwith arbitrary directional selectivity provides a key to the analysis of higherdimensional singularities. Unfortunately, the ridgelet transform is only applicableto objects with global straight-line singularities, which are rarely observed in realapplications. In the area of image processing, most of the image edges are curvedrather than straight lines. Hence, ridgelets are not able to efficiently represent suchimages. In order to analyze local line or curve singularities, a natural idea is toconsider a partition for the image, and then to apply the ridgelet transform to theobtained sub-images. This block ridgelet based transform, which is named curvelettransform, was first proposed by Candes and Donoho [11]. This is also called first-generation curvelet transform or CurveletG1 (Candes and Donoho [11]).

Despite these interesting properties, however, the application of or CurveletG1 islimited because the geometry of ridgelets is itself unclear, as they are not true ridgefunctions in digital images. First, the construction involves a complicated seven-index structure among which we have parameters for scale, location and orientation.In addition, the parabolic scaling ratio width≈ length2 is not completely true. In fact,CurveletG1 assumes a wide range of aspect ratios. These facts make mathematicaland quantitative analysis especially delicate. Second, the spatial partitioning of theCurveletG1 transform uses overlapping windows to avoid blocking effects. Thisleads to an increase of the redundancy of the discrete curveletG1 (DCTG1). Thecomputational cost of the DCTG1 algorithm may also be a limitation for large-scaledata, especially if the FSS-based DRT implementation is used.

Later, the second-generation curvelet transform or CurveletG2, introduced byCand‘es andDonoho [12], exhibit amuch simpler and natural indexing structurewiththree parameters: scale, orientation or angle and location, hence simplifying math-ematical analysis. The CurveletG2 transform also implements a tight frame expan-sion and has a much lower redundancy. Unlike the DCTG1, the discrete CurveletG2(DCTG2) implementation will not use ridgelets yielding a faster algorithm (Candeset al. [18]).

4.3.1 The First Generation Curvelet Transform

The CurveletG1 transform [11, 26] opens the possibility to analyze an image withdifferent block sizes, but with a single transform. The main idea is to first decomposethe image into a set of wavelet bands, and then analyze each band by using a localridgelet transform. The block size can be changed at each scale level. Differentlevels of the multiscale ridgelet pyramid are used to represent different sub-bandsof a filter bank output. At the same time, this sub-band decomposition imposes arelationship between the width and length of the important frame elements so that

4.3 Curvelets 103

they are anisotropic and obey approximately the parabolic scaling law width ≈length2.

The first-generation discrete curvelet transform (DCTG1) of a continuum functionf (x)makes use of a dyadic sequence of scales, and a bank of filters with the propertythat the bandpass filter � j is concentrated near the frequencies [22 j , 22 j+2], forexample

� j ( f ) = �2 j ∗ f, �2 j (ξ) = �(2−2 jξ). (4.3.1)

In wavelet theory, one uses a decomposition into dyadic sub-bands [2 j , 2 j+1]. In con-trast, the subbands used in the discrete curvelet transform of continuum functionshave the nonstandard form [22 j , 22 j+2]. This is nonstandard feature of the DCTG1well worth remembering (this is where the approximate parabolic scaling law comesinto play).

The DCTG1 decomposition is the sequence of the following steps:

1. Sub-band Decomposition: The object f is decomposed into sub-bands.2. Smooth Partitioning: Each sub-band is smoothly windowed into squares of an

appropriate scale (of side-length ∼ 2 j ).3. Ridgelet Analysis: Each square is analyzed via the DRT.

In this definition, the two dyadic sub-bands [22 j , 22 j+1] and [22 j+1, 22 j+2] aremerged before applying the ridgelet transform.

4.3.2 Sparse Representation by First Generation Curvelets

The CurveletG1 elements can form either a frame or a tight frame for L2(R2) (Can-des and Donoho [14]), depending on the 2D-WT used and the DRT implementation(rectopolar or FSS Radon transform). The frame elements are anisotropic by con-struction and become successively more anisotropic at progressively higher scales.These curvelets also exhibit directional sensitivity and display oscillatory compo-nents across the ridge. A central motivation leading to the curvelet construction wasthe problem of non-adaptively representing piecewise smooth (e.g. C2) images fwhich have discontinuity along a C2 curve. Such a model is called cartoon modelof non-textured images. With the CurveletG1 tight frame construction, it was shownby Candes and Donoho [14] that for such f , the M-term non-linear approximationsfM of f obey, for each κ > 0,

|| f − fM ||2 ≤ CκM−2+κ, M → +∞.

The M-term approximations in the CurveletG1 are almost rate optimal, much betterthan M-term Fourier or wavelet approximations for such images.


4.3.3 The Second-Generation Curvelet Transform

The second-generation curvelet transform has been shown to be a very efficienttool for many different applications in image processing, seismic data exploration,fluid mechanics, and solving PDEs. The second generation curvelets or CurveletG2are defined at scale 2− j , orientation θl and position x j,l

k = R−1θl

(2− j k1, 2− j/2k2) bytranslation and rotation of a mother curvelet ϕ j as

ϕ j,l,k(x) = ϕ j (Rθl (x − x j,lk )), (4.3.2)

where Rθl is the rotation by θl radians. θl is the equispaced sequence of rotationangles θl = 2π2−| j/2|l, with integer l such that 0 ≤ θl ≤ 2π (note that the numberof orientations varies as 1/

√scale. k = (k1, k2) ∈ Z

2 is the sequence of translationparameters. The waveform ϕ j is defined by means of its Fourier transform ϕ j (ξ),written in polar coordinates in the Fourier domain

ϕ j (r, θ) = 2−3 j/4ω(2− j r)v

(2� j/2�θ2π

). (4.3.3)

The support of ϕ j is a polar parabolic wedge defined by the support of ω and v. Theradial and angular windows, both are smooth, nonnegative and real-valued, appliedwith scale-dependent window widths in each direction. ω and v must also satisfy thepartition of unity property (See Candes et al. [18] for more details).

In continuous frequency ξ, the CurveletG2 coefficients of data f (x) are definedas the inner product

c j,l,k := 〈 f,ϕ j,l,k〉 =∫R2

f (ξ)ϕ j (Rθl ξ)eix j,l

k ·ξdξ. (4.3.4)

This construction implies a few properties:

(i) the CurveletG2 defines a tight frame of L2(R2),(ii) the effective length and width of these curvelets obey the parabolic scaling

relation (2 j = width) = (length = 2 j/2)2,(iii) the curvelets exhibit an oscillating behavior in the direction perpendicular to

their orientation.

Curvelets as just constructed are complex-valued. It is very easy to obtain real-valued curvelets by working on the symmetrized version ϕ j (r, θ) + ϕ j (r, θ + π).

The discrete transform takes as input data defined on a Cartesian grid and outputsa collection of coefficients. The continuous-space definition of the CurveletG2 usescoronae and rotations that are not especially adapted to Cartesian arrays. It is thenconvenient to replace these concepts by their Cartesian counterparts. That is, we useconcentric squares instead of concentric circles and shears instead of rotations.

The Cartesian equivalent to the radial window ω j (ξ) = ω j (2− jξ) would be abandpass frequency-localized window which can be derived from the difference of

4.3 Curvelets 105

separable low-pass windows Hj (ξ) = h(2− jξ1)h(2− jξ2), where h is a 1D low-passfilter, we have:

ω j (ξ) =√H 2

j+1(ξ) − H 2j (ξ) ∀ j ≥ 0, and, ω0(ξ) = h(ξ1)h(ξ2) (4.3.5)

Another possible choice is to select these windows inspired by the constructionof Meyer wavelets, for more details see Candes and Donoho [12]. For more detailsabout the construction of the Cartesian ω j ’s see Candes et al. [18].

Each cartesian coronae has four quadrants: east, west, north and south. Eachquadrant is separated into 2� j/2� wedges with the same area. For example take theeast quadrant −π/4 ≤ θl < π/4 and by symmetry around the origin, we wouldproceed for west quadrant and finally for north and south quadrant by exchangingthe roles of ξ1 and ξ2. Define the angular window for the l-th direction as

v j,l(ξ) = v

(2� j/2� ξ2 − ξ1tanθl

ξ1

), (4.3.6)

with the sequence of equispaced slopes (not angles) tanθl = 2−� j/2�l, for l =−2� j/2�, . . . , 2� j/2� − 1. Now, we are defining the window which is the Cartesiananalog of φ j above,

u j,l(ξ) = ω j (ξ)v j,l(ξ) = ω j (ξ)v j,0(Sθl ξ), (4.3.7)

where Sθl is the shear matrix. From the definition, it can be seen that u j,l is supportednear the trapezoidal wedge {ξ = (ξ1, ξ2) : 2 j ≤ ξ1 ≤ 2 j1 ,−2− j/2 ≤ ξ2/ξ1−tanθl ≤2− j/2}. The collection of u j,l(ξ) gives rise to the frequency tiling. From u j,l(ξ), thedigital CurveletG2 construction suggests cartesian curvelets that are translated andsheared versions of a mother cartesian curvelet

φDj (ξ) = u j,0(ξ), φD

j,k,l(x) = 23 j/4φDj

(STθl x − m

)

where m = (k12− j , k22− j/2).

4.3.4 Sparse Representation by Second Generation Curvelets

It has been shown by Candes and Donoho [12] that with the CurveletG2 tight frameconstruction, the M-term non-linear approximation error of C2 images except atdiscontinuities along C2 curves obey

|| f − fM ||2 ≤ CM2(logM)3.


This is an asymptotically optimal convergence rate (up to the (logM)3 factor), andholds uniformly over the C2-C2 class of functions. This is a remarkable result sincethe CurveletG2 representation is non-adaptive. However, the simplicity due to thenon-adaptivity of curvelets has a cost: curvelet approximations loose their near opti-mal properties when the image is composed of edges which are not exactly C2.Additionally, if the edges are Cα-regular with α > 2, then the curvelets convergencerate exponent remain 2. Other adaptive geometric representations such as bandletsare specifically designed to reach the optimal decay rate O(Mα) (Peyre and Mallat[54]).

4.4 Contourlet

In the field of geometrical image transforms, there are many 1D transforms designedfor detecting or capturing the geometry of image information, such as the Fouriertransform and wavelet transform. However, the ability of 1D transform processing ofthe intrinsic geometrical structures, such as smoothness of curves, is limited in onedirection, thenmore powerful representations are required in higher dimensions. Thecontourlet transform which was proposed by Do and Vetterli [20, 21], is a new two-dimensional transform method for image representations. The contourlet transformhas properties of multiresolution, localization, directionality, critical sampling andanisotropy. Its basic functions are multiscale and multidimensional. The contours oforiginal images, which are the dominant features in natural images, can be capturedeffectively with a few coefficients by using contourlet transform.

The contourlet transform is inspired by the human visual system and curvelettransform which can capture the smoothness of the contour of images with differentelongated shapes and in variety of directions. However, it is difficult to sampling ona rectangular grid for curvelet transform since curvelet transform was developed incontinuous domain and directions other than horizontal and vertical are very differenton rectangular grid. Therefore, the contourlet transform was proposed initially as adirectional multiresolution transform in the discrete domain and then studies itsconvergence to an expansion in the continuous domain.

The contourlet transform is one of the new geometrical image transforms, whichcan efficiently represent images containing contours and textures (Mallat [50], Sko-dras et al. [56]). This transform uses a structure similar to that of curvelets (Donohoand Vetterli [29], that is, a stage of subband decomposition followed by a directionaltransform. In the contourlet transform, a Laplacian pyramid is employed in the firststage, while directional filter banks (DFB) are used in the angular decompositionstage. Due to the redundancy of the Laplacian pyramid, the contourlet transform hasa redundancy factor of 4/3 and hence, it may not be the optimum choice for imagecoding applications. The discrete contourlet transform has a fast iterated filter bankalgorithm that requires an order N operations for N -pixel images.

4.5 Contourlet Transform 107

4.5 Contourlet Transform

Wavelets in 2D are good at isolating the discontinuities at edge points, but will notsee the smoothness along the contours. In addition, separable wavelets can captureonly limited directional information i.e. an important and unique feature of mul-tidimensional signals. These disappointing behaviors indicate that more powerfulrepresentations are needed in higher dimensions. Consider the following scenario tosee how one can improve the 2D separable wavelet transform for representing imageswith smooth contours. Imagine that there are two painters, one with a wavelet styleand the other with a new style, both wishing to paint a natural scene. Both paintersapply a refinement technique to increase resolution from coarse to fine. Here, effi-ciency is measured by how quickly, that is with how few brush strokes, one canfaithfully reproduce the scene. Consider the situation when a smooth contour isbeing painted. Because 2D wavelets are constructed from tensor products of 1Dwavelets, the wavelet style painter is limited to using square-shaped brush strokesalong the contour, using different sizes corresponding to themultiresolution structureof wavelets. As the resolution becomes finer, we can clearly see the limitation of thewavelet style painter who needs to usemany fine dots to capture the contour. The newstyle painter, on the other hand, exploits effectively the smoothness of the contour bymaking brush strokes with different elongated shapes and in a variety of directionsfollowing the contour. This intuition was formalized by Candes and Donoho in thecurvelet construction.

Curvelet transform can capture the smoothness of the contour of images withdifferent elongated shapes and in variety of directions. However, it is difficult tosampling on a rectangular grid for curvelet transform since it was developed in con-tinuous domain and directions other than horizontal and vertical are very differenton rectangular grid. Therefore, Do and Vetterli in 2001 [23] proposed a double filterbank structure for obtaining sparse expansions for typical images having smoothcontours. In this double filter bank, the Laplacian pyramid constructed by Burt andAdelson in 1983 [7], is first used to capture the point discontinuities, and then fol-lowed by a directional filter bank provided by Bamberger and Smith in 1992 [3], tolink point discontinuities into linear structures. The overall result is an image expan-sion using basic elements like contour segments and thus are named contourlets.In particular, contourlets have elongated supports at various scales, directions, andaspect ratios. This allows contourlets to efficiently approximate a smooth contour atmultiple resolutions in much the same way as the new scheme shown in Fig. 4.2.

The contourlet transform was proposed initially as a directional multiresolutiontransform in the discrete domain and then studies its convergence to an expansion inthe continuous domain. In the frequency domain, the contourlet transform providesa multiscale and directional decomposition. In this section, we will provide only thediscrete domain construction using filter banks. For detail study of its convergenceto an expansion in the continuous domain, see Do and Vetterli [21].


Fig. 4.2 Wavelet versus new scheme: illustrating the successive refinement by the two systemsnear a smooth contour, which is shown as a thick curve separating two smooth regions

4.5.1 Multiscale Decomposition

For obtaining a multiscale decomposition, we use a Laplacian pyramid (LP) as intro-duced by Burt and Adelson in1983 [7]. The LP decomposition at each step generatesa sampled lowpass version of the original and the difference between the originaland the prediction, resulting in a bandpass image. The process can be iterated on thecoarse version.

A drawback of the LP is the implicit oversampling. However, in contrast to thecritically sampled wavelet scheme, the LP has the distinguishing feature that eachpyramid level generates only one bandpass image [even for multidimensional cases]which does not have scrambled frequencies. This frequency scrambling happens inthe wavelet filter bank when a highpass channel, after downsampling, is folded backinto the low frequency band, and thus its spectrum is reflected. In the LP, this effectis avoided by downsampling the lowpass channel only.

Do and Vellerli in 2003 [20] had studied the LP using the theory of frames andoversampled filter banks and they showed that the LP with orthogonal filters (that is,h[n] = g[−n] and g[n] is orthogonal to its translates with respect to the subsamplinglattice byM) is a tight framewith frame bounds equal to 1. In this case, they suggestedthe use of the optimal linear reconstruction using the dual frame operator, which issymmetricalwith the forward transform.Note that this new reconstruction is differentfrom the usual reconstruction and is crucial for our contourlet expansion.

Figure4.3a shows the decomposition process and Fig. 4.3b shows the new recon-struction scheme for the Laplacian pyramid, where H and G are the analysis andsynthesis filters, respectively and M is the sampling matrix.


Fig. 4.3 Laplacian Pyramid a decomposition scheme b reconstruction scheme

Fig. 4.4 The multichannel view of an l-level tree structured directional filter bank

4.5.2 Directional Decomposition

Bamberger and Smith in 1992 [3] introduced a 2D directional filter bank (DFB) thatcan be maximally decimated while achieving perfect reconstruction. The DFB isefficiently implemented using a l-level tree-structured decomposition that leads to2l subbands with wedge-shaped frequency partition.

The original construction of the DFB involves modulating the input signal andusing quincunx filter banks with diamond-shaped filters. An involved tree expandingrule has to be followed to obtain the desired frequency partition. As a result, thefrequency regions for the resulting subbands do not follow a simple ordering asshown in Fig. 4.4 based on the channel indices, see Park et al. [52] for more details.

By using the multirate identities, we can transform a l-level tree-structured DFBinto a parallel structure of 2l channels with directional filters and overall samplingmatrices. Denote these directional synthesis filters as D(l)

k , 0 ≤ k < 2l , which corre-spond to the subbands indexed. The oversampling matrices have diagonal form as:

S(l)k =

{diag(2l−1, 2), for 0 ≤ k < 2l−1,

diag(2, 2l−1), for 2l−1 ≤ k < 2l ,(4.5.1)

which correspond to the basically horizontal and basically vertical subbands, respec-tively.

With this, it is easy to see that the family

{g(l)k [n − S(l)

k m]}0≤k<2l ,m∈Z2 (4.5.2)


obtained by translating the impulse responses of the directional synthesis filtersD(l)

k over the sampling lattices S(l)k , is a basis for discrete signals in l2(Z2). This

basis exhibits both directional and localization properties. These basis functionshave linear supports in space and span all directions. Therefore Eq. (4.5.2) resemblesa local Radon transform and the basis functions are referred to as Radonlets.

Furthermore, Do [25], in his Ph.D. thesis, had shown that if the building blockfilter uses orthogonal filters, then the resulting DFB is orthogonal and Eq. (4.5.2)becomes an orthogonal basis.

4.5.3 The Discrete Contourlet Transform

The directional filter bank (DFB) is designed to capture the high frequency compo-nents of images and is representing directionality. Therefore, low frequency com-ponents are handled poorly by the DFB. In fact, low frequencies would leak intoseveral directional subbands, hence DFB does not provide a sparse representationfor images. To improve the situation, low frequencies should be removed beforethe DFB. This provides another reason to combine the DFB with a multiresolutionscheme.

Therefore, the LP permits further subband decomposition to be applied on itsbandpass images. Those bandpass images can be fed into a DFB so that directionalinformation can be captured efficiently. The scheme can be iterated repeatedly on thecoarse image (see Fig. 4.5). The end result is a double iterated filter bank structure,named pyramidal directional filter bank (PDFB) or contourlet filter bank, whichdecomposes images into directional subbands at multiple scales. The scheme isflexible since it allows for a different number of directions at each scale.

Fig. 4.5 Pyramidal directional filter bank


Precisely, let a0[n] be the input image. After the LP stage the output is a lowpassimageaJ [n] and J bandpass imagesb j [n], j = 1, 2, ···, J in thefine-to-coarse order.In detail, the j-th level of the LP decomposes the image a j−1[n] into a coarser imagea j [n] and a detail image b j [n]. Each bandpass image b j [n] is further decomposed by

an l j -level DFB into 2l j bandpass directional images c(l j )j,k [n], k = 0, 1, · · ·, 2l j − 1.

The main properties of the discrete contourlet transform are stated as follows:

1. If, the LP and the DFB, both use perfect-reconstruction filters, then the discretecontourlet transform succeeds perfect reconstruction, i.e. it provides a frame operator.

With orthogonal filters, the LP is a tight frame with frame bounds equal to1 (Do and Vetterli [24]), which means it preserves the l2-norm, or ||a0||22 =∑J

j=1 ||b j ||22 + ||aJ ||22. Similarly, with orthogonal filters the DFB is an orthogo-

nal transform (Do [23]), which means ||b j ||22 = ∑2l j −1k=0 ||c(l j )

j,k ||22. Combining thesetwo stages, the discrete contourlet transform satisfies the norm preserving or tightframe condition. We have the following:

2. If, the LP and the DFB, both use orthogonal filters, then the discrete contourlettransform forms a tight frame with frame bounds equal to 1.

Since the DFB is critically sampled, the redundancy of the discrete contourlettransform is equal to the redundancy of the LP, which is 1 + ∑J

j=1(1/4)j < 4/3.

Hence, we have the following result:

3. The discrete contourlet transform has a redundancy ratio that is less than 4/3.

By using multirate identities, the LP bandpass channel corresponding to thepyramidal level j is approximately equivalent to filtering by a filter of size aboutC12 j × C12 j , followed by downsampling by 2 j−1 in each dimension. For the DFB,from (3) we see that after l j levels (l j ≥ 2) of tree-structured decomposition, theequivalent directional filters have support of width about C22 and length aboutC22l j−1. Combining these two stages, again usingmultirate identities, into equivalentcontourlet filter bank channels, we see that contourlet basis images have support ofwidth about C2 j and length about C2 j+l j−2. Hence, we have the following result:

4. Suppose an l j -level DFB is applied at the pyramidal level j of the LP, then the basisimages of the discrete contourlet transform (i.e. the equivalent filters of the contourletfilter bank) have an essential support size of width ≈ C2 j and length ≈ C2 j+l j−2.

Let L p and Ld be the number of taps of the pyramidal and directional filters usedin the LP and the DFB, respectively (without loss of generality we can suppose thatlowpass, highpass, analysis and synthesis filters have same length). With a poly-phase implementation, the LP filter bank requires L p/2 + 1 operations per inputsample. Thus, for an N -pixel image, the complexity of the LP stage in the contourletfilter bank is:

J∑j=1

N

(1

4

) j−1 (L p

2+ 1

)<

4

3N

(L p

2+ 1

)(operations). (4.5.3)


For the DFB, its building block two-channel filter banks requires Ld operations perinput sample. With an l-level full binary tree decomposition, the complexity of theDFB multiplies by l. This holds because the initial decomposition block in the DFBis followed by two blocks at half rate, four blocks at quarter rate and so on. Thus,the complexity of the DFB stage for an N -pixel image is:

J∑j=1

N

(1

4

) j−1

Ld l j <4

3NLd max{l j }(operations). (4.5.4)

Combining (4.5.3) and (4.5.4) we obtain the following result:

5. Using FIR filters, the computational complexity of the discrete contourlet trans-form is O(N ) for N -pixel images.

Since the multiscale and directional decomposition stages are decoupled in thediscrete contourlet transform, we can have a different number of directions at differ-ent scales, thus offering a flexible multiscale and directional expansion. Moreover,the full binary tree decomposition of the DFB in the contourlet transform can begeneralized to arbitrary tree structures, similar to the wavelet packets generalizationof the wavelet transform (Coifman et al. [19]). The result is a family of directionalmultiresolution expansions, which we call contourlet packets.

Furthermore, similar to the wavelet filter bank, the contourlet filter bank has anassociated continuous domain expansion in L2(R2) using contourlet functions andthe new elements in this framework are multidirection and its combination withmultiscale.

4.6 Shearlet

One of the most useful features of wavelets is their ability to efficiently approxi-mate signals containing pointwise singularities. However, wavelets fail to capturethe geometric regularity along the singularities of surfaces, because of their isotropicsupport. To exploit the anisotropic regularity of a surface along edges, the basismust include elongated functions that are nearly parallel to the edges. Several imagerepresentations have been proposed to capture the geometric regularity of a givenimage. They include ridgelet, curvelets, contourlets and bandelets etc. In particular,the construction of curvelets is not built directly in the discrete domain and they donot provide a multiresolution representation of the geometry. In consequence, theimplementation and the mathematical analysis are more involved and less efficient.Contourlets are bases constructed with elongated basis functions using the combina-tion of a multiscale and a directional filter bank. However, contourlets have less cleardirectional features than curvelets, which leads to artifacts in denoising and compres-sion. Bandelets are bases adapted to the function that is represented. Asymptotically,the resulting bandelets are regular functions with compact support, which is not thecase of contourlets. However, in order to find basis optimally adapted to an image

4.6 Shearlet 113

of size N , the bandelet transform searches for the optimal geometry. For an imageof N pixels, the complexity of this best bandelet basis algorithm is O(N 3/2) whichrequires extensive computation.

Recently, a new representation scheme has been introduced by Labate et al. in2005 [46] is called the shearlet representation which yields nearly optimal approxi-mation properties (Guo and Labate [36]). Shearlets are frame elements used in thisrepresentation scheme. This new representation is based on a simple and rigorousmathematical framework which not only provides a more flexible theoretical tool forthe geometric representation of multidimensional data, but is also more natural forimplementation. As a result, the shearlet approach can be associated to amultiresolu-tion analysis (MRA) and this leads to a unified treatment of both the continuous anddiscrete world (Labate et. al. [46]). However, all known constructions of shearlets sofar are band-limited functions which have an unbounded support in space domain. Infact, in order to capture the local features of a given image efficiently, representationelements need to be compactly supported in the space domain. Furthermore, thisproperty often leads to more convenient framework for practically relevant discreteimplementation.

Before defining the system of shearlets in a formal way, let us introduce intuitivelythe ideas which are at the core of its construction. in order to achieve optimally sparseapproximations of signals exhibiting anisotropic singularities such as cartoon-likeimages, the analyzing elements must consist of waveforms ranging over severalscales, orientations, and locations with the ability to become very elongated. Thisrequires a combination of an appropriate scaling operator to generate elements atdifferent scales, an orthogonal operator to change their orientations, and a translationoperator to displace these elements over the 2D plane.

A family of vectors {ψn}n∈� form a frame for a Hilbert space H if there exist twopositive constants A, B such that for each f ∈ H we have

A|| f ||2 ≤∑n∈�

|〈 f,ψn〉|2 ≤ B|| f ||2. (4.6.1)

If A = B, the frame is called tight frame. The constants A and B are called lowerand upper frame bounds, respectively. If A = B = 1, the frame is called normalizedtight frame or Parseval frame.

Let f ∈ L2(Rd) and GLd(R) be the set of all d × d invertible matrices with realentries. The dilation operator DA, for A ∈ GLd(R) is defined by

DA f (x) = |det A|−1/2 f (A−1x), (4.6.2)

and the translation operator Tt for t ∈ Rd is defined by

Tt f (x) = f (x − t). (4.6.3)

Since the scaling operator is required to generatewaveformswith anisotropic support,we utilize the family of dilation operators DAa , a > 0, based on parabolic scaling


matrices Aa of the form

Aa =[a 00 a1/2

].

This type of dilation resembles to the parabolic scaling, which has a elongated historyin the literature of harmonic analysis and can be outlined back to the second dyadicdecomposition from the theory of oscillatory integrals. It should be point out that,rather than Aa , the more general matrices diag(a, aα) with the parameter α ∈ (0, 1)controlling the degree of anisotropy could be used. Though, the valueα = 1/2 showsa special role in the discrete setting. In fact, parabolic scaling is essential in orderto obtain optimally sparse approximations of cartoon-like images, since it is bestadapted to the C2-regularity of the curves of discontinuity in this model class.

Now, we need an orthogonal transformation to change the orientations of thewaveforms. The best obvious choice seems to be the rotation operator. Though,rotations destroy the structure of the integer lattice Z2 whenever the rotation angle isdifferent from 0, π

2 ,π, 3π2 . This issue becomes a thoughtful problem for the transition

from the continuum to the digital setting. we choose the shearing operator DSs , s ∈R, where the shearing matrix Ss as an alternative orthogonal transformation, is givenby

Ss =[1 s0 1

].

The shearing matrix parameterizes the orientations using the variable s associatedwith the slopes rather than the angles, and has the advantage of leaving the integerlattice invariant, provided s is an integer. Now, we define continuous shearlet systemsby combining these three operators.

Definition 4.1 Let ψ ∈ L2(R2). The continuous shearlet system SH(ψ) is definedby

SH(ψ) = {ψa,s,t = Tt DAa DSsψ : a > 0, s ∈ R, t ∈ R2}.

Similar to the relation of wavelet systems to group representation theory, thetheory of continuous shearlet systems can also be developed within the theory ofunitary representations of the affine group and its generalizations. For this, we definethe shearlet group S, as the semi-direct product

(R+ × R) × R2,

equipped with group multiplication given by

(a, s, t) · (a′, s ′, t ′) = (aa′, s + s ′√a, t + Ss Aat′).

A left-invariant Haar measure of this shearlet group is daa3 dsdt . The unitary repre-

sentation σ : S → U(L2(R2)) is defined by

4.6 Shearlet 115

σ(a, s, t)ψ = Tt DAa DSsψ,

where U(L2(R2)) denotes the group of unitary operators on L2(R2). A continuousshearlet system SH(ψ) can be re-written as

SH(ψ) = {σ(a, s, t)ψ : (a, s, t) ∈ S}.

The representation σ is unitary but not irreducible. If this additional property isdesired, the shearlet groupneeds to be extended to (R∗×R)×R

2,whereR∗ = R−{0},yielding the continuous shearlet system

SH(ψ) = {σ(a, s, t)ψ : a ∈ R∗, s ∈ R, t ∈ R

2}.

4.7 Shearlet Transform

4.7.1 Continuous Shearlet Transform

Similar to the continuous wavelet transform, the continuous shearlet transformdefines a mapping of f ∈ L2(R2) to the components of f associated with theelements of S.

Definition 4.2 Let ψ ∈ L2(R2). The continuous shearlet transform (CST) of f ∈L2(R2) is defined by

f �−→ SH ψ( f )(a, s, t) = 〈 f,σ(a, s, t)ψ〉=

∫f (x)σ(a, s, t)ψ(x)dx, (a, s, t) ∈ S.

Thus, SH ψ maps the function f to the coefficients SH ψ( f )(a, s, t) associatedwith the scale variable a > 0, the orientation variable s ∈ R, and the locationvariable t ∈ R

2.

Of specific importance are the conditions on ψ under which the continuous shear-let transform is an isometry, since this is automatically associated with a reconstruc-tion formula. For this, we define the notion of an admissible shearlet is also calledcontinuous shearlet.

Definition 4.3 If ψ ∈ L2(R2) satisfies

∫R2

|ψ(ξ1, ξ2)|2ξ21

dξ2dξ1 < ∞, (4.7.1)


it is called an admissible shearlet.

It is necessary to mention that any function ψ such that ψ is compactly supportedaway from the origin is an admissible shearlet.

Definition 4.4 Let ψ ∈ L2(R2) be defined by

ψ(ξ) = ψ(ξ1, ξ2) = ψ1(ξ1)ψ2

(ξ2

ξ1

),

where ψ1 ∈ L2(R2) is a discrete wavelet in the sense that it satisfies the discreteCalderon condition, given by

∑j∈Z

|ψ1(2− jξ)|2 = 1 for a.e. ξ ∈ R, (4.7.2)

with ψ1 ∈ C∞(R) and supp ψ1 ⊆ [− 12 ,− 1

16

]and ψ2 ∈ L2(R) is a bump function in

the sense that1∑

k=−1

|ψ2(ξ + k)|2 = 1 for a.e. ξ ∈ [−1, 1], (4.7.3)

satisfying ψ2 ∈ C∞(R) and supp ψ2 ⊆ [−1, 1]. Then ψ is called a classical shearlet.

Thus, a classical shearlet ψ is a function which is wavelet-like along one axis andbump-like along another one. Note that there exist several choices of ψ1 and ψ2

satisfying conditions (4.7.2) and (4.7.3). One possible choice is to set ψ1 to be aLemarie-Meyer wavelet and ψ2 to be a spline (Easley et al. [31], Guo and Labate[36]). This example was originally introduced in (Guo et al. [40]) and later slightlymodified by Guo et al. [36] and Labate et al. [46].

Let ψ ∈ L2(R2) be an admissible shearlet. Define

C+ψ =

∫ ∞

0

∫R

|ψ(ξ1, ξ2)|2ξ21

dξ2dξ1 and C−ψ =

∫ 0

−∞

∫R

|ψ(ξ1, ξ2)|2ξ21

dξ2dξ1.

(4.7.4)If C−

ψ = C+ψ = 1, then SH ψ is an isometry.

4.7.2 Discrete Shearlet Transform

The discrete shearlet systems are formally defined by sampling continuous shearletsystems on a discrete subset of the shearlet group S.

Definition 4.5 Let ψ ∈ L2(R2) and � ⊆ S. An irregular discrete shearlet systemassociated with ψ and �, denoted by SH(ψ,�), is defined by

4.7 Shearlet Transform 117

SH(ψ,�) = {ψa,s,t = a−3/4ψ(A−1a S−1

s (· − t)) : (a, s, t) ∈ �}.

A (regular) discrete shearlet system associated withψ, denoted by SH(ψ), is definedby

SH(ψ) = {ψ j,k,m = 23 j/4ψ(Sk A2 j · −m) : j, k ∈ Z, m ∈ Z2}.

By choosing � = {(2 j , k, Sk A2 j m) : j, k ∈ Z,m ∈ Z2}, we can find the regular

versions of discrete shearlet systems from the irregular systems. Furthermore, in thedefinition of a regular discrete shearlet system, the translation parameter is sometimeschosen to belong to c1Z × c2Z for some (c1, c2) ∈ (R+)2. This provides someadditional flexibility which is very useful for some other constructions.

We are particularly interested not only in finding generic generator functions ψbut also in selecting a generator ψ with special properties, e.g., regularity, vanishingmoments, and compact support similar to the wavelet case, so that the correspondingbasis or frame of shearlets has satisfactory approximation properties.

Theorem 4.1 Let ψ ∈ L2(R2) be a classical shearlet. Then SH(ψ) is a Parsevalframe for L2(R2).

Proof For a.e. ξ ∈ R2, Using the properties of classical shearlets, we have,

∑j∈Z

∑k∈Z

|ψ(ST−k A2− j ξ)|2 =∑j∈Z

∑k∈Z

|ψ1(2− jξ1)|2|ψ2

(2 j/2 ξ2

ξ1− k

)|2

=∑

j∈Z |ψ1(2− jξ1)|2

∑k∈Z |ψ2

(2 j/2 ξ2

ξ1+ k

)|2 = 1.

The result follows immediately from this observation and the fact that supp ψ ⊂[− 12 ,

12

]. ��

Since a classical shearlet ψ is a well-localized function, by Theorem 4.1, thereexit Parseval frames SH(ψ) ofwell-localized discrete shearlets. Thewell localizationproperty is critical for deriving superior approximation properties of shearlet systemsand it will be required for deriving optimally sparse approximations of cartoon-likeimages. Without the assumption that ψ is well localized, one can construct discreteshearlet systems which form not only tight frames but also orthonormal bases. Thus,a well localized discrete shearlet system can form a frame or a tight frame but not anorthonormal basis.

Now, we define a discrete shearlet transform as follows. We state the definitiononly for regular case with obvious extension to the irregular shearlet systems.

Definition 4.6 Let ψ ∈ L2(R2). The discrete shearlet transform of f ∈ L2(R2) isthe mapping defined by

f �−→ SH ψ f ( j, k,m) = 〈 f,ψ j,k,m〉 ( j, k,m) ∈ Z × Z × Z2.


Thus, SH ψ maps the function f to the coefficients SH ψ f ( j, k,m) associatedwith the scale index j , the orientation index k, and the position index m.

4.7.3 Cone-Adapted Continuous Shearlet Transform

Even though the continuous shearlet systems defined as above display an elegantgroup structure, they do have a directional bias, which can be easily seen in Fig. 4.6.

For better understanding, consider a function which is mostly concentrated alongthe ξ2 axis in the frequency domain. Then the energy of f is more and more con-centrated in the shearlet components SHψ f (a, s, t) as s → ∞. In the limiting case,f is a delta distribution supported along the ξ2 axis, the typical model for an edgealong the x1 axis in spatial domain, f can only be detected in the shearlet domain ass → ∞. It is clear that this behavior can be a serious limitation for some applications.

Fig. 4.6 Fourier domain support of several elements of the shearlet system


Fig. 4.7 Resolving the problem of biased treatment of directions by continuous shearlet systems

Oneway to address this problem is to partition the Fourier domain into four cones,while separating the low-frequency region by cutting out a square centered aroundthe origin. This yields a partition of the frequency plane as shown in Fig. 4.7. Noticethat, within each cone, the shearing variable s is only allowed to vary over a finiterange, hence producing elements whose orientations are distributed more uniformly.

Now, we are defining the following variant of continuous shearlet systems, knownas cone-adapted continuous shearlet system.

Definition 4.7 Letφ,ψ, ψ ∈ L2(R2). The cone-adapted continuous shearlet system,denoted by SH(φ,ψ, ψ), is defined by

SH(φ,ψ, ψ) = �(φ) ∪ �(ψ) ∪ �(ψ),


where

�(φ) = {φt = φ(· − t) : t ∈ R2},

�(ψ) = {ψa,s,t = a−3/4φ(A−1a S−1

s (· − t)) : a ∈ (0, 1], |s| ≤ 1 + a1/2, t ∈ R2},

�(ψ) = {ψa,s,t = a−3/4ψ( A−1a S−1

s (· − t)) : a ∈ (0, 1], |s| ≤ 1+ a1/2, t ∈ R2},

and Aa = diag(a1/2, a).

In the following definition, the function φ will be chosen to have compact fre-quency support near the origin, which ensures that the system�(φ) is associatedwiththe low-frequency regionR = {(ξ1, ξ2) : |ξ1|, |ξ2| ≤ 1}.By choosingψ to satisfy thecondition of definition 4.4, the system �(ψ) is associated with the horizontal conesC1 ∪ C3 = {(ξ1, ξ2) : |ξ2/ξ1| ≤ 1, |ξ1| > 1}. The shearlet ψ can be chosen likewisewith the roles of ξ1 and ξ2 reversed, i.e., ψ(ξ1, ξ2) = ψ(ξ1, ξ2).Then the system �(ψ)

is associated with the vertical cones C2 ∪ C4 = {(ξ1, ξ2) : |ξ2/ξ1| > 1, |ξ2| > 1}.Similar to the situation of continuous shearlet systems, an associated cone-adapted

continuous shearlet transform can be defined.

Definition 4.8 Let

Scone = {(a, s, t) : a ∈ (0, 1], |s| ≤ 1 + a1/2, t ∈ R2}.

Then, for φ,ψ, ψ ∈ L2(R2), the cone-adapted continuous shearlet transform off ∈ L2(R2) is the mapping

f �−→ SH φ,ψ,ψ f (t ′, (a, s, t), (a, s, t)) = (〈 f,φt ′ 〉, 〈 f,ψa,s,t 〉, 〈 f, ψa ,s ,t 〉)

where(t ′, (a, s, t), (a, s, t)) ∈ R

2 × S2cone.

Similar to the situation above, conditions on ψ, ψ and φ can be formulated forwhich the mapping SH φ,ψ,ψ is an isometry.

Theorem 4.2 Let ψ, ψ ∈ L2(R2) be admissible shearlets satisfying C+ψ = C−

ψ = 1and C+

ψ= C−

ψ= 1, respectively, and let � ∈ L2(R2) be such that, for a.e. ξ =

(ξ1, ξ2) ∈ R2,

|φ(ξ)|2 + χC1∪C3(ξ)

∫ 1

0|ψ1(aξ1)|2 da

a+ χC2∪C4(ξ)

∫ 1

0|ψ1(aξ2)|2 da

a= 1.

Then, for each f ∈ L2(R2)

|| f ||2 =∫R

|〈 f, Ttφ〉|2dt +∫Scone

|〈( f χC1∪C3) ,ψa,s,t 〉|2 daa3

dsdt

+∫Scone

|〈( f χC2∪C4) , ψa,s ,t 〉|2 daa3

dsdt


In this result, the function φ, ψ and ˆψ can in fact be chosen to be in C∞0 (R2). In

addition, the cone-adapted shearlet system can be designed so that the low-frequencyand high-frequency parts are smoothly combined. A more detailed analysis of thecontinuous shearlet transform and cone-adapted continuous shearlet transform andits generalizations can be found in Grohs [35].

4.7.4 Cone-Adapted Discrete Shearlet Transform

Similar to the situation of continuous shearlet systems, discrete shearlet systemsalso suffer from a biased treatment of directions. For the sake of generality, let usstart by defining cone-adapted discrete shearlet systems with respect to an irregularparameter set.

Definition 4.9 Let φ,ψ, ψ ∈ L2(R2), � ⊂ R2 and �, � ⊆ Scone. The irregular

cone-adapted discrete shearlet system SH(φ,ψ, ψ;�,�, �) is defined by

SH(φ,ψ, ψ;�,�, �) = �(φ;�) ∪ �(ψ;�) ∪ �(ψ; �),

where�(φ;�) = {φt = φ(· − t) : t ∈ �},�(ψ;�) = {ψa,s,t = a−3/4ψ(A−1

a S−1s (· − t)) : (a, s, t) ∈ �},

�(ψ; �) = {ψa,s,t = a−3/4ψ( A−1a S−1

s (· − t)) : (a, s, t) ∈ �}.The regular variant of the cone-adapted discrete shearlet systems is much more

frequently used. To allow more flexibility and enable changes to the density of thetranslation grid, we introduce a sampling factor c = (c1, c2) ∈ (R+)2 in the transla-tion index. Hence, we have the following definition.

Definition 4.10 For φ,ψ, ψ ∈ L2(R2), � ⊂ R2 and c = (c1, c2) ∈ (R+)2, the

regular cone-adapted discrete shearlet system SH(φ,ψ, ψ; c) is defined by

SH(φ,ψ, ψ; c) = �(φ; c1) ∪ �(ψ; c) ∪ �(ψ; c),

where�(φ; c1) = {φm = φ(· − c1m) : m ∈ Z

2},�(ψ; c) = {ψ j,k,m = 23 j/4ψ(Sk A2 j · −Mcm) : j ≥ 0, |k| ≤ �2 j/2�,m ∈ Z

2},�(ψ; c) = {ψ j,k,m = 23 j/4ψ(STk A2 j · −Mcm) : j ≥ 0, |k| ≤ �2 j/2�,m ∈ Z

2}with

Mc =[c1 00 c2

]and Mc =

[c2 00 c1

].

If c = (1, 1), the parameter c is omitted in the formulae above.


The generating functionsφwill be called shearlet scaling functions and the gener-ating functions ψ, ψ are called shearlet generators. Notice that the system �(φ; c1)is associated with the low-frequency region, and the systems �(ψ; c) and �(ψ; c)are associated with the conic regions C1 ∪ C3 and C2 ∪ C4, respectively.

Since the construction of discrete shearlet orthonormal basis is impossible, oneaims to derive Parseval frames. For that we first see that a classical shearlet is ashearlet generator of a Parseval frame for the subspace of L2(R2) of functions whosefrequency support lies in the union of two cones C1 ∪ C3.

Theorem 4.3 Let ψ ∈ L2(R2) be a classical shearlet. Then the shearlet system

�(ψ) = {ψ j,k,m = 23 j/4ψ(Sk A2 j · −Mcm) : j ≥ 0, |k| ≤ �2 j/2�,m ∈ Z2}

is a Parseval frame for L2(C1 ∪ C3) = { f ∈ L2(R2) : supp f ⊂ C1 ∪ C3}.Proof Since ψ be a classical shearlet, Eq. (4.7.3) implies that, for any j ≥ 0,

∑|k|≤�2 j/2�

|ψ2(2j/2ξ + k)|2 = 1, |ξ| ≤ 1.

By using this observation together with Eq. (4.7.2), a direct computation gives that,for a.e. ξ = (ξ1, ξ2) ∈ C1 ∪ C3,

∑j≥0

∑|k|≤�2 j/2�

|ψ(ST−k A2− j ξ)|2 =∑j≥0

∑|k|≤�2 j/2�

|ψ1(2− jξ1)|2|ψ2

(2 j/2 ξ2

ξ1− k

)|2

=∑j≥0

|ψ1(2− jξ1)|2

∑|k|≤�2 j/2�

|ψ2

(2 j/2 ξ2

ξ1+ k

)|2 = 1.

The claim follows immediately from this observation and the fact that suppψ ⊂[− 12 ,

12

]. ��

A result very similar to Theorem 4.3 holds for the subspace of L2(C2 ∪ C4) ifψ is a replaced by ψ. This indicates that one can build up a Parseval frame for thewhole space L2(R2) by piecing together Parseval frames associated with differentcones on the frequency domain together with a coarse scale system which takes careof the low-frequency region. Using this idea, we have the following result.

Theorem 4.4 Letψ ∈ L2(R2) be a classical shearlet, and let φ ∈ L2(R2) be chosenso that, for a.e. ξ ∈ R

2,

|φ(ξ)|2 +∑j≥0

∑|k|≤�2 j/2�

|ψ(ST−k A2− j ξ)|2χC +∑j≥0

∑|k|≤�2 j/2�

| ˆψ(S−k A2− j ξ)|2χC = 1.

Let PC�(ψ) denote the set of shearlet elements in�(ψ) after projecting their FouriertransformsontoC = {(ξ1, ξ2) ∈ R

2 : |ξ2/ξ1| ≤ 1},with a similar definitionholding


for PC�(ψ) where C = R2 \ C. Then, the modified cone-adapted discrete shearlet

system �(φ) ∪ PC�(ψ) ∪ PC�(ψ) is a Parseval frame for L2(R2).

Notice that, despite its simplicity, the Parseval frame construction above has onedrawback. When the cone-based shearlet systems are projected onto C and C , theshearlet elements overlapping the boundary lines ξ1 = ±ξ2 in the frequency domainare cut so that the boundary shearlets lose their regularity properties. To avoid thisproblem, it is possible to redefine the boundary shearlets in such a way that their reg-ularity is preserved. This requires to slightly modifying the definition of the classicalshearlet. Then the boundary shearlets are obtained, essentially, by piecing togetherthe shearlets overlapping the boundary lines ξ1 = ±ξ2 which have been projectedonto C and C . This modified construction yields smooth Parseval frames of band-limited shearlets and can be found in Guo and Labate [37], where also the higherdimensional versions are discussed. The shearlet transform associated with regularcone-adapted discrete shearlet systems is defined as follows:

Definition 4.11 Set � = N0 × {−�2 j/2�, . . . , �2 j/2�} ×Z2. For φ,ψ, ψ ∈ L2(R2),

the cone-adapted discrete shearlet transform of f ∈ L2(R2) is the mapping definedby

f �−→ SH φ,ψ,ψ f (m ′, ( j, k,m), ( j , k, m)) = (〈 f,φm ′ 〉, 〈 f,ψ j,k,m〉, 〈 f, ψ j ,k,m〉)

where(m ′, ( j, k,m), ( j , k, m)) ∈ Z

2 × � × �.

4.7.5 Compactly Supported Shearlets

The shearlet systems, which are generated by classical shearlets, are band-limited,i.e., they have compact support in the frequency domain and, hence, cannot be com-pactly supported in the spatial domain. Thus, a different approach is needed for theconstruction of compactly supported shearlet systems.

Before stating the main result, let us first introduce the following notation.Let φ,ψ, ψ ∈ L2(R2). Define � : R2 × R

2 → R by

�(ξ,ω) = |φ(ξ)||φ(ξ + ω)| + �1(ξ,ω) + �2(ξ,ω), (4.7.5)

where�1(ξ,ω) =

∑j≥0

∑|k|≤�2 j/2�

|ψ(STk A2− j ξ)||ψ(STk A2− j ξ + ω)| (4.7.6)

and�2(ξ,ω) =

∑j≥0

∑|k|≤�2 j/2�

|ψ(STk A2− j ξ)||ψ(STk A2− j ξ + ω)|. (4.7.7)


Also, for c = (c1, c2) ∈ (R+)2, let

R(c) =∑

m∈Z2\{0}(�0(c

−11 m)�0(−c−1

1 m))1/2 + (�1(M−1c m)�1(−M−1

c m))1/2

+ (�2(M−1c m)�1(−M−1

c m))1/2, (4.7.8)

where

�0(ω) = ess supξ∈R2 |φ(ξ)||φ(ξ + ω)| and �i (ω) = ess supξ∈R2 |�i (ξ,ω)| for i = 1, 2.

Using these notation, we can now state the following theorem from Kittipoom etal. [44].

Theorem 4.5 Let φ,ψ ∈ L2(R2) be such that

φ(ξ1, ξ2) ≤ C1 · min{1, |ξ1|−γ} · min{1, |ξ2|−γ}

and|ψ(ξ1, ξ2)| ≤ C2 · min{1, |ξ1|α} · min{1, |ξ1|−γ} · min{1, |ξ2|−γ}

for some positive constants C1,C2 < ∞ and α > γ > 3. Define ψ(x1, x2) =ψ(x1, x2), and let Lin f , Lsup be defined by

Lin f = essinfξ∈R2�(ξ, 0) and Lsup = esssupξ∈R2�(ξ, 0).

Then there exists a sampling parameter c = (c1, c2) ∈ (R+)2 with c1 = c2 such thatSH(φ,ψ, ψ; c) forms a frame for L2(R2) with frame bounds A and B satisfying

0 <1

|detMc| [Lin f − R(c)] ≤ A ≤ B ≤ 1

|detMc| [Lsup + R(c)] < ∞.

It can be easily verified that the conditions imposed on φ and ψ by Theorem 4.5are satisfied by many suitably chosen scaling functions and classical shearlets. Inaddition, one can construct various compactly supported separable shearlets thatsatisfy these conditions. The difficulty, however, arises when aiming for compactlysupported separable functions φ and ψ which ensure that the corresponding cone-adapted discrete shearlet system is a tight or almost tight frame. Separability is usefulto achieve fast algorithmic implementations. In fact, it was shown by Kittipoom etal. [44] that there exists a class of functions generating almost tight frames, whichhave (essentially) the form

ψ(ξ) = m1(4ξ1)φ(ξ1)φ(2ξ2), ξ = (ξ1, ξ2) ∈ R2, (4.7.9)


where m1 is a carefully chosen bandpass filter and φ an adaptively chosen scalingfunction. The proof of this fact is highly technical and will be omitted.

4.7.6 Sparse Representation by Shearlets

One of the main motivations for the introduction of the shearlet framework is thederivation of optimally sparse approximations of multivariate functions. Before sta-ting the main results, it is enlightening to present a heuristic argument in order todescribe how shearlet expansions are able to achieve optimally sparse approxima-tions of cartoon-like images.

For this, consider a cartoon-like function f and let SH(φ,ψ, ψ; c) be a shearletsystem. Since the elements of SH(φ,ψ, ψ; c) are effectively or in case of compactlysupported elements, exactly supported inside a box of size 2− j/2 × 2− j/2, it fol-lows that at scale 2− j there exist about O(2 j/2) such waveforms whose support istangent to the curve of discontinuity. Similar to the wavelet case, for j sufficientlylarge, the shearlet elements associated with the smooth region of f , as well as theelements whose overlap with the curve of discontinuity is non-tangential, yield neg-ligible shearlet coefficients 〈 f,ψ j,k,m〉 (or 〈 f, ψ j,k,m〉). Each shearlet coefficient canbe controlled by

|〈 f,ψ j,k,m〉| ≤ || f ||∞||ψ j,k,m ||L1 ≤ C2−3 j/4,

similarly for 〈 f, ψ j,k,m〉. Using this estimate and the observation that there exist atmost O(2 j/2) significant coefficients, we can conclude that the N th largest shearletcoefficient, which we denote by |sN ( f )|, is bounded by O(N 3/2). This implies that

|| f − fN ||2L2 ≤∑l>N

|sl( f )|2 ≤ CN−2,

where fN denotes the N -term shearlet approximation using the N largest coefficientsin the shearlets expansion. Indeed, the following result holds.

Theorem 4.6 Let �(φ) ∪ PC�(ψ) ∪ PC�(ψ) be a Parseval frame for L2(R2) asdefined in Theorem 4.4, where ψ ∈ L2(R2) is a classical shearlet and φ ∈ C∞

0 (R2).

Let f ∈ ε2(R2) and fN be its nonlinear N-term approximation obtained by selectingthe N largest coefficients in the expansion of f with respect to this shearlet system.Then there exists a constant C > 0, independent of f and N, such that

|| f − fN ||22 ≤ CN−2(log N )3 as N → ∞.

Since a log-like factor is negligible with respect to the other terms for large N , theoptimal error decay rate is essentially achieved. It is remarkable that an approximationrate which is essentially as good as the one obtained using an adaptive construction


can be achieved using a nonadaptive system. The same approximation rate with thesame additional log-like factor is obtained using a Parseval frame of curvelets.

Interestingly, the same error decay rate is also achieved using approximationsbased on compactly supported shearlet frames, as stated below.

Theorem 4.7 Let SH(φ,ψ, ψ; c) be a frame for L2(R2), where c > 0, andφ,ψ, ψ ∈ L2(R2) are compactly supported functions such that, for all ξ = (ξ1, ξ2) ∈R

2, the shearlet ψ satisfies

(i) |ψ(ξ)| ≤ C1 min{1, |ξ1|α} min{1, |ξ1|−γ} min{1, |ξ2|−γ} and

(ii) | ∂∂ ξ

ψ(ξ)| ≤ |h(ξ1)|(1 + |ξ2|

|ξ1|)−γ

,

where α > 5, γ ≥ 4, h ∈ L1(R),C1 is a constant and the shearlet ψ satisfies (i)and (ii) with the roles of ξ1 and ξ2 reversed.

Let f ∈ ε2(R2) and fN be its nonlinear N-term approximation obtained byselecting the N largest coefficients in the expansion of f with respect to this shearletframe SH(φ,ψ, ψ; c). Then there exists a constant C > 0, independent of f and N,such that

|| f − fN ||22 ≤ CN−2(logN )3 as N → ∞.

Conditions (i) and (ii) are rather mild conditions and might be regarded as a weakversion of directional vanishing moment conditions.

References

1. Arivazhagan, S., Ganesan, L., Kumar, T.S.: Texture classification using curvelet statisticaland co-occurrence feature. In: Proceedings of the 18th International Conference on PatternRecognition (ICPR 2006), vol. 2, pp. 938–941 (2006)

2. Averbuch, A., Coifman, R.R., Donoho, D.L., Israeli, M., Walden, J.: Fast slant stack: a notionof radon transform for data in a cartesian grid which is rapidly computible, algebraically exact.geometrically faithful and invertible. SIAM J. Sci. Comput. (2001)

3. Bamberger, R.H., Smith, M.J.T.: A filter bank for the directional decomposition of images:theory and design. IEEE Trans. Signal Proc. 40(4), 882–893 (1992)

4. BeamLab 200. http://www-stat.stanford.edu/beamlab/ (2003)5. Bobin, J., Moudden, Y., Starck, J.L., Elad, M.: Morphological diversity and source separation.

IEEE Trans. Signal Process. 13(7), 409–412 (2006)6. Bobin, J., Starck, J.L., Fadili, J., Moudden, Y.: Sparsity, morphological diversity and blind

source separation. IEEE Trans. Image Process. 16(11), 2662–2674 (2007)7. Burt, P.J., Adelson, E.H.: The laplacian pyramid as a compact image code. IEEE Trans. Com-

mun. 31(4), 532–540 (1983)8. Candes E.: Harmonic Analysis of Neural Network. Report No. 509, Department of Statistics,

Standford University (1996)9. Candes, E.J., Demanet, L.: Curvelets and fourier integral operators. C. R. Acad. Sci. Paris, Ser.

I (2003)10. Candes, E.J., Demanet, L.: The curvelet representation of wave propagators is optimally sparse.

Commun. Pure Appl. Math. 58(11), 1472–1528 (2005)11. Candes, E.J., Donoho, D.L.: Curvelets and curvilinear integrals. J. Approx. Theory. 113, 59–90

(2001)

http://www-stat.stanford.edu/beamlab/

References 127

12. Candes, E.J., Donoho, D.L.: New Tight Frames of Curvelets and Optimal Representations ofObjects with Smooth Singularities. Technical Report, Statistics, Stanford University (2002)

13. Candes, E.J., Donoho, D.L.: Ridgelets: the key to high dimensional intermittency? Philos.Trans. R. Soc. Lond. A. 357, 2495–2509 (1999)

14. Candes, E.J., Donoho, D.L.: Curvelets-a surprisingly effective nonadaptive representation forobjects with edges. In: Cohen, A., Rabut, C., Schumaker, L.L. (eds.) Curve and Surface Fitting:Saint-Malo, 1999. Vanderbilt University Press, Nashville, TN (1999)

15. Candes, E.J.: Ridgelets: Theory and Applications. Ph.D thesis, Stanford University (1998)16. Candes, E.J.: Harmonic analysis of neural networks. Appl. Comput. Harmonic Anal. 6, 197–

218 (1999)17. Candes, E.J.: Ridgelets and the representation of mutilated sobolev functions. SIAM J. Math.

Anal. 33, 197–218 (1999)18. Candes, E., Demanet, L., Donoho, D., Ying, L.: Fast discrete curvelet transforms. SIAMMul-

tiscale Model Simul. 5(3), 861–899 (2006)19. Coifman, R.R., Meyer, Y., Wickerhauser, M.V.: Wavelet analysis and signal processing. In:

Wavelets and their Applications, M.B.R. et al. (ed.) Boston: Jones and Barlett, pp. 153–178(1992)

20. Do, M.N., Vetterli, M.: Contourlets. In: Stoeckler, J., Welland, G.V. (eds.) Beyond Wavelets.Academic Press, San Diego (2003)

21. Do,M.N., Vetterli, M.: The contourlet transform: an efficient directional multiresolution imagerepresentation. IEEE Trans. Image Process. 14(12), 2091–2106 (2005)

22. Do, M.N., Vetterli, M.: Framing pyramids. IEEE Trans. Signal Proc. 51(9), 2329–2342 (2003)23. Do, M.N., Vetterli, M.: Pyramidal directional filter banks and curvelets. In: Proceedings of the

IEEE International Conference on Image Proc. Thessaloniki, Greece (2001)24. Do, M.N., Vetterli, M.: The finite ridgelet transform for image representation. IEEE Trans.

Image Process. 12(1), 16–28 (2003)25. Do, M.N.: Directional Multiresolution Image Representations. Ph.D. dissertation, Swiss Fed-

eral Institute of Technology, Lausanne, Switzerland (2001)26. Donoho, D.L., Duncan, M.R.: Digital curvelet transform: strategy, implementation and experi-

ments. In: Szu, H.H., Vetterli,M., Campbell,W., Buss, J.R. (eds.) Proceedings of the Aerosense2000, Wavelet Applications VII, SPIE, vol. 4056, pp. 12–29 (2000)

27. Donoho,D.L., Flesia,A.G.:Digital ridgelet transformbasedon true ridge functions. In: Schmei-dler, J., Welland, G.V. (eds.) Beyond Wavelets. Academic Press, San Diego (2002)

28. Donoho, D.L.: Orthonormal ridgelets and linear singularities. SIAM J. Math Anal. 31(5),1062–1099 (2000)

29. Donoho, D.L., Vetterli, M., DeVore, R.A., Daubechies, I.: Data compression and harmonicanalysis. IEEE Trans. Inf. Theory 44(6), 2435–2476 (1998)

30. Douma, H., DeHoop,M.V.: Leading-order seismic imaging using curvelets. Geophysics 72(6),5221–5230 (2007)

31. Easley, G., Labate, D., Lim, W.Q.: Sparse directional image representations using the discreteshearlet transform. Appl. Comput. Harmon. Anal. 25, 25–46 (2008)

32. Elad,M., Starck, J.L., Donoho, D., Querre, P.: Simultaneous cartoon and texture image inpaint-ing using morphological component analysis (MCA). J. Appl. Comput. Harmonic Anal. 19,340–358 (2006)

33. Fadili, M.J., Starck, J.L.: Sparse representation-based image deconvolution by iterative thresh-olding. In: Murtagh, F., Starck, J.L. (eds.) Astronomical Data Analysis IV. Marseille, France(2006)

34. Fadili, M.J., Starck, J.L., Murtagh, F.: Inpainting and zooming using sparse representations.Comput. J. 52(1), 64–79 (2007)

35. Grohs, P.: Continuous shearlet tight frames. J. Fourier Anal. Appl. 17, 506–518 (2011)36. Guo, K., Labate, D.: Optimally sparse multidimensional representation using shearlets. SIAM

J. Math. Anal. 39, 298–318 (2007)37. Guo, K., Labate, D.: The construction of smooth parseval frames of shearlets. Math. Model.

Nat. Phenom. 8(1), 82–105 (2013)


38. Guo, K., Kutyniok, G., Labate, D.: Sparse multidimensional representations using anisotropicdilation and shear operators. In: Wavelets and Splines (Athens, GA, 2005), pp. 189–201. Nash-boro Press, Nashville, TN (2006)

39. Guo, K., Labate, D., Lim, W.: Edge analysis and identification using the continuous shearlettransform. Appl. Comput. Harmon. Anal. 27(1), 24–46 (2009)

40. Guo, K., Labate, D., Lim, W.Q., Weiss, G., Wilson, E.: Wavelets with composite dilations.Electron. Res. Announc. Am. Math. Soc. 10, 78–87 (2004)

41. Hennenfent, G., Herrmann, F.J.: Seismic denoisingwith nonuniformly sampled curvelets. IEEEComput. Sci. Eng. 8(3), 16–25 (2006)

42. Herrmann, F.J., Moghaddam, P.P., Stolk, C.C.: Sparsity and continuity-promoting seismicimage recovery with curvelet frames. Appl. Comput. Harmonic Anal. 24(2), 150–173 (2008)

43. Jin, J., Starck, J.L., Donoho, D.L., Aghanim, N., Forni, O.: Cosmological non-Gaussian sig-natures detection: comparison of statistical tests. Eurasip J. 15, 2470–2485 (2005)

44. Kittipoom, P., Kutyniok, G., Lim, W.Q.: Construction of compactly supported shearlet frames.Constr. Approx. 35, 21–72 (2012)

45. Kutyniok, G., Labate, D.: The construction of regular and irregular shearlet frames. J. WaveletTheory Appl. 1, 1–10 (2007)

46. Labate, D., Lim, W.Q., Kutyniok, G., Kutyniok, G.: Sparse multidimensional representationusing shearlets. Wavelets XI, SPIE. 5914, 254–262 (2005)

47. Labate, D., Lim, W.Q., Kutyniok G., Weiss G.: Sparse Multidimensional Representation usingShearlets: in Wavelets XI. In: Papadakis, M., Laine, A.F., Unser, M.A. (eds.) SPIE Proc. vol.5914, SPIE, Bellingham, WA, pp. 254–262 (2005)

48. Labate, D., Lim, W., Kutyniok, G., Weiss, G.: Sparse Multidimensional Representation usingShearlets. Wavelets XI, San Diego, CA, 254–262 (2005), SPIE Proc. 5914, SPIE, Bellingham,WA (2005)

49. Lambert, P., Pires, S., Ballot, J., Garca, R.A., Starck, J.L., Turck-Chieze, S.: Curvelet analysisof asteroseismic data I. Method description and application to simulated sun-like stars. Astron.Astrophys. 454, 1021–1027 (2006)

50. Mallat, S.: A Wavelet Tour of Signal Processing, 2nd ed. Academic Press, San Diego (1999)51. Matus, F., Flusser, J.: Image representations via a finite radon transform. IEEE Trans. Pattern

Anal. Mach. Intell. 15(10), 996–1006 (1993)52. Park, S.I., Smith, M.J.T., Mersereau, R.M.: Improved structures of maximally decimated direc-

tional filter banks for spatial image analysis. IEEE Trans. Image Proc. 13, 1424–1431 (2004)53. Pennec, E.L., Mallat S.: Image processing with geometrical wavelets. In: International Con-

ference on Image Processing (2000)54. Peyre, G., Mallat, S.: A review of bandlet methods for geometrical image representation.

Numer. Algorithms 44(3), 205–234 (2007)55. Saevarsson, B., Sveinsson, J., Benediktsson, J.: Speckle reduction of SAR images using adap-

tive curvelet domain. In: Proceedings of the IEEE International Conference on Geoscience andRemote Sensing Symposium, IGARSS 03, vol. 6, pp. 4083–4085 (2003)

56. Skodras, A., Christopoulos, C., Ebrahimi, T.: The JPEG 2000 still image compression standard.IEEE Signal Process. Mag. 18, 36–58 (2001)

57. Starck, J.L., Abrial, P., Moudden, Y., Nguyen, M.: Wavelets, ridgelets and curvelets on thesphere. Astron. Astrophys. 446, 1191–1204 (2006)

58. Starck, J.L., Bijaoui, A., Lopez, B., Perrier, C.: Image reconstruction by the wavelet transformapplied to aperture synthesis. Astron. Astrophys. 283, 349–360 (1994)

59. Starck, J.L., Candes, E., Donoho, D.L.: The curvelet transform for image denoising. IEEETrans. Image Process. 11(6), 131–141 (2002)

60. Starck, J.L., Candes, E., Donoho, D.L.: Astronomical image representation by the curvelettansform. Astron. Astrophys. 398, 785–800 (2003)

61. Starck, J.L., Elad, M., Donoho, D.L.: Redundant multiscale transforms and their applicationfor morphological component analysis. Adv. Imaging Electron Phys. 132 (2004)

62. Starck, J.L., Elad, M., Donoho, D.: Image decomposition via the combination of sparse repre-sentation and a variational approach. IEEE Trans. Image Process. 14(10), 1570–1582 (2005)

References 129

63. Starck, J.L., Murtagh, F., Bijaoui, A.: Image Processing and Data Analysis: The MultiscaleApproach. Cambridge University Press, Cambridge (1998)

64. Starck, J.L.,Murtagh, F.,Candes,E.,Donoho,D.L.:Gray andcolor image contrast enhancementby the curvelet transform. IEEE Trans. Image Process. 12(6), 706–717 (2003)

65. Starck, J.L., Nguyen, M.K., Murtagh, F.: Wavelets and curvelets for image deconvolution: acombined approach. Signal Process. 83(10), 2279–2283 (2003)

66. The Curvelab Toolbox. http://www.curvelet.org (2005)67. Velisavljevic, V., Beferull-Lozano, B., Vetterli, M., Dragotti, P.L.: Directionlets: anisotropic

multi-directional representation with separable filtering. ITIP 15(7), 1916–1933 (2006)68. Zhang, Z., Huang, W., Zhang, J., Yu, H., Lu, Y.: Digital image watermark algorithm in the

curvelet domain. In: Proceedings of the International Conference on Intelligent InformationHiding and Multimedia Signal Processing (IIH-MSP06), pp. 105–108 (2006)

http://www.curvelet.org

Part IIIApplication of Multiscale Transforms

to Image Processing

Chapter 5Image Restoration

There are various applications of image restoration in todaysworld. Image restorationis an important process in the field of image processing. It is a process to recoveroriginal image from distorted image. Image restoration is a task to improve thequality of image via estimating the amount of noises and blur involved in the image.To restore image its too important to know a prior knowledge about an image i.e.the knowledge about how an image was degraded or distorted. It is must to find outthat which type of noise is added in an image and how image gets blurred. So theprior knowledge about an image is a one of the important part in image restoration.Image gets degraded due to different conditions such as atmospheric conditions andenvironmental conditions, so it is required to restore the original image by usingdifferent image restoration algorithms. The main application of image restorationi.e. image reconstruction is in radio astronomy, radar imaging and tomography inmedical imaging.

The main aim of restoration process is to remove the degradation from the imageand obtain the imageF(x, y)which is close to the original image f (x, y). This processis processed in two domains: spatial domain and frequency domain.

Themain aim of this chapter is to focus on image restoration techniques using var-ious transforms such as Fourier transform, wavelet transform, undecimated wavelettransform, complex wavelet transform, quaternion wavelet transform, ridgelet trans-form, curvelet transform, contourlet transform and shearlet transform for naturalimages.

5.1 Model of Image Degradation and Restoration Process

First of all we will see that how an image f (x, y) gets degraded and then how it canbe restored by using different image restoration algorithms. Consider the originalimage f (x, y). If noise η(x, y) operates on original input image then a degraded


133

134 5 Image Restoration

image g(x, y) is produced. The main objective of image restoration is that the outputto be as same as possible to the original image. In mathematical form, the degradedimage g(x, y) can be represented as:

g(x, y) = h(x, y) ∗ f (x, y) + η(x, y), (5.1.1)

where h(x, y) is the degradation function and the symbol ∗ represents convolution ofh(x, y)with f (x, y). FromEq. (5.1.1), it is clear that the original image gets convolvedwith the degraded image i.e. the original image f (x, y) gets convolved with thedegradation function h(x, y).

As we know, taking convolution of two functions in spatial domain is equivalentto the product of the Fourier transform of the two functions in frequency domain.Hence, to convert the convolutions into multiplication take DFT of above equationin frequency domain. We have:

G(u, v) = H (u, v)F(u, v) + N (u, v) (5.1.2)

where the terms in capital letters are theFourier transformsof the corresponding termsin Eq. (5.1.1). To reduce the effect of noise from degraded image inverse filteringor pseudo inverse filtering can be used. In the next section, we are providing briefdiscussion about noise models, types of noise and image denoising techniques.

5.2 Image Quality Assessments Metrics

In the digital imageprocessing, the image restoration and enhancement techniques areperformed to improve the quality of the degraded image. In addition, since the originalimage corresponding to an degraded image is unknown, the quality assessment isperformed by simulating the degradation model to the original (reference) image. Todemonstrate the performance of the image restoration and enhancement techniques,objective image quality assessments metrics have been used in the literature.

The fundamental quality assessment metric is the mean squared error (MSE),which represents the loss between the observed and predicted in statistics and regres-sion analysis. In digital image processing, theMSEmeans that how close the restoredimage to the latent image. The MSE is defined as

MSE = 1

MN

M∑

x=1

N∑

y=1

(f (x, y) − f (x, y)

)2, (5.2.1)

where f (x, y) and f (x, y) respectively represent the restored and reference imageof the size M × N . The MSE is computed by averaging the square of the errorsbetween the restored and reference images. The small MSE value represents betterperformance of the techniques and high-quality of the restored image. The root mean

5.2 Image Quality Assessments Metrics 135

square error (RMSE) is the square root of the MSE and is defined as

RMSE =√√√√ 1

MN

M∑

x=1

N∑

y=1

(f (x, y) − f (x, y)

)2. (5.2.2)

The mean absolute error (MAE) is the average of the absolute of the differencebetween the observed and predicted signals. The MAE between the original andrestored images is computed as

MAE = 1

MN

M∑

x=1

N∑

y=1

∣∣∣f (x, y) − f (x, y)∣∣∣ . (5.2.3)

The peak signal-to-noise ratio (PSNR) represents the ratio between the maximumpower of an image (signal) and the power of the errors between the reference andrestored images. The PSNR is written using the MSE as

PSNR(dB) = 10 log10

(L2

MSE

),

= 20 log10

(L√MSE

),

= 20 log10 (L) − 10 log10 (MSE) ,

(5.2.4)

where L is the maximum intensity level in the original image f (x, y). If the image isquantized by 8 bits, the L is 255. If the MSE value is close to 0, the PSNR value isinfinite. The higher PSNR value represents better image quality.

The ameasure of structural similarity (SSIM) is computed based on the propertiesof the human visual system (HVS) [183]. Since the HVS is sensitive to the variationin the luminance, contrast, and structural information of a scene, the SSIM assessesthe image quality by using the similarity measurement in luminance, contrast, andstructural information on the local patterns between the two images. The SSIM isdefined as

SSIM (p, q) =(

2μpμq + C1

μ2p + μ2

q + C1

)α

×(

2σpσq + C2

σ2p + σ2

q + C2

)β

×(

σpq + C3

σpσq + C3

)γ

(5.2.5)where p and q respectively represent the window of the size n × n in the originaland restored images. μ and σ are the average and variance of the window, andσpq the covariance between the window p and q. The first term compares the localluminance change relative to background luminance, which is related toWeber’s law.The second terms is a function to compare the change in the low-contrast region thanhigh-contrast region. The third term compares the structural similarity by computingthe correlation coefficient between p and q.


The overall SSIM index of the entire image is evaluated by averaging the SSIMindex of each local window as

SSIM (P,Q) = 1

N

N∑

k=1

SSIM (pk , qk), (5.2.6)

where P and Q respectively represent the reference and restored images, pk and qkthe k-th window, and N the total number of windows.

5.3 Image Denoising

Digital images play an important role both in daily life applications such as satellitetelevision, magnetic resonance imaging, computer tomography as well as in areas ofresearch and technology such as geographical information systems and astronomy.Data sets collected by image sensors are generally contaminated by noise. Imperfectinstruments, problems with the data acquisition process, and interfering natural phe-nomena can all degrade the data of interest. Furthermore, noise can be introduced bytransmission errors and compression. Thus, denoising is often a necessary and thefirst step to be taken before the images data is analyzed. It is necessary to apply anefficient denoising technique to compensate for such data corruption.

Removing noise from the original signal is still a challenging problem forresearchers. There have been several published algorithms and each approach has itsassumptions, advantages, and limitations. Image denoising still remains a challengefor researchers because noise removal introduces artifacts and causes blurring of theimages.

Noise modeling in images is greatly affected by capturing instruments, data trans-mission media, image quantization and discrete sources of radiation. Different algo-rithms are used depending on the noisemodel.Most of the natural images are assumedto have additive random noise which is modeled as a Gaussian. Speckle noise isobserved by Guo et al. [72] in 1994 in ultrasound images whereas Jain [80] in 2006observed Rician noise affects MRI images.

5.4 Noise Models

In order to restore an image we need to know about the degradation functions.Different models for the noise are described in this section. The set of noise modelsare defined by specific probability density functions (PDFs). Noise can be present inimage in two ways; either in additive or multiplicative form.

5.4 Noise Models 137

5.4.1 Additive Noise Model

Noise signal that is additive in nature gets added to the original signal to generate acorrupted noisy signal and follows the following rule:

g(x, y) = f (x, y) + η(x, y), (5.4.1)

where f (x, y) is the original image intensity and η(x, y) denotes the noise introducedto produce the corrupted signal g(x, y) at (x, y) pixel location.

5.4.2 Multiplicative Noise Model

In this model, noise signal gets multiplied to the original signal. The multiplicativenoise model follows the following rule:

g(x, y) = f (x, y) × η(x, y). (5.4.2)

5.5 Types of Noise

There are various types of noise. They have their own characteristics and are inherentin images through different ways. Some commonly found noise models and theircorresponding PDFs are given below.

5.5.1 Amplifier (Gaussian) Noise

The typical model of amplifier noise is additive, independent at each pixel and inde-pendent of the signal intensity, called Gaussian noise. In color cameras, blue colorchannels are more amplified than red or green channel, hence, more noise can bepresent in the blue channel. Amplifier noise is a major part of the read noise of animage sensor, that is, of the consistent noise level in dark areas of the image. This typeof noise has a Gaussian distribution, which has a bell shaped probability distributionfunction (PDF) is given by

P(z) = 1√2πσ2

e−(z−μ)2/2σ2(5.5.1)

where z is the Gaussian random variable representing noise, μ is the mean or aver-age of the function and σ is the standard deviation of the noise. Graphically, it isrepresented as shown in Fig. 5.1.


Fig. 5.1 Gaussian distribution function

5.5.2 Rayleigh Noise

The PDE function of Rayleigh noise is given by

P(z) ={

2b (z − a)e−(z−a)2/b if z ≥ a,

0 if z ≤ a.(5.5.2)

The mean is given by

z = a + b

2,

and variance by

σ2 = (b − a)2

12.

5.5.3 Uniform Noise

It is another commonly found image noise i.e. uniform noise. Here the noise can takeon values in an interval [a, b] with uniform probability. The PDF is given by

P(z) ={

1(b−a) if a ≤ z ≤ b,

0 otherwise.(5.5.3)

5.5 Types of Noise 139

5.5.4 Impulsive (Salt and Pepper) Noise

Impulsive noise is sometimes called as salt and pepper noise or spike noise. Thiskind of noise is usually seen on images. It represents itself as arbitrarily occurringwhite and black pixels. An image that contains impulsive noise will have dark pixelsin bright regions and bright pixels in dark regions. It can be caused by dead pixels,analog-to-digital converter errors and transmitted bit errors. The PDF of impulsenoise is given by

P(z) =⎧⎨

⎩

Pa for z = a,Pb for z = b,0 otherwise.

(5.5.4)

If b > a, intensity b will appear as a light dot in the image and the vice-versa, levela will appear like a dark dot. If Pa or Pb is zero, the impulse noise is called unipolar.

5.5.5 Exponential Noise

The PDF of exponential noise is given by

P(z) ={ae−az if z ≥ 0,0 if z < 0,

(5.5.5)

where a > 0. The mean is given by

z = 1

a,

and variance by

σ2 = 1

a2.

5.5.6 Speckle Noise

Speckle noise is considered asmultiplicative noise. It is a granular noise that degradesthe quality of images obtained by active image devices such as active radar and syn-thetic aperture radar (SAR) images. Due to random fluctuations in the return signalfrom an object in conventional radar that is not big as single image processing ele-ment, speckle noise occurs. It increases the mean grey level of a local area. Specklenoisemakes image interpretation difficult in SAR images causedmainly due to coher-ent processing of backscattered signals from multiple distributed targets. Speckle


Fig. 5.2 Gamma distribution function

noise follows a gamma distribution and its probability distribution function (PDF) isgiven by:

P(z) = zα−1

(α − 1)!ααe−z/α (5.5.6)

where σ2 is variance and z is the gray level. The gamma distribution is given inFig. 5.2.

5.6 Image Deblurring

Image deblurring (or restoration) is an old problem in image processing, but it con-tinues to attract the attention of researchers and practitioners alike. A number ofreal-world problems from astronomy to consumer imaging find applications forimage restoration algorithms and image restoration is an easily visualized exam-

5.6 Image Deblurring 141

ple of a larger class of inverse problems that arise in all kinds of scientific, medical,industrial and theoretical problems.

Sometimesblurmaybeproducedby thephotographer to strengthenphotos expres-siveness, but unintentional blur will decrease the image quality, which is caused byincorrect focus, object motion, hand shaking and so on. Convolution of the idealimage with 2D point-spread function (PSF) h(x, y) in Eq. (5.1.1) is blurring of theimage. Different types of blurs are also responsible for degradation, blurred imagesare restored by restoration model in the frequency domain. There are mainly fourtypes of deblurring problem:

5.6.1 Gaussian Blur

It is a filter that blends a specific number of pixels incrementally, following a bell-shaped curve. Blurring is dense in the center and feathers at the edge. In remotesensing, atmospheric turbulence is a severe limitation. The occurrence of Gaussianblur depends on a variety of factors such as temperature, wind speed, exposuretime. It is type of image blurring filter which use Gaussian function for calculatingtransformation applied on each pixel. The equation of Gaussian function is

G(x) = 1√2πσ

e(−x)2

2σ2 , (5.6.1)

where x is distance from origin in horizontal axis and σ is standard deviation ofGaussian distribution.

5.6.2 Motion Blur

There is relativemotion between camera and object due to the image capturing calledmotion blur. Many types of motion blur can be distinguished all of which are due torelative motion between the recording device and the scene. This can be in the formof a translation, a rotation, a sudden change of scale, or some combinations of these.

5.6.3 Rectangular Blur

This is blurring in imagewith specific rectangular area.Blur in image canbe identifiedat any part based on this it can be circular and rectangular.


5.6.4 Defocus Blur

Defocus blur occurs in image when camera is improperly focused on image. Theresolution of imagemedium depends on amount of defocus. If there ismore toleranceof image there is low resolution in image. For good resolution of image defocus inimage should be minimize.

5.7 Superresolution

In most digital imaging applications, high resolution images or videos are usuallydesired for later image processing and analysis. The desire for high image resolutionstems from two principal application areas: improvement of pictorial informationfor human interpretation; and helping representation for automatic machine percep-tion. Image resolution describes the details contained in an image, the higher theresolution, the more image details. The resolution of a digital image can be classi-fied in many different ways: pixel resolution, spatial resolution, spectral resolution,temporal resolution, and radiometric resolution.

Superresolution image restoration is the process of producing a high- resolutionimage (or a sequence of high-resolution images) from a set of low-resolution images.The process requires an image acquisitionmodel that relates A high-resolution imageto multiple low resolution images and involves solving the resulting inverse problem.In addition to the degradation processes in single image restoration, super resolutionimage restoration incorporatesmotion anddownsamplingoperations into the imagingprocess.

Superresolution (SR) are techniques that construct high-resolution (HR) imagesfrom several observed low-resolution (LR) images, thereby increasing the high fre-quency components and removing the degradations caused by the imaging process ofthe low resolution camera. The basic idea behind SR is to combine the non-redundantinformation contained inmultiple low-resolution frames to generate a high-resolutionimage.

5.8 Classification of Image Restoration Algorithms

There are various restoration techniques as well as spatial domain filter for noiseremoval. The spatial domain methods are used for removing additive noise only.Sometimes blur helps to increase photos expressiveness but it decreases the qualityof image unintentionally. In image restoration, the improvement in the quality of therestored image over the recorded blurred one is measured by the signal-to-noise ratioimprovement. Image restoration techniques are used to make the corrupted image assimilar as that of the original image.

5.8 Classification of Image Restoration Algorithms 143

Image Restoration in Presence of Noise

The noise removal is done by filtering of the degraded image. There are two basicapproaches to image denoising; (a) spatial filteringmethods and (b) transformdomainfiltering methods. The Objective of any filtering approach are:

i. To suppress the noise effectively in uniform regions.ii. To preserve edges and other similar image characteristics.iii. To provide a visually natural appearance.

5.8.1 Spatial Filtering

Themost widely used filtering techniques in Image Processing are the spatial domainfiltering techniques. Spatial filtering is the method of choice in situations when onlyadditive noise is present in the image. Themain idea behind Spatial Domain Filteringis to convolve a mask with the image. Spatial filters can be further classified intomean filters, order-statistics filters, Weiner filters and adaptive filters.

I. Mean Filters

A mean filter is the optimal linear filter for Gaussian noise in the sense of meansquare error. Mean filtering is a simple, intuitive and easy to implement methodof smoothing images i.e. reducing the amount of intensity variation between onepixel and the next. It is often used to reduce noise in images. It reduces the intensityvariations between the adjacent pixels. Mean filter is nothing just a simple slidingwindow spatial filter that replaces the centre value of the window with the originalsignal and it works well only if the underlying signal is smooth.

The idea of mean filtering is simply to replace each pixel value in an imagewith the mean (average) value of its neighbors, including itself. This has the effectof eliminating pixel values which are unrepresentative of their surroundings. Meanfiltering is usually thought of as a convolution filter. Like other convolutions it isbased around a kernel, which represents the shape and size of the neighborhoodto be sampled when calculating the mean. Often a 3 × 3 square kernel is used, asalthough larger kernels (e.g. 5 × 5 squares) can be used for more severe smoothing.(Note that a small kernel can be applied more than once in order to produce a similarbut not identical effect as a single pass with a large kernel.) averaging kernel oftenused in mean filtering.

An imagewith themean (average) value of its neighbors, including itself. This hasthe effect of eliminatingpixel valueswhich are unrepresentative of their surroundings.next. It is often used to reduce noise in images. The idea of mean filtering is simplyto replace each pixel value in amount of intensity variation between one pixel andthe The two main problems with mean filtering, which are:

i. A single pixel with a very unrepresentative value can significantly affect themean value of all the pixels in its neighborhood.


ii. When the filter neighborhood straddles an edge, the filter will interpolate newvalues for pixels on the edge and so will blur that edge. This may be a problem ifsharp edges are required in the output.

There are four types of mean filters:

A. Arithmetic Mean Filter

In this type of mean filter the middle pixel value of the mask is replaced with thearithmetic mean of all the pixel values within the filter window. A mean filter simplysmoothes local variations in an image. Noise is reduced and as a result the imagesmoothens, but edges within the image get blurred.

If Sxy represent a rectangular subimage window of size m × n, centered at point(x, y), then the value of restored image f at the point (x, y) is defined as

f (x, y) = 1

mn

∑

(s,t)∈Sxyg(s, t), (5.8.1)

where g(x, y) is the corrupted image.

B. Geometric Mean Filter

The working of a geometric mean filter is same as the arithmetic mean filter; the onlydifference is that instead of taking the arithmetic mean the geometric mean is taken.The restored image is given by the expression

f (x, y) =⎡

⎣∏

(s,t)∈Sxyg(s, t)

⎤

⎦

1mn

, (5.8.2)

Value of each restored pixel is the product of pixels in the mask, raised to a power1mn .

C. Harmonic Mean Filter

In the harmonic mean method, the gray value of each pixel is replaced with theharmonic mean of gray values of the pixels in a surrounding region. The harmonicmean is defined as

H = n1x1

+ 1x2

+ · · · + 1xn

.

The restored image is given by the function:

f (x, y) = mn∑

(s,t)∈Sxy1

g(s,t)

. (5.8.3)


D. Contra-Harmonic Mean Filter

The restored image is given by the equation

f (x, y) =∑

(s,t)∈Sxy g(s, t)Q−1

∑(s,t)∈Sxy g(s, t)Q

. (5.8.4)

whereQ is the order of the system. IfQ = 0, it behaves as arithmetic mean filter andif Q = 1, it behaves as harmonic mean filter.

II. Order-Statistic Filter

order-statistic filters are spatial filters whose response is based on ordering (ranking)the values of the pixels contained in the image area encompassed by the filter. Theranking result determines the response of the filter. There are four types of order-statistic filters.

A. Median Filter

The best-known order-statistic filter is the median filter and it is also belongs to theclass of non-linear filter. The median filter is normally used to reduce noise in animage, somewhat like the mean filter. However, it often does a better job than themean filter of preserving useful detail in the image. Median filter replaces the valueof a pixel with the median value of the gray levels within the filter window. Medianfilters are very effective for impulse noise.

f (x, y) = median(s,t)∈Sxy {g(s, t)}. (5.8.5)

Median filters are popular because they provide excellent noise-reduction capabilitiesfor certain types of random noise, with considerably less blurring than linear smooth-ing filters of similar size. Median filters are particularly effective in the presence ofbipolar and unipolar impulse noise.

B. Max and Min Filters

The maximum filter is defined as the maximum of all pixels within a local region ofan image. So it replaces the center pixel value with the maximum value of pixel inthe subimage window. Similarly the minimum filter is defined as the minimum of allpixels within a local region of an image and the center pixel value is replaced withthe minimum value of pixel in the subimage window.

f (x, y) = max(s,t)∈Sxy {g(s, t)} for max filter (5.8.6)

f (x, y) = min(s,t)∈Sxy {g(s, t)} for min filter (5.8.7)

C. Midpoint Filter

This filter computes the midpoint between the maximum and minimum values in thearea encompassed by the filter.


f (x, y) = 1

2[max(s,t)∈Sxy {g(s, t)} + min(s,t)∈Sxy {g(s, t)}]. (5.8.8)

Note that this filter combines order statistics and averaging. Itworks best for randomlydistributed noise, like Gaussian or uniform noise.

D. Alpha-Trimmed Mean Filter

In Alpha-trimmed filter the d/2 lowest and d/2 highest intensity values of g(s, t) inthe neighborhood Sxy are deleted and the remaining (mn − d) pixels are averaged.The center pixel value is replaced with this averaged value.

The estimation function for the restored image is given by

f (x, y) = 1

(mn − d)

∑

(s,t)∈Sxygr(s, t). (5.8.9)

III. Wiener Filter

Wiener filter incorporate both the degradation function and statistical characteristicsof noise in to the restoration process. The scientist Wiener proposed this the conceptin the year 1942. The filter, which consists of the terms inside the brackets, also iscommonly referred as the minimum mean square error filter or least square errorfilter The Wiener filter is used to signal estimation problem for stationary signals.The Wiener filter is the MSE-optimal stationary linear filter for images degraded byadditive noise and blurring. In analysis of the Wiener filter requires the assumptionthat the signal and noise processes are second order stationary (in the random processsense). Wiener filters are also applied in the frequency domain.

IV. Adaptive Filters

The behavior of the Adaptive filters changes with the statistical characteristics of theimage inside the filter window. Therefore the performance of Adaptive filters is muchbetter in comparison with the non-adaptive filters. But the improved performance isat the cost of added filter complexity.

Mean and variance are two important statistical measures on which the adap-tive filtering is depends upon. For example if the local variance is high comparedto the overall image variance, the filter should return a value close to the presentvalue. Because high variance is usually associated with edges and edges should bepreserved.

5.8.2 Frequency Domain Filtering

Image smoothing and image sharpening can be achieved by frequency domain fil-tering. Smoothing is done by high frequency attenuation i.e. by low pass filtering.Sharpening is done by high pass filtering which attenuates the low frequency com-ponents without disturbing the high frequency components.


The frequency domain filteringmethods can be subdivided according to the choiceof the basis functions. The basis functions can be further classified as data adaptiveand non-adaptive. Non-adaptive transforms are discussed first since they are morepopular.

I. Spatial-Frequency Filtering

Spatial-frequency filtering refers use of low pass filters using fast Fourier transform(FFT). In frequency smoothing methods [51] the removal of the noise is achievedby designing a frequency domain filter and adapting a cut-off frequency when thenoise components are decorrelated from the useful signal in the frequency domain.These methods are time consuming and depend on the cut-off frequency and thefilter function behavior. Furthermore, they may produce artificial frequencies in theprocessed image.

II. Wavelet Domain

Filtering operations in thewavelet domain can be subdivided into linear and nonlinearmethods.

A. Linear Filters

Linear filters such as Wiener filter in the wavelet domain yield optimal results whenthe signal corruption can be modeled as a Gaussian process and the accuracy crite-rion is the mean square error (MSE) [36, 165]. However, designing a filter based onthis assumption frequently results in a filtered image that is more visually displeas-ing than the original noisy signal, even though the filtering operation successfullyreduces the MSE. In a wavelet-domain spatially adaptive FIR Wiener filtering forimage denoising is proposed by Zhang et al. [202] in 2000, where wiener filtering isperformed only within each scale and intrascale filtering is not allowed.

B. Non-Linear Threshold Filters

Themost investigated domain in denoising usingWavelet Transform is the non-linearcoefficient thresholding based methods. The procedure exploits sparsity property ofthe wavelet transform and the fact that the Wavelet Transform maps white noise inthe signal domain to white noise in the transform domain. Thus, while signal energybecomes more concentrated into fewer coefficients in the transform domain, noiseenergy does not. It is important principle that enables the separation of signal fromnoise.

The procedure in which small coefficients are removed while others are leftuntouched is called Hard Thresholding introduced by Donoho [53]. But the methodgenerates spurious blips, better known as artifacts, in the images as a result of unsuc-cessful attempts of removing moderately large noise coefficients. To overcome thedemerits of hard thresholding, wavelet transform using soft thresholding was alsointroduced by Donoho in 1995 [53]. In this scheme, coefficients above the thresholdare shrunk by the absolute value of the threshold itself. Similar to soft threshold-ing, other techniques of applying thresholds are semi-soft thresholding and Garrote


thresholding.Most of the wavelet shrinkage literature is based onmethods for choos-ing the optimal threshold which can be adaptive or non-adaptive to the image.

i. Non-adaptive Thresholds

VISUShrink is non-adaptive universal threshold introduced by Donoho andJhonstone [51] in 1994, which depends only on number of data points. It has asymp-totic equivalence suggesting best performance in terms of MSE when the numberof pixels reaches infinity. VISUShrink is known to yield overly smoothed imagesbecause its threshold choice can be unwarrantably large due to its dependence on thenumber of pixels in the image.

ii. Adaptive Thresholds

SUREShrink uses a hybrid of the universal threshold and the SURE [Steins UnbiasedRisk Estimator] threshold and performs better than VISUShrink, is also introducedby Donoho and Jhonstone [51] in 1994. BayesShrink, introduced by Simoncelli andAdelson [157] in 1996 and by Chipmann et al. in 1997 [32], minimizes the BayesRisk Estimator function assuming Generalized Gaussian prior and thus yielding dataadaptive threshold. BayesShrink outperforms SUREShrink most of the times. Crossvalidation [81] replaces wavelet coefficient with the weighted average of neighbor-hood coefficients tominimize generalized cross validation (GCV) function providingoptimum threshold for every coefficient.

The assumption that one can distinguish noise from the signal solely based oncoefficient magnitudes is violated when noise levels are higher than signal magni-tudes. Under this high noise circumstance, the spatial configuration of neighboringwavelet coefficients can play an important role in noise-signal classifications. Signalstend to formmeaningful features (e.g. straight lines, curves), while noisy coefficientsoften scatter randomly.

C. Non-orthogonal Wavelet Transforms

Undecimated Wavelet Transform (UDWT) has also been used for decomposing thesignal to provide visually better solution. Since UDWT is shift invariant it avoidsvisual artifacts such as pseudo-Gibbs phenomenon. Though the improvement inresults is much higher, use of UDWT adds a large overhead of computations thusmaking it less feasible. Lang et al. [99] extended normal hard/soft thresholdingto shift invariant discrete wavelet transform. Cohen [38] exploited Shift InvariantWavelet Packet Decomposition (SIWPD) to obtain number of basis functions. Thenusing Minimum Description Length principle the best basis function was found outwhich yielded smallest code length required for description of the given data. Then,thresholding was applied to denoise the data.

In addition toUDWT, use ofmulti-wavelets is exploredwhich further enhances theperformance but further increases the computation complexity. The multi-waveletsare obtained by applying more than one mother function (scaling function) to givendataset. Multi-wavelets possess some important properties such as short support,symmetry, and the most importantly higher order of vanishing moments.


D. Wavelet Coefficient Model

This approach focuses on exploiting themulti-resolution properties ofWavelet Trans-form. This technique identifies close correlation of signal at different resolutions byobserving the signal across multiple resolutions. This method produces excellentoutput but is computationally much more complex and expensive. The modeling ofthe wavelet coefficients can either be deterministic or statistical.

i. Deterministic Modeling of Wavelet Coefficients

The Deterministic method of modeling involves creating tree structure of waveletcoefficients with every level in the tree representing each scale of transformation andnodes representing the wavelet coefficients. This approach is adopted by Baraniuk[14] in 1999. The optimal tree approximation displays a hierarchical interpretationof wavelet decomposition. Wavelet coefficients of singularities have large waveletcoefficients that persist along the branches of tree. Thus if a wavelet coefficient hasstrong presence at particular node then it being signal, its presence should be morepronounced at its parent nodes. If it is noisy coefficient, for instance spurious blip,then such consistent presence will be missing. Lu et al. [106] tracked wavelet localmaxima in scale space, by using a tree structure. Other denoising method based onwavelet coefficient trees was proposed by Donoho [55] in 1997.

ii. Statistical Modeling of Wavelet Coefficients

This approach focuses on some more interesting and appealing properties of thewavelet transform such as multiscale correlation between the wavelet coefficients,local correlation between neighborhood coefficients etc. This approach has an inher-ent goal of perfecting the exactmodeling of image datawith use ofwavelet transform.A good review of statistical properties of wavelet coefficients can be found by Bueci-grossi and Smoncelli [18] and by Romberg et al. [146]. The following two techniquesexploit the statistical properties of the wavelet coefficients based on a probabilisticmodel.

a. Marginal Probabilistic Model

A number of researchers have developed homogeneous local probability models forimages in thewavelet domain. Specifically, themarginal distributions ofwavelet coef-ficients are highly kurtosis, and usually have a marked peak at zero and heavy tails.The Gaussian mixture model (GMM) [32] and the generalized Gaussian distribu-tion (GGD) [120] are commonly used to model the wavelet coefficients distribution.Although GGD is more accurate, GMM is simpler to use. Mihcak et al. [115] in 1999proposed a methodology in which the wavelet coefficients are assumed to be condi-tionally independent zero-mean Gaussian random variables, with variances modeledas identically distributed, highly correlated random variables. An approximate Max-imum A Posteriori (MAP) Probability rule is used to estimate marginal prior distri-bution of wavelet coefficient variances. All these methods mentioned above requirea noise estimate, which may be difficult to obtain in practical applications. Simon-celli and Adelson [157] used a two parameter generalized Laplacian distribution for


the wavelet coefficients of the image, which was estimated from the noisy observa-tions. Chang et al. [25] proposed the use of adaptive wavelet thresholding for imagedenoising, by modeling the wavelet coefficients as a generalized Gaussian randomvariable, whose parameters are estimated locally i.e. within a given neighborhood.

b. Joint Probabilistic Model

Hidden Markov Models (HMM) [146] models are efficient in capturing inter-scaledependencies,whereasRandomMarkovField [110]models aremore efficient to cap-ture intrascale correlations. The complexity of local structures is not well describedby RandomMarkov Gaussian densities whereas HiddenMarkovModels can be usedto capture higher order statistics. The correlation between coefficients at same scalebut residing in a close neighborhood are modeled by Hidden Markov Chain Modelwhere as the correlation between coefficients across the chain is modeled by HiddenMarkov Trees. Once the correlation is captured byHMM, ExpectationMaximizationis used to estimate the required parameters and from those, denoised signal is esti-mated from noisy observation using well known MAP estimator. Portilla et al. [137]described a model in which each neighborhood of wavelet coefficients is describedas a Gaussian scale mixture (GSM) which is a product of a Gaussian random vector,and an independent hidden random scalar multiplier. Strela et al. [165] describedthe joint densities of clusters of wavelet coefficients as a Gaussian scale mixture,and developed a maximum likelihood solution for estimating relevant wavelet coef-ficients from the noisy observations. Another approach that uses a Markov randomfield model for wavelet coefficients was proposed by Jansen and Bulthel [81] in2001.A disadvantage of HMT is the computational burden of the training stage. Inorder to overcome this computational problem, a simplified HMT, named as UHMT,was proposed by Romberg et al. [146] in 2001.

III. New Multiscale Transforms

Wavelet denoising is performed by taking the wavelet transform of the noisy imageand then removing out the detail (typically high-pass) coefficients that fall below acertain threshold. The thresholding can be either soft or hard. An inverse wavelettransform is then applied to the thresholded wavelet coefficients to yield the finalreconstructed image. As in classical low-pass filtering, zeroing out detail coefficientsremoves high-frequencynoise.However, inwavelet denoising, if the signal itself has ahigh-pass feature, such as a sharp discontinuity, the corresponding detail coefficientwill not be removed out in the thresholding. In this way, wavelet denoising canlow-pass filter the signal while preserving the high-frequency components. Sameis the case with other types of Xlet transforms. Recently, a new method known ascycle spinning has been proposed as an improvement on threshold based on waveletdenoising.

The wavelet transform is not time-invariant. So the estimate of signal which isshifted and then denoised is different than that obtained by without shifting. Thecycle spinning estimate is obtained by linearly averaging these shifted estimates. Theerrors in the individual estimates will not be statistically dependent, and therefore,the averaging will reduce the noise power but will introduce blur and artifacts to


some extent. Wavelet transform in two dimensions is obtained by a tensor productof one dimensional wavelet transform. Wavelet is good at isolating discontinuitiesacross horizontal or vertical edges but it will not achieve the smoothness along thecurved edges.

To overcome the drawback of the wavelet transform, Candes [22] presented anew technique named as ridgelet transform. The basis of ridgelet transform is radontransform. Radon transform has been widely used for tomographic reconstructionusingRadon projections taken at an angle θ ∈ [0, 2π) introduce a redundancy of four.Hence the projection in one quadrant θ ∈ [0, 2π) are used for tomographic represen-tation to reduce computational overheads. Though projections at an angle θ ∈ [0, 2π)do not introduce any redundancy, the achieved tomographic representation is verypoor. The conventional discrete wavelet transform (DWT) introduces artifacts duringprocessing of curves. Finite Ridgelet Transform (FRIT) solved this problem by map-ping the curves in terms of ridges. However, blind application of FRIT all over theimage is computationally very heavy. Finite Curvelet Transform (FCT) selectivelyapplies FRIT only on the tiles containing small portions of a curve. This work aimsat presenting denoising techniques for digital images using different transforms suchas wavelet, ridgelet, curvelet transform contourlet transform and shearlet transform;jointly represented as Xlet transforms or new multiscale transforms.

Other Image Restoration Techniques

5.8.3 Direct Inverse Filtering

The blurring function of the corrupted image is known or can be developed then ithas been proved as quickest and easiest way to restore the distorted image. Blurringcan be considered as low pass filtering in inverse filtering approach and use highpass filtering action to reconstruct the blurred image without much effort. Supposefirst that the additive noise is negligible. A problem arises if it becomes very smallor zero for some point or for a whole region in the plane then in that region inversefiltering cannot be applied.

5.8.4 Constraint Least-Square Filter

Regularized filtering is used in a better way when constraints like smoothness areapplied on the recovered image and very less information is known about the additivenoise. The blurred and noisy image is regained by a constrained least square restora-tion algorithm that uses a regularized filter. Regularized restoration provides almostsimilar results as the wiener filtering but viewpoint of both the filtering techniquesare different. In regularized filtering less previous information is required to apply


restoration. The regularization filter is frequently chosen to be a discrete Laplacian.This filter can be understood as an approximation of a Weiner filter. It is also calledRegularized filter, which is a vector matrix form of linear degradation model.

5.8.5 IBD (Iterative Blind Deconvolution)

Iterative Blind Deconvolution (IBD) was put forward by Ayers and Dainty and it isone of the methods used in blind Deconvolution. This method is based on FourierTransformation causes less computation. Iterative Blind Deconvolution has goodanti-noise capability. In this method image restoration is difficult process whereimage recovery is performed with little or no prior knowledge of the degrading PSF.The Iterative Blind Deconvolution algorithm has higher resolution and better quality.The main drawback of this method is that convergence of the iterative process is notguaranteed. But the original image can have effect on the final result.

5.8.6 NAS-RIF (Nonnegative and Support ConstraintsRecursive Inverse Filtering)

The aim of blind Deconvolution is to reconstruct a reliable estimated image from ablurred image.D.Kundur put forwardNAS-RIF algorithm (Nonnegative andSupportConstraints Recursive Inverse Filtering) to achieve this aim.

NAS-RIF algorithm based on given image make an estimation of target image.The estimation ismade byminimizing an error functionwhich contains the domain ofimage andnonnegative informationof pixels of image.There is a feasible solution thatmakes the error function globally optimized. In theory, the estimation is equivalent tothe real image. The advantage of this algorithm is that we don’t need to know aboutthe parameters of PSF and the priori information of original image, all we have todetermine support domain of target area and to make sure the estimation of imageis nonnegative. Another advantage is that this algorithm contains a process whichmakes sure the function can convergent to global least. The disadvantage of NAS-RIF is that it is sensitive to noise, so it is only proper for images with symmetricalbackground.

5.8.7 Superresolution Restoration Algorithm Basedon Gradient Adaptive Interpolation

The basic idea of the gradient-based adaptive interpolation is that the interpolatedpixel value is affected by the local gradient of a pixel, mainly in edge areas of theimage. The more influence it should have on the interpolated pixel the smaller thelocal gradient of a pixel. The method involves three subtasks: registration, fusion and


deblurring. Firstly we utilize the frequency domain registration algorithm to estimatethe motions of the low resolution images. According to the motion the low resolutionimages are mapped to the uniform high resolution grid, and then the gradient basedadaptive interpolation is used to form a high resolution image. Finally, wiener filteris applied to reduce the effects of blurring and noise caused by the system. The mainadvantage of this algorithm is low computation complexity.

5.8.8 Deconvolution Using a Sparse Prior

This algorithm formulates theDeconvolution problem as given the observation deter-mining the maximum a-posterior estimate of the original image. Furthermore, thealgorithm exploits a prior enforcing spatial-domain sparsely of the image deriva-tives. The resulting non-convex optimization problem is solved using an iterativere-weighted least square method. Although this algorithm has not been nativelydevised for Poisoning observations, it has been rather successfully applied to rawimages. By the selection of the smoothness-weight parameter allowing a sufficientnumber of iterations we can get better result.

5.8.9 Block-Matching

Block-matching is employed to find blocks that contain high correlation becauseits accuracy is significantly impaired by the presence of noise. We utilize a block-similarity measure which performs a coarse initial denoising in local 2D transformdomain. In this method image is divided into blocks and noise or blur is removedfrom each block.

5.8.10 LPA-ICI Algorithm

The LPA-ICI algorithm is nonlinear and spatially-adaptive with respect to thesmoothness and irregularities of the image and blurs operators. Simulation experi-ments demonstrate efficiency and good performance of the proposed Deconvolutiontechnique.

5.8.11 Deconvolution Using Regularized Filter (DRF)

Deconvolution by Regularized filtering is another category of Non-Blind Decon-volution technique. When constraints like smoothness are applied on the recoveredimage and limited information about the noise is known, then regularized Decon-


volution is used effectively. The degraded image is actually restored by constrainedleast square restoration by using a regularized filter. In regularized filtering less priorinformation is required to apply the restoration. Regularization can be a useful tool,when statistical data is unavailable. Moreover, this framework can be extended toadapt to edges of image, noise that is varying spatially and other challenges.

5.8.12 Lucy-Richardson Algorithm

Image restoration method is divided into two types one is blind and other is nonblind deconvolution. The non-blind deconvolution is one in which the PSF is known.The RichardsonLucy deconvolution algorithm has become popular in the fields ofmedical imaging and astronomy. Initially it was found in the early 1970s from byestheorem by Lucy and Richardson. In the early 1980s it was redeliver by Verdi asan algorithm to solve emission tomography imaging problems, in which Poisoningstatistics are dominant. Lucy Richardson is nonlinear iterative method. During thepast two decades, this method have been gaining more acceptance as restorationtool that result in better than those obtained with linear methods. Thus for restoredimage of good quality the Number of iterations is determined manually fore veryimage as per the PSF size. The RichardsonLucy algorithm is an iterative procedurefor recovering a latent image that has been the blurred by Known PSF.

5.8.13 Neural Network Approach

Neural network is a form of multiprocessor computer system, with simple process-ing elements, interconnected group of nodes. These Interconnected components arecalled neurons, which send message to each other. When an element of the neuralnetwork fails, it can continue without any problem by their parallel nature.

ANN provides a robust tool for approximating a target function given a set inputoutput example and for the reconstruction function from a class images. Algorithmsuch as the Back propagation and the Perception use gradient-decent techniques totune the network parameters to best-fit a training set of input output examples. Backpropagation neural network approach for image restoration is capable of learningcomplex non-linear function this method calculate gradient of function with respectto all weight in function.

5.9 Application of Multiscale Transform in ImageRestoration

In this section we will detail review of applications of multiscale transform one byone in image restoration.

5.9 Application of Multiscale Transform in Image Restoration 155

5.9.1 Image Restoration Using Wavelet Transform

Image restoration from corrupted image is a classical problem in the field of imageprocessing. Mainly, image denoising has remained a basic problem in the field ofimage processing. Wavelets give a superior performance in image denoising dueto properties such as sparsity and multiresolution structure. With Wavelet Trans-form gaining popularity in the last two decades various algorithms for denoising inwavelet domain were introduced. The focus was shifted from the Spatial and Fourierdomain to the Wavelet transform domain. Ever since Donohos [53] Wavelet basedthresholding approach was published in 1995, there was a surge in the denoisingpapers being published. Although Donohos concept was not revolutionary, his meth-ods did not require tracking or correlation of the wavelet maxima and minima acrossthe different scales as proposed by Mallat. Thus, there was a renewed interest inwavelet based denoising techniques since Donoho demonstrated a simple approachto a difficult problem. Researchers published different ways to compute the param-eters for the thresholding of wavelet coefficients. Data adaptive thresholds wereintroduced to achieve optimum value of threshold. Later efforts found that substan-tial improvements in perceptual quality could be obtained by translation invariantmethods based on thresholding of anUndecimatedWavelet Transform. These thresh-olding techniques were applied to the nonorthogonal wavelet coefficients to reduceartifacts. Multiwavelets were also used to achieve similar results. Probabilistic mod-els using the statistical properties of the wavelet coefficient seemed to outperform thethresholding techniques. Recently, much effort has been devoted to Bayesian denois-ing in Wavelet domain. Hidden Markov Models and Gaussian Scale Mixtures havealso become popular and more research continues to be published. Tree Structuresordering the wavelet coefficients based on their magnitude, scale and spatial loca-tion have been researched. Data adaptive transforms such as independent componentanalysis (ICA) have been explored for sparse shrinkage. The trend continues to focuson using different statistical models to model the statistical properties of the waveletcoefficients and its neighbors. Future trend will be towards finding more accurateprobabilistic models for the distribution of non-orthogonal wavelet coefficients.

The multi-resolution analysis performed by the wavelet transform has provedto be particularly efficient in image denoising. Since the early use of the classicalorthonormal wavelet transform for removing additive white Gaussian noise throughthresholding, a lot of work has been done leading to some important observations:

1. Better performances can be achieved with shift invariant transformations.2. Directionality of the transform is important in processing geometrical images.3. Further improvements can be obtained with more sophisticated thresholding

functions which incorporate inter-scale and intra-scale dependencies.Variousmethods have been attempted to take advantage of these observations such

as the undecimated discrete wavelet transform (UDWT) (becomes shift-invariantthrough the removal of the down sampling found in the DWT), the double densitydiscrete wavelet transform (DDDWT) (which uses oversampled filters), the dual treediscrete wavelet transform (DT-DWT) (uses two sets of critically sampled filters that


form Hilbert transform pairs) and the double density dual tree wavelet transform(DDDTDWT) (uses two sets of oversampled filters forming complex sub bands togive spatial feature information alongmultiple directions). For estimating the optimalthreshold many procedures have been developed, such as VisuShrink, BayesShrink,SUREshrink, NeighSURE etc. These methods model the denoising problem assum-ing various distributions for the signal coefficients and noise components. The opti-mal threshold is determined based on estimators such as maximum apriority (MAP),maximum absolute deviation (MAD), maximum likelihood estimation (MLE) etc.

Mallat and Hwang [111], in 1992, demonstrated that wavelet transform is particu-larly suitable for the applications of non-stationary signals whichmay spontaneouslyvary in time. They developed an algorithm that removes white noises from signalsby analyzing the evolution of the wavelet transform maxima across scales. In two-dimensions, the wavelet transform maxima indicate the location of edges in images.In addition, they also extended the denoising algorithm for image enhancement.

Donoho [53], in 1995, recovered objects from noisy and incomplete data. Themethod utilized nonlinear operation in wavelet domain and they concluded thatthe recovery of the signals with noisy data is not possible by traditional Fourierapproaches. He proposed the heuristic principles, theoretical foundations and possi-ble application area for denoising.

Xu et al. [192], in 1994, introduced a spatially selective noise filtration techniquebased on the direct spatial correlation of the wavelet transform at several adjacentscales. They used a high correlation to infer that there was a significant feature at theposition that should be passed through the filter.

Donoho and Stone [51, 52] established the thresholding by coining soft-thresholdand hard-threshold wavelet denoising methods. The basic idea was comparisonbetween different scale coefficients module a certain threshold and obtained thede-noised signal by the inverse transform. Although the implementation of thresh-oldingmethodwas simple, it did not take into account the correlation betweenwaveletcoefficients.

Donoho [53], in 1995, proposed a method for signal recovery from noisy data.The reconstruction is defined in the wavelet domain by translating all the empiricalwavelet coefficients of noisy data towards 0 by an amount σ · √

(2ln(n)/n). However,the method lacks the smoothing property. The method of adaption and the method ofproof are bothmore technically discussed by Donoho using soft thresholding whichuses a pyramid filtering. It acts as an unconditional basis for a very wide range ofsmoothness spaces.

Malfait and Roose [110], in 1997, proposed a new method for the suppressionof noise in images via the wavelet transform. The method had relies on two mea-sures. The first is a classic measure of smoothness of the image and is based on anapproximation of the local Holder exponent via the wavelet coefficients. The second,novel measure takes into account geometrical constraints, which are generally validfor natural images. The smoothness measure and the constraints are combined in aBayesian probabilistic formulation, and are implemented as a Markov random field(MRF) image model. The manipulation of the wavelet coefficients is consequentlybased on the obtained probabilities. A comparison of quantitative and qualitative


results for test images demonstrates the improved noise suppression performancewith respect to previous wavelet-based image denoising methods.

Wood and Johnson [186], in 1998, did denoising of synthetic, phantom, and volun-teer cardiac images either in the magnitude or complex domains. Authors suggesteddenoising prior to rectification for superior edge resolution of real and imaginaryimages. Magnitude and complex denoising significantly improved SNR.

Crouse et al. [42], in 1998, developed a novel algorithm for image denoising. Thealgorithm used wavelet domain hidden Markov models (HMM) for statistical signalprocessing. This algorithm was compared with Sureshrink and Bayesian algorithmand showed better result on 1024 length test signal. Least MSE obtained was 0.081.This algorithm was not validated on image but rather on a 1-D test signal so the MSEwas very less.

Strela and Walden [166], in 1998, studied wavelet thresholding in the context ofscalar orthogonal, scalar biorthogonal, multiple orthogonal and multiple biorthogo-nal wavelet transforms. This led to a careful formulation of the universal thresholdfor scalar thresholding and vector thresholding. Multi-wavelets generally outper-formed scalar wavelets for image denoising for all four noisy 1D test images andthe results were visually very impressive. Chui-Lian scaling functions and waveletscombined with repeated row preprocessing appears to be a good general method. Forboth 1D and 2D cases, the reconstructed signals derived from such a good generalmethod demonstratedmuch reduced noise levels, typically 50 percent of the standarddeviation of the original noise.

Mihcak et al. [115], in 1999, introduced a simple spatially adaptive statisticalmodel for wavelet image coefficients, called LAWML and LAWMAP methods,and applied it to image denoising. Their model was inspired by a recent waveletimage compression algorithm, the estimation-quantization (EQ) coder. They pro-duced model wavelet image coefficients as zero-mean Gaussian random variableswith high local correlation. They assumed a marginal prior distribution on waveletcoefficients variances and estimate them using an approximate maximum a posteri-ori probability rule. Then they applied an approximate minimummean squared errorestimation procedure to restore the noisy wavelet image coefficients. Despite thesimplicity of their method, both in its concept and implementation, their denoisingresults were among the best reported in the literature. However, retained too smallwavelet coefficients, severe burr phenomenon appeared in the reconstructed image.

Chang et al. [25], in 2000, organized the paper in two parts. The first part of thispaper proposed an adaptive, data-driven threshold for image denoising via waveletsoft-thresholding. The threshold was derived in a Bayesian framework, and the priorused on the wavelet coefficients was the generalized Gaussian distribution (GGD)widely used in image processing applications. The proposed threshold was simpleand closed-form, and it was adaptive to each subband because it depends on data-driven estimates of the parameters. Experimental results had shown that the proposedmethod, called BayesShrink, was typically within 5 percent of the MSE of the bestsoft-thresholding benchmark with the image assumed known. It also outperformedDonoho and Johnstones SureShrink most of the time.


The second part of the paper attempted to further validate recent claimed thatlossy compression could be used for denoising. The BayesShrink threshold couldaided in the parameter selection of a coder designed with the intention of denoising,and thus achieving simultaneous denoising and compression. Specifically, the zero-zone in the quantization step of compression was analogous to the threshold valuein the thresholding function. The remaining coder design parameters were chosenbased on a criterion derived from Rissanens minimum description length (MDL)principle. Experiments show that this compression method did indeed remove noisesignificantly, especially for large noise power. However, it introduced quantizationnoise and should be used only if bitrate were an additional concern to denoising.

Rosenbaum et al. [147], in 2000, used wavelet shrinkage denoising algorithmsand Nowak’s algorithm for denoising the magnitude images. The wavelet shrinkagedenoising methods were performed using both soft and hard thresholding and it wassuggested that changes in mean relative SNR are statistically associated with typeof threshold and type of wavelet. The data-adaptive wavelet filtering was found toprovide the best overall performance as compared to direct wavelet shrinkage.

Chang et al. [26, 27] recovered image frommultiple noisy copies by combining thetwo operations of averaging and thresholding. Averaging followed by thresholding orthresholding followed by averaging produces different estimators. The signalwaveletcoefficients are modeled as Laplacian and the noise is modeled as Gaussian. Fourstandard images are used for performance comparison with other standard denoisingtechniques. Barbara image denoised by the proposed algorithm have least MSE of51.27. Along with Barbara image, three other test images are used to evaluate theperformance of this algorithm.

Pizurica et al. [134], in 2002, presented a new wavelet-based image denoisingmethod, which extended a recently emerged geometrical Bayesian framework. Thenew method combined three criteria for distinguishing supposedly useful coeffi-cients from noise: coefficient magnitudes, their evolution across scales and spatialclustering of large coefficients near image edges. These three criteria were com-bined in a Bayesian framework. The spatial clustering properties were expressedin a prior model. The statistical properties concerning coefficient magnitudes andtheir evolution across scales were expressed in a joint conditional model. The threemain novelties with respect to related approaches were (1) the interscale-ratios ofwavelet coefficients were statistically characterized and different local criteria fordistinguishing useful coefficients from noise were evaluated, (2) a joint conditionalmodel was introduced, and (3) a novel anisotropic Markov random field prior modelwas proposed. The results demonstrated an improved denoising performance overrelated earlier techniques.

Sender and Selesnick [152], in 2002, proposed new non-Gaussian bivariate dis-tributions only for the dependencies between the coefficients and their parents indetail since most simple nonlinear thresholding rules for wavelet-based denoisingassume that the wavelet coefficients are independent, however, wavelet coefficientsof natural images have significant dependencies and corresponding nonlinear thresh-old functions (shrinkage functions) were derived from the models using Bayesianestimation theory. The new shrinkage functions did not assume the independence of


wavelet coefficients. They showed three image denoising examples in order to showthe performance of these new bivariate shrinkage rules. In the second example, asimple subband-dependent data-driven image denoising system was described andcompared with effective data-driven techniques in the literature, namely VisuShrink,SureShrink, BayesShrink, and hiddenMarkovmodels. In the third example, the sameidea was applied to the dual-tree complex wavelet coefficients.

Regularization is achieved by promoting a reconstruction with low-complexity,expressed in the wavelet coefficients, taking advantage of the well-known sparsity ofwavelet representations. Previous works have investigated wavelet-based restorationbut, except for certain special cases, the resulting criteria are solved approximatelyor require demanding optimization methods.

High-resolution image reconstruction refers to the reconstruction ofhigh-resolution images from multiple low-resolution, shifted, degraded samples ofa true image. Chan et al. [28], in 2003 analyzed this problem from the wavelet pointof view. By expressing the true image as a function in L(R2), they derived itera-tive algorithms which recover the function completely in the L sense from the givenlow-resolution functions. These algorithms decomposed the function obtained fromthe previous iteration into different frequency components in the wavelet transformdomain and add them into the new iterate to improve the approximation. We applywavelet (packet) thresholding methods to denoised the function obtained in the pre-vious step before adding it into the new iterate. Their numerical results showed thatthe reconstructed images from our wavelet algorithms are better than that from theTikhonov least-squares approach.

Figueiredo and Nowak [64], in 2003, introduced an expectation maximization(EM) algorithm for image restoration based on a penalized likelihood formulated inthe wavelet domain. The proposed EM algorithm combines the efficient image repre-sentation offered by the discrete wavelet transform (DWT) with the diagonalizationof the convolution operator obtained in the Fourier domain. The algorithm substi-tutes between an E-step based on the fast Fourier transform (FFT) and a DWT-basedM-step, resulting in an efficient iterative process requiringO(N logN ) operations periteration. This approach performed competitively with, in some cases better than, thebest existing methods in benchmark tests.

Argenti and Torricelli [10], in 2003, assumedWiener-like filtering and performedin a shift-invariant wavelet domain by means of an adaptive rescaling of the coef-ficients of undecimated octave decomposition calculated from the parameters ofthe noise model, and the wavelet filters. The proposed method found in excellentbackground smoothing as well as preservation of edge sharpness and well details.LLMMSE evaluation in an undecimated wavelet domain tested on both ultrasonicimages and synthetically speckled images demonstrated an efficient rejection of thedistortion due to speckle.

Portilla et al. [137], in 2003, proposed a method for removing noise from digitalimages, based on a statisticalmodel of the coefficients of an over-completemultiscaleoriented basis. Neighborhoods of coefficients at adjacent positions and scales weremodeled as the product of two independent random variables: a Gaussian vector anda hidden positive scalar multiplier. The latter modulated the local variance of the


coefficients in the neighborhood, and was thus able to account for the empiricallyobserved correlation between the coefficients amplitudes. Under this model, theBayesian least squares estimate of each coefficient reduced to a weighted average ofthe local linear estimates over all possible values of the hidden multiplier variable.They demonstrated through simulations with images contaminated by additive whiteGaussian noise that the performance of this method substantially surpassed that ofpreviously published methods, both visually and in terms of mean squared error.

Synthetic aperture radar (SAR) images are inherently affected by multiplicativespeckle noise, which is due to the coherent nature of the scattering phenomenon.Achim et al. [3], in 2003, proposed a novel Bayesian-based algorithm within theframework of wavelet analysis, which reduces speckle in SAR images while preserv-ing the structural features and textural information of the scene. Firstly they showedthat the subband decompositions of logarithmically transformed SAR images wereaccurately modeled by alpha-stable distributions, a family of heavy-tailed densities.Consequently, they exploited this a priori information by designing a maximum aposteriori (MAP) estimator. They used the alpha-stable model to develop a blindspeckle-suppression processor that performed a nonlinear operation on the data andthey related this nonlinearity to the degree of non-Gaussianity of the data. Finally,they compared their proposed method to current state-of-the-art soft thresholdingtechniques applied on real SAR imagery and they quantified the achieved perfor-mance improvement.

Xie et al. [189], in 2004, using theMDL principle, provided the denoising methodbased on a doubly stochastic process model of wavelet coefficients that gave a newspatially varying threshold. This method outperformed the traditional thresholdingmethod in both MSE error and compression gain.

Wink and Roerdink [185], in 2004, estimated two denoising methods for thesimulation of an fMRI series with a time signal in an active spot by the averagetemporal SNR inside the original activated spot and by the shape of the spot detectedby thresholding the temporal SNR maps. These methods were found to be bettersuited for low SNRs but they were not preferred for reasonable quality images asthey introduced heavy decompositions. Therefore, wavelet based denoising methodswere used since they preserved sharpness of the images, from the original shapes ofactive regions as well and produced a smaller total number of errors than Gaussiannoise. But both Gaussian and wavelet based smoothing methods introduced severedeformations and blurred the edges of the active mark. For low SNR both techniquesare found to be on similarity. For high SNRWavelet methods are better thanGaussianmethod, giving a maximum output of above 10db.

Choi and Baranuik [35], in 2004, defined Besov Balls (a convex set of imageswhose Besov norms are bounded from above by their radii) in multiple waveletdomains and projected them onto their intersection using the projection onto convexsets (POCS) algorithm. It resembled to a type of wavelet shrinkage for image denois-ing. This algorithm provided significant improvement over conventional waveletshrinkage algorithm, based on a single wavelet domain such as hard thresholding ina single wavelet domain.


Yoon and Vaidyanathan [199], in 2004, offered the custom thresholding schemeand demonstrated that it outperformed the traditional soft and hard-thresholdingschemes.

Hsung et al. [75], in 2005, improved the traditional wavelet method by apply-ing Multivariate Shrinkage on multiwavelet transform coefficients. Firstly a simple2nd order orthogonal pre filter design method was used for applying multiwaveletof higher multiplicities (preserving orthogonal pre-filter for any multiplicity). Thenthreshold selections were studied using Steins unbiased risk estimator (SURE) foreach resolution point, provided the noise constitution is known. Numerical experi-ments showed that a multivariate shrinkage of higher multiplicity usually gave betterperformance and the proposed LSURE substantially outperformed the traditionalSURE in multivariate shrinkage denoising, mainly at high multiplicity.

Zhang et al. [201], in 2005, proposed a wavelet-based multiscale linear minimummean square-error estimation (LMMSE) scheme for image denoising and the deter-mination of the optimal wavelet basis with respect to the proposed scheme was alsodiscussed. The overcompletewavelet expansion (OWE), which ismore effective thanthe orthogonal wavelet transform (OWT) in noise reduction, was used. To explore thestrong interscale dependencies of OWE, they combined the pixels at the same spatiallocation across scales as a vector and apply LMMSE to the vector. Compared withthe LMMSE within each scale, the interscale model exploited the dependency infor-mation distributed at adjacent scales. The performance of the proposed scheme wasdependent on the selection of the wavelet bases. Two criteria, the signal informationextraction criterion and the distribution error criterion, were proposed to measure thedenoising performance. The optimal wavelet that achieves the best tradeoff betweenthe two criteria could be determined from a library of wavelet bases. To estimatethe wavelet coefficient statistics precisely and adaptively, they classified the waveletcoefficients into different clusters by context modeling, which exploited the waveletintrascale dependency and yields a local discrimination of images. Experiments showthat the proposed scheme outperforms some existing denoising methods.

Selesnick et al. [151], in 2005, developed a dual-tree complex wavelet transformwith important additional properties such as shift invariant and directional selectiv-ity at higher dimensions. The dual-tree complex wavelet transform is non-separablebut is based on computationally efficient separable filter bank. Kingsbury provedhow complex wavelets with good properties illustrate the range of applications suchas image denoising, image rotation, estimating image geometrical structures, esti-mating local displacement, image segmentation, image sharpening and many moreapplications.

Cho and Bui [34], in 2005, proposed the multivariate generalized Gaussian dis-tribution model, which adjusts different parameters and can include Gaussian, gen-eralized Gaussian, and non-Gaussian model, but parameters estimation was morecomplex in image denoising process.

Deconvolution of images is an ill-posed problem, which is very often tackledby using the diagonalization property of the circulant matrix in the discrete Fouriertransform (DFT) domain. On the other hand, the discrete wavelet transform (DWT)


has shown significant success for image denoising because of its space-frequencylocalization.

Koc and Ercelebi [89], in 2006, proposed a method of applying a lifting basedwavelet domain e-median filter (LBWDEMF) for image restoration. The proposedmethod transforms an image into the wavelet domain using lifting-based waveletfilters, then applies an e-median filter in the wavelet domain, transforms the resultinto the spatial domain, and finally goes through one spatial domain e-median filterto produce the final restored image. They compared the result obtained using theproposed method to those using a spatial domain median filter (SDMF), spatialdomain e-median filter (SDEMF), and wavelet thresholding method. Experimentalresults showed that the proposedmethodwas superior to SDMF,SDEMF, andwaveletthresholding in terms of image restoration.

Sudha et al. [167], in 2007, described a new method for suppression of noisein image by fusing the wavelet Denoising technique with optimized thresholdingfunction, improving the denoised results significantly. Simulated noise images wereused to evaluate the denoising performance of proposed algorithm alongwith anotherwavelet-based denoising algorithm. Experimental result showed that the proposeddenoising method outperformed standard wavelet denoising techniques in terms ofthe PSNR and the preservation of edge information. Thy had compared this withvarious denoising methods like wiener filter, Visu Shrink, Oracle Shrink and BayesShrink.

Luisier et al. [108], in 2007, introduced a new approach to orthonormal waveletimage denoising. Instead of postulating a statistical model for the wavelet coeffi-cients, we directly parametrize the denoising process as a sum of elementary non-linear processes with unknown weights. We then minimize an estimate of the meansquare error between the clean image and the denoised one. The key point wasthat we had at our disposal a very accurate, statistically unbiased, MSE estimate,Steins unbiased risk estimate, that depends on the noisy image alone, not on theclean one. Like the MSE, this estimate was quadratic in the unknown weights, andits minimization amounts to solving a linear system of equations. The existenceof this a priori estimate made it unnecessary to devise a specific statistical modelfor the wavelet coefficients. Instead, and contrary to the custom in the literature,these coefficients were not considered random anymore. We described an interscaleorthonormal wavelet thresholding algorithm based on this new approach and showedits near-optimal performance, both regarding quality and CPU requirement, by com-paring it with the results of three state-of-the-art nonredundant denoising algorithmson a large set of test images. An interesting fallout of this study was the developmentof a new, group-delay-based, parentchild prediction in a wavelet dyadic tree.

Giaouris and Finch [68], in 2008, presented that the denoising scheme based onthe Wavelet Transform did not distort the signal and the noise component after theprocess was found to be small.

Rahman et al. [138], in 2008, proposed a hybrid-type image restoration algorithmthat takes the advantage of the diagonalization property of the DFT of a circulantmatrix during the deconvolution process and space-frequency localization propertyof the DWT during the denoising process. The restoration algorithm was operated


iteratively and switched between two different domains, viz., the DWT and DFT.For the DFT-based deconvolution, they used the least-squares method, wherein reg-ularization parameter was estimated adaptively. For the DWT-based denoising, aMAP estimator that modeled the local neighboring wavelet coefficients by the Gram-Charlier PDF, had been used. The proposed algorithm showed a well convergenceand experimental results on standard images showed that the proposed method pro-vides better restoration performance than that of several existing methods in termsof signal-to-noise ratio and visual quality.

Poornachandra [136], in 2008, used the wavelet-based denoising for the recoveryof signal contaminated by white additive Gaussian noise and investigated the noisefree reconstruction property of universal threshold.

Gnanadurai and Sadasivam [70], in 2008, proposedmethod in which the choice ofthe threshold estimation was carried out by analysing the statistical parameters of thewavelet subband coefficients like standard deviation, arithmetic mean and geometri-cal mean. This frame work described a computationally more efficient and adaptivethreshold estimation method for image denoising in the wavelet domain based onGeneralized Gaussian Distribution (GGD) modeling of subband coefficients. Thenoisy image was first decomposed into many levels to obtain different frequencybands. Then soft thresholding method was used to remove the noisy coefficients, byfixing the optimum thresholding value by the proposedmethod. Experimental resultson several test images by using this method showed that this method yields signifi-cantly superior image quality and better Peak signal-to-noise ratio (PSNR). To provethe efficiency of this method in image denoising, they compared this with variousdenoising methods like wiener filter, Average filter, VisuShrink and BayesShrink.

Anbarjafari and Demirel [6] proposed a new super-resolution technique basedon interpolation of the high-frequency subband images obtained by discrete wavelettransform (DWT) and the input image. The proposed technique usedDWT to decom-pose an image into different subband images and then the high-frequency subbandimages and the input low-resolution image had been interpolated, followed by com-bining all these images to generate a new super-resolved image by using inverseDWT. The proposed technique had been tested on Lena, Elaine, Pepper, and Baboonand the quantitative peak signal-to-noise ratio (PSNR) and visual results showed thesuperiority of the proposed technique over the conventional and state-of-art imageresolution enhancement techniques. For Lenas image, the PSNR was 7.93dB higherthan the bicubic interpolation.

Firoiu et al. [65] presented a Bayesian approach of wavelet based image denois-ing. They proposed the denoising strategy in two steps. In the first step, the image isdenoised using association of bishrink filter hyper analyticwavelet transform (HWT)computed with intermediate wavelets, for example Daubechies-12. In the secondstep, the same denoising approach is followed with only the difference in the applica-tion of the bishrink filter HWT using the mother wavelets Daubechies-10. Orthonor-mal bases of compactly supported wavelets, with arbitrarily high regularity are con-structed. The order of high regularity increases linearly with the support width givenby Daubechies (1998). PSNR improvement of 6.67dB is observed for Lena image.


Motwani et al. [119] described different methodologies for noise reduction (ordenoising) giving an insight as to which algorithm should be used to find the mostreliable estimate of the original image data given its degraded version.

Mathew and Shibu [114] developed a technique for Reconstruction of super res-olution image using low resolution natural color image. The presented techniqueidentified local features of low resolution image and then enhanced its resolutionappropriately. It is noticed that the higher PSNR was observed for the developedtechnique than the existing methods.

Vohra and Tayal [177] analyzed the image de-noising using discrete wavelet trans-form. The experiments were conducted to study the suitability of different waveletbases and also different window sizes. Among all discrete wavelet bases, coiflet per-forms well in image de-noising. Experimental results also showed that Sureshrinkprovided better result than Visushrink and Bayesshrink as compared to Weiner filter.

Later [2010, 2011], many effective image denoising methods based on this modelcombining different transforms are obtained.

Wang et al. [180] in 2011 proposed a simple model of pixel pattern classifier withorientation estimation module to strengthen the robustness of denoising algorithm.Moreover, instead of determining the transform strategy, sub-blocks, robust adap-tive directional lifting (RADL) algorithm is performed at each pixel level to pursuebetter denoising results. RADL is performed only on pixels belonging to textureregions thereby reducing artifacts and improving performance of the algorithm. ThePeak Signal to Noise Ratio (PSNR) improvement on Barbara image is 6.66dB andSSIM index improvement is 0.355. Six different images are used to evaluate theperformance of this algorithm.

Mohideen et al. [118] in 2011 compared the wavelet and multi wavelet techniqueto produce the best denoised mammographic images using efficient multi waveletalgorithm. Mammographic images are denoised and enhanced using multi waveletwith hard thresholding. Initially the images are pre-processed to improve its localcontrast and discrimination of faint details. Image suppression and edge enhancementare performed. Edge enhancement is performed based onmulti wavelet transform. Ateach resolution, coefficients associated with the noise are modeled and generalizedby Laplacian random variables. The better denoising results depend on the degreeof noise; generally its energy is distributed over low frequency band while both itsnoise and details are distributed over high frequency band. Also the applied hardthreshold in different scale of frequency sub bands limits the performance of imagedenoising algorithms.

Ruikar and Doye [148] in 2011 proposed different approaches of wavelet basedimage denoising methods. Themain aim of authors was to modify the wavelet coeffi-cients in the new basis, the noise could be removed from the data. They extended theexisting technique and providing a comprehensive evaluation of the proposedmethodby using different noise, such as Gaussian, Poissons, Salt and Pepper, and Speckle.A signal to noise ratio as a measure of the quality of denoising was preferred.

Liu et al. [104] in 2012 proposed a denoising method based on wavelet thresh-old and subband enhancement method for image de-noising. This method used softthreshold method for the minimum scale wavelet coefficients, takes further decom-


posing for other wavelet coefficient and takes effective enhancement and mixingthreshold processing for each subband after being decomposed. Thus making fulluse of high frequency information of each of the multi-dimension could add imagedetails and got a better enhancement and de-noising effectively.

Kumar and Saini [94] in 2012 suggested some new thresholdingmethod for imagedenoising in the wavelet domain by keeping into consideration the shortcomings ofconventional methods and explored the optimal wavelet for image denoising. Theyproposed a computationally more efficient thresholding scheme by incorporating theneighbouring wavelet coefficients, with different threshold value for different subbands and it was based on generalized Gaussian Distribution (GGD)modeling of subband coefficients. In here proposedmethod, the choice of the threshold estimationwascarried out by analyzing the statistical parameters of thewavelet sub band coefficientslike standard deviation, arithmetic mean and geometrical mean. It was demonstratedthat their proposed method performs better than: VisuShrink, Normalshrink andNeighShrink algorithms in terms of PSNR ratio. Further a comparative analysis hadbeen made between Daubechies, Haar, Symlet and Coiflet wavelets to explore theoptimum wavelet for image denoising with respect to Lena image. It had been foundthat with Coiflet wavelet higher PSNR ratio was achieved than others.

Naik and Patel [122] in 2013 presented single image super resolution algorithmbased on both spatial and wavelet domain. Their algorithm was iterative and usedback projection tominimize reconstruction error. They also introducedwavelet baseddenoising method to remove noise. PSNR ratio and visual quality of images werealso showed the effectiveness of algorithm.

Wavelets gave a superior performance in image denoising due to its propertiessuch asmulti-resolution.Non-linearmethods especially those based onwavelets havebecome popular due to its advantages over linear methods. Abdullah Al Jumah [1] in2013, applied non-linear thresholding techniques in wavelet domain such as hard andsoft thresholding, wavelet shrinkages such as Visu-shrink (nonadaptive) and SURE,Bayes and Normal Shrink (adaptive), used Discrete Stationary Wavelet Transform(DSWT) for different wavelets at different levels, denoised an image and determinedthe best one out of them. Performance of denoising algorithm was measured usingquantitative performance measures such as Signal-to-Noise Ratio (SNR) and MeanSquare Error (MSE) for various thresholding techniques.

There are two disadvantages in variational regularization based image restorationmodel. Firstly, the restored image is susceptible to noise because the diffusion coef-ficient depends on image gradient. Secondly, in the process of energy minimization,the selection of Lagrange multiplier λ which is used to balance the regular term andthe fidelity term can directly affects the quality of the restored image.

Li et al. [102] in 2014 introduced multiresolution feature of multiscale waveletinto the energy minimization model and proposed a wavelet based image restora-tion model to solve the above problems. They replaced Lagrange multiplier λ by anadaptive weighting function λj in their proposed model, which is constructed by theimage wavelet transform coefficients. Experimental results and theoretical analysisshowed that the proposed model reduced iterations in the energy minimization pro-cess overcome the cartoon effects in the variational model and pseudo-Gibbs effect


in traditional wavelet thresholdmethods and canwell protect the detail features whiledenoising.

Neha and Khera [124] in 2014 proposed new technique which is combinationof Enhanced Empirical Mode Decomposition (EEMD), has been presented alongwith the standard wavelet thresholding techniques like hard thresholding to denoisethe image. Authors presented a comparative analysis of various image denoisingtechniques using wavelet transforms and a lot of combinations had been applied inorder to find the best method that can be followed for denoising intensity images.

Varinderjit et al. [174] in 2014 defined a general mathematical and experimentalmethodology to compare and classify classical image de-noising algorithms andproposed a nonlocal means (NL-means) algorithm addressing the preservation ofstructure in a digital image. The mathematical analysis was based on the analysis ofthe method noise, defined as the difference between a digital image and its de-noisedversion.

Chouksey et al. [37] in 2015 provided denoising scheme with a wavelet interscalemodel based on Minimum mean square error (MMSE) and discussed the optimizewavelet basis from family of wavelet. In their proposed method, the wavelet coef-ficients at the same spatial locations at two adjacent scales were represented as avector with orthogonal wavelet transform and the orthogonal wavelet based withmmse was applied to the vector for enhancing the peak signal to noise ratio (PSNR).The New algorithm filter also showed reliable and stable performance across a dif-ferent range of noise densities varying from 10% to 90%. The performance of theproposed method had been tested at low, medium and high noise densities on grayscales and at high noise density levels the new proposed algorithm provided betterperformance as compare with other existing denoising filters.

By considering the problem of generating a super-resolution (SR) image from asingle low resolution (LR) input image in the wavelet domain, Rakesh et al. [141]in 2015 proposed an intermediate stage for estimating the high frequency (HF) subbands to achieve a sharper image. Experimental results indicated that the proposedapproach outperforms existing methods in terms of resolution.

Gadakh and Thorat [67] in 2015 presented a new and fast method for removal ofnoise and blur from Magnetic Resonance Imaging (MRI) using wavelet transform.They utilized a fact that wavelets can represent magnetic resonance images well,with relatively few coefficients. They used this property to improve MRI restorationwith arbitrary k-space trajectories. They showed that their non-linear method wasperforming fast than other regularization algorithms.

Wagadre and Singh [178] in 2016 described a method to remove the motion blurpresent in the image taken from any cameras by which motion blurred. They restorednoisy image using Wiener and Lucy Richardson method then applied wavelet basedfusion Technique for restoration. The performance of the every stage was tabulatedfor the parameters like SNR and RMSE of the restored images and it has beenobserved that image fusion technique provided better results as compared to previoustechniques.

Sowmya et al. [160] in 2016 proposed a new image resolution enhancementalgorithm based on discrete wavelet transform (DWT), lifting wavelet transform


(LWT) and sparse recovery of the input image. Firstly, a single low resolution (LR)was decomposed into different subbands using two operators DWT and LWT. Inparallel, the LR image was subjected to a sparse representation interpolation. Finally,The higher frequency sub-bands in addition to the sparse interpolated LR imagewere combined to give a high resolution (HR) image using inverse discrete wavelettransform (IDWT). The qualitative and quantitative analysis of our method showedprominence over the conventional and various state-of-the art super resolution (SR)techniques.

Rakheja and Vig [142] in 2016 combined neighborhood processing techniqueswith wavelet Transform for image denoising and simulated results showed that thecombined algorithm performs better than both individually. They obtained simulatedresults for Gaussian, Speckle and Salt & Pepper noise, for denoising median filter ofsize 3X3, 5X5 and discrete wavelet Transformwere used. Then results obtained wereevaluated on the basis of Peak signal to noise ratio which has improved remarkably.

Leena et al. [100] in 2016 presented a methodology to denoise an image based onleast square approach using wavelet filters. This work was the extension of the onedimensional signal denoising approach based on least square (proposed bySelesnick)to two dimensional image denoising. In their proposed work, the matrix constructedusing second order filter in the least square problem formulation was replaced withthe wavelet filters. The performance of the proposed algorithmwas validated throughPSNR. From the results of PSNR values, it was evident that the proposedmethod per-forms equally well as the existing second order filter. The advantage of the proposedmethod lies in the fact that it was simple and involves low mathematical complexity.

Thangadurai and Patrick [170] in 2017 proposed new set of blur invariant descrip-tors. These descriptors have been advanced in the wavelet domain 2D and 3D imagesto be invariant to centrally symmetric blur. First, Image registration was done usingwavelet domain blur invariants. The method uses Daubuchies and B-spline waveletfunction which was used to construct blur invariants. The template image was cho-sen from the degraded image. The template images and the original images werematched with its similarities. This wavelet domain blur invariants accurately registeran image compared to spatial domain blur invariants which might result in misfocusregistration of an image. Despite of the presence of harmful blurs, the image regis-tration has been correctly performed. The experiments carried out by using SDBIs,were failed in some of the image registration. Then regression based process is doneabout to produce an image convolved with near diffraction limited PSF, which canbe shown as blur invariant. Eventually a blind deconvolution algorithm is carriedout to remove the diffraction limited blur from fused image the final output. Finally,image was restored by using blind deconvolution algorithm and also PSNR valuesare calculated. Hence the image quality was improved by using proposed method.

Sushil Kumar [97] in 2017 proposed a comparative study of image denoisingmethod using BlockShrink algorithm between theWavelet transform (DWT) and theSlantlet transform (SLT). Slantlet transform, which is also a wavelet-like transformand a better candidate for signal compression compared to the DWT based schemeand which can provide better time localization. BlockShrink was found to be a bettermethod than other conventional image denoising methods. It was found that DWT


based BlockShrink thresholding was better option than BiShrink thresholding interms of PSNR for the Gaussian noise. The PSNR for the SLT based BlockShrink,though found to be less than DWT based method, was better option as it providedbetter time localization and better signal compression compared to the classicalDWT.

Generally the Gaussian and salt Pepper noise occurred in images of differentquality due to random variation of pixel values. It is necessary to apply variousfiltering techniques to denoise these images. There are lots of filtering methods pro-posed in literature which includes the haar, sym4, and db4 Wavelet Transform basedsoft and hard thresholding approach to denoise such type of noisy images. Chaud-hari and Mahajan [29] in 2017 analysed exiting literature on haar, db4 and sym4Wavelet Transform for image denoising with variable size images from self gen-erated grayscale database generated from various image sources such as satelliteimages(NASA), Engineering Images and medical images. However this new pro-posed Denoising method showed satisfactory performances with respect to existingliterature on standard indices like Signal-to-Noise Ratio (SNR), Peak Signal to NoiseRatio (PSNR) and Mean Square Error (MSE). Wavelet coefficient could be used toimprove quality of true image and from noise since wavelet transform representsnatural image better than any other transformations. The main aim of this work wasto eliminate the Gaussian and salt Pepper noise in wavelet transform domain. Sub-sequently a soft and hard threshold based denoising algorithm had been developed.Finally, the denoised image was compared with original image using some quanti-fying statistical indices such as MSE, SNR and PSNR for different noise variancewhich the experimental results demonstrate its effectiveness over previous method.

Wavelet transform is an effective method for removal of noise from image. Buttraditional wavelet transform cannot improve the smooth effect and reserve imagesprecise details simultaneously, even false Gibbs phenomenon can be produced.Wanget al. [179] in 2017 proposed a new image denoising method based on adaptive mul-tiscale morphological edge detection beyond the above limitation. Firstly, the noisyimage was decomposed by using one wavelet base, then the image edge was detectedby using the adaptive multiscale morphological edge detection based on the waveletdecomposition. On this basis, wavelet coefficients belonging to the edge positionwere dealt with the improved wavelet domain wiener filtering and the others weredealt with the improved Bayesian threshold and the improved threshold function.Finally, wavelet coefficients were inversely processed to obtain the denoised image.This method provided the better result from existing and this method can effectivelyremove the image noise without blurring edges and highlight the characteristics ofimage edge at the same time.

Ramadhan et al. [144] in 2017 proposed and tested a new method of imagede-noising based on usingmedian filter (MF) in the wavelet domain. Various types ofwavelet transform filters were used in conjunction with median filter in experiment-ing with the proposed approach in order to obtain better results for image de-noisingprocess, and, consequently to select the best suited filter. Wavelet transform workingon the frequencies of sub-bands split from an image was a powerful method foranalysis of images. Experimental work showed that the proposed method provided


better results than using only wavelet transform or median filter alone. The MSE andPSNR values were used for measuring the improvement in de-noised images.

Neole et al. [125] in 2017 presented a novel approach to image denoising usingedge profile detection and edge preservation in spatial domain in presence of zeromean additive Gaussian noise. A Noisy image was initially preprocessed using theproposed local edge profile detection and subsequent edge preserving filtering inspatial domain followed further by the modified threshold bivariate shrinkage algo-rithm. The proposed technique did not require any estimate of standard deviation ofnoise present in the image. Performance of the proposed algorithm was presentedin terms of PSNR and SSIM on a variety of test images containing a wide range ofstandard deviation starting from 15 to 100. The performance of the proposed algo-rithm was much better than NL means and Bivariate Shrinkage while its comparablewith BM3D.

5.9.2 Image Restoration Using Complex Wavelet Transform

The classical discrete wavelet transform (DWT) provides a means of implementinga multiscale analysis, based on a critically sampled filter bank with perfect recon-struction. However, questions arise regarding the good qualities or properties of thewavelets and the results obtained using these tools, the standard DWT suffers fromthe following problems described as below:

1. Shift sensitivity: It has been observed that DWT is seriously disadvantaged bythe shift sensitivity that arises from down samples in the DWT implementation.

2. Poor directionality: An m-dimension transform (m > 1) suffers poor direc-tionality when the transform coefficients reveal only a few feature in the spatialdomain.

3. Absence of phase information: Filtering the image with DWT increases itssize and adds phase distortions; human visual system is sensitive to phase distortion.Such DWT implementations cannot provide the local phase information.

It is found that the above problems can be solved effectively by the complexwavelet transform (CWT). The structure of the CWT is the same as in DWT, exceptthat the CWT filters have complex coefficients and generate complex output sam-ples. However, a further problem arises here because perfect reconstruction becomesdifficult to achieve for complex wavelet decompositions beyond level 1, when theinput to each level becomes complex. For many applications it is important that thetransformmust be perfectly invertible. A few authors, such as Lawton [98] andBelzeret al. [15], have experimentedwith complex factorizations of the standardDaubechiespolynomials and obtained PR complex filters, but these do not give filters with goodfrequency-selectivity properties. To provide shift invariance and directional selec-tivity, all of the complex filters should emphasize positive frequencies and rejectnegative frequencies, or vice versa. Unfortunately, it is very difficult to design aninverse transform, based on complex filters which has good frequency selectivityand PR at all levels of the transform.


To overcome this problem, in 1998, Kingsbury [86, 87] developed the dual-treecomplex wavelet transform (DTCWT), which added perfect reconstruction to theother attractive properties of complex wavelets: shift invariance; good directionalselectivity; limited redundancy; and efficient order-N computation.

The dual-tree transformwas developed by noting that approximate shift invariancecan be achieved with a real DWT by doubling the sampling rate at each level of thetree. Now we are going to survey of image denoising techniques using dual treecomplex wavelet transforms.

Achim and Kuruoglu [2] in 2005, described a noval for removing noise fromdigital images in the dual-tree complex wavelet transform framework. They designeda bivariate maximum a posteriori estimator, which relies on the family of isotropicα-stable distributions.Using this relatively new statisticalmodel theywere able to bettercapture the heavy-tailed nature of the data as well as the interscale dependencies ofwavelet coefficients.

Chitchian et al. [33] in 2009, applied a locally adaptive denoising algorithm toreduce speckle noise in time-domain optical coherence tomography (OCT) imagesof the prostate. The algorithmwas illustrated using DWT and DTCWT. Applying theDTCWT provided improved results for speckle noise reduction in OCT images. Thecavernous nerve and prostate gland could be separated from discontinuities due tonoise, and image quality metrics improvements with a signal-to-noise ratio increaseof 14dB14dB were attained.

Xingming and Jing [189] in 2009 proposed a novel method based on HMTmodelby the use of Fourier Wavelet Regulation Deconvolution (ForWaRD) algorithmand compared with some conventional image restoration algorithms using com-plex wavelets. In the proposed method, they first applied the Wiener filter on theblurring image in the Fourier domain, and then used the hidden Markov tree model(HMT) to remove the unwanted noise in wavelet domain. Simulations for solvingthe typical convolution and noised linear degraded model were made, in which theperformances based complex wavelets and real orthogonal wavelets were comparedin detail. Experimental results showed that the suggested method using complexwavelets performed better in the view of visual effects and objective criterion thanthe conventional methods.

Wang et al. [182] in 2010, proposed a technique based on the dual-tree complexwavelet transform (DTCWT) to enhance the desired features related to some specialtype of machine fault. Since noise inevitably exists in the measured signals, theydeveloped an enhanced vibration signals denoising algorithm incorporatingDTCWTwith NeighCoeff shrinkage. Denoising results of vibration signals resulting from acrack gear indicate the proposed denoising method can effectively remove noise andretain the valuable information as much as possible compared to those DWT- andSGWT-based NeighCoeff shrinkage denoising methods.

Sathesh and Manoharan [150] in 2010, proposed a image denoising techniqueusing Dual Tree Complex Wavelet Transform (DTCWT) along with soft threshold-ing.

In Medical diagnosis operations such as feature extraction and object recognitionwill play the key role. These tasks will become difficult if the images are corrupted


with noise. Raja and Venkateswarlub [139] in 2012 proposed the denoising methodwhich used dual tree complex wavelet transform to decompose the image and shrink-age operation to eliminate the noise from the noisy image. In the shrinkage stepThey used semi-soft and stein thresholding operators along with traditional hard andsoft thresholding operators and verified the suitability of dual tree complex wavelettransform for the denoising of medical images. Their results proved that the denoisedimage using Dual Tree Complex Wavelet Transform (DTCWT) had a better balancebetween smoothness and accuracy than the DWT and less redundant than Undeci-matedWavelet Transform (UDWT). They used the SSIM along with PSNR to assessthe quality of denoised images.

Kongo et al. [90] in 2012 presented a new denoising method for ultrasound med-ical image in restoration domain. The approach was based on analysis in Dual-treewavelet Transform (DT-CWT). Various methods had been developed in the liter-ature; most of them used only the standard wavelet transform (DWT). However,the Discrete Wavelet Transform (DWT) had some disadvantages that undermine itsapplication in image processing. They investigated a performances complex wavelettransform (DT-CWT) combined with Bivariate Shrinkage and Visu-shrinkage. Theproposed method was tested on a noisy ultrasound medical image, and the restoredimages show a great effectiveness of DT-CWT method compared to the classicalDWT.

Vijay and Mathurakani [176] in 2014, proposed a image denoising techniqueusing Dual Tree Complex Wavelet Transform (DTCWT) along with Byes thresh-olding. Convolution based 2D processing was employed for simulation resulted inimprovement in PSNR.

Mitiche et al. [116] in 2013 proposed a denoising approach basing on dual treecomplex wavelet and shrinkage (where either hard and soft thresholding operators ofdual tree complex wavelet transform for the denoising of medical images are used).The results proved that the denoised images using DTCWT (Dual Tree ComplexWavelet Transform) have a better balance between smoothness and accuracy thanthe DWT and are less redundant than SWT (Stationary Wavelet Transform). Theyused the SSIM (Structural Similarity Index Measure) along with PSNR (Peak Signalto Noise Ratio) and SSIM Map to assess the quality of denoised images.

Naimi et al. [123] in 2015, proposed a denoising approach basing on dual treecomplex wavelet and shrinkage with the Wiener filter technique (where either hardor soft thresholding operators of dual tree complex wavelet transform for the denois-ing of medical images are used). The results proved that the denoised images usingDTCWT (Dual Tree Complex Wavelet Transform) with Wiener filter have a betterbalance between smoothness and accuracy than the DWT and were less redundantthan SWT (StationaryWavelet Transform). They used the SSIM (Structural Similar-ity Index Measure) along with PSNR (Peak Signal to Noise Ratio) and SSIM mapto assess the quality of denoised images.

Rao and Ramakrishna [145] in 2015 proposed a new algorithm for image denois-inig based on DTCWT. In this algorithm, the decomposed coefficients combinedwith the bivariate shrinkage model for the estimation of coefficients in high fre-quency sub bands and Bayesian shrinkage method was applied in order to remove


the noise in highest frequency sub-band coefficients. The experimental results werecompared with the existing shrinkage methods Visu and Bayes shrinkage methods interms of peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM).The experimental outcome showed that the proposed technique could eliminate thenoise efficiently and protected the edge information well, this algorithm providedgood denoising effect as well as peak signal to noise ratio and structural similarityindex was better than traditional denoising methods.

Yaseena et al. [195] in 2016 compared image denoising techniques based onreal and complex wavelet-transforms. Possibilities provided by the classical discretewavelet transform (DWT)with hard and soft thresholdingwere considered, and influ-ences of the wavelet basis and image resizing were discussed. The quality of imagedenoising for the standard 2-D DWT and the dual-tree complex wavelet transform(DT-CWT) was studied. It was shown that DT-CWT outperforms 2-D DWT at theappropriate selection of the threshold level.

Kumar and Reddy [96] in 2017 assumed the noisy image was to be compleximage and its real part and imaginary parts were separated. These were subjected toBi-shrink filter separately into different stages of decomposition depending upon theseverity of noise. The obtained de-noise image was compared with original imageusing different parametric measures like Peak Signal to Noise Ratio, Structural sim-ilarity Index measure, Covariance and Root mean square Error whose values weretabulated. The values of retrieved image obtained yields much better visual effectand hence this method was said to be a better one when compared with de-noisingmethods using Weiner Filter and various Local Adaptive Filters.

5.9.3 Image Restoration Using Quaternion WaveletTransform

As a mathematical tool, wavelet transform is a major breakthrough of the Fouriertransform and Fourier transform window known to many people since it has goodtime-frequency features andmultiple resolution.Wavelet analysis theory has becomeone of the most useful tools in signal analysis, image processing, pattern recognitionand other fields. In image processing, the basic idea of the wavelet transform isto decompose image multiresolution that is the original image is decomposed intodifferent space and different frequency sub-image, and then coefficients of sub-imageare processed. Mostly used wavelet transforms are real discrete wavelet transformand complex wavelet transform and so on.

The discrete wavelet transform (DWT) and dual-tree complex wavelet transform(DTCWT) however suffer from two major drawbacks. The first drawback is the realdiscrete wavelet transform signal small shift will produce the energy of wavelet coef-ficient distribution change, making it difficult to extract or model signal informationfrom the coefficient values. Dual-tree complex wavelet although overcame the firstproblem but it can generate signal phase ambiguity when represented two dimen-


sional images features. While the quaternion wavelet transform is a new multiscaleanalysis image processing tool, it is based on the Hilbert two-dimensional transformtheory, which has approximate shift invariance and can well overcome the abovedrawbacks as a result the signal to noise ratio was improved to a greater extent ascompared with DWT.

Quaternion wavelet research is divided into two branches: first is based on quater-nion numerical function multiresolution analysis theory of quaternion wavelet, usinga single tree structure. Mitrea [117] in 1994 gave quaternion wavelet form concept.Traversoni [172] in 2001 used real wavelet transform and complex wavelet transformby quaternion Haar kernel and proposed discrete quaternion wavelet transform andgave some applications in image processing. He and Yu [74] in 2004, used matrixvalue function multiresolution analysis structure for consecutive quaternion wavelettransform and provided some properties. Bahri [12] constructed discrete quaternionwavelet transform (DQWT) through complexmatrix function and proved some basicproperties. Bahri et al. [11] in 2011 introduced through quaternion wavelet admissi-bility conditions, systematically extended the consecutive wavelet transform conceptto consecutive quaternion wavelet concept and provided the reconstruction theoremand continuous quaternion wavelet basic properties. But these are mainly the con-cepts and properties of promotion, because its filters structure and implementationare difficulties, it has not made any progress in application at present.

Another branch is based on Bulow quaternion analytic signal, by using real fil-ter and dual-tree structure Bulow [19] in 1999 constructed the quaternion wavelettransform. The filter has the advantages of simple structure, relatively easy, and therewas quaternion signal application background. Corrochano [41] in 2006 constructedquaternion wavelet transform (QWT) through quaternion Gabor filter and discussedthe QWT properties and wavelet pyramid algorithm. He pointing out that the DWTis without phase, CWT only has one phase, while QWT can provide three phasesand putting forward the image multiresolution disparity estimation method basedon the theory of QWT. Based on the dual-tree complex wavelet, Chan et al. [24]in 2008 used the concepts and properties of Bulow quaternion analytic signal andquaternion Fourier transform, constructed dual-tree quaternion wavelet transform,and worked out the meaning of three phases, two of which represent the image oflocal displacement information, another as image texture feature, which can be usedto estimate the image of the local geometric features. Xu et al. [191] in 2010 usedquaternion wavelet transforms amplitude and phase method and applied it to the facerecognition also obtained certain result. Soulard and Carre [159] in 2011 applied thequaternion wavelet transform to image texture analysis and proved the feasibilityof this method. Now we are going to provide some applications quaternion wavelettransform in the field of image denoising.

Yin et al. [197] in 2012, mainly studied some of the concepts and properties ofquaternion wavelet Transform and applied the quaternion wavelet in image denois-ing. They provided forward Bayesian denoising method based on quaternion wavelettransform, considering wavelet coefficients correlation, and generalized Gaussiandistribution was used to model the probability distribution function of wavelet coef-ficients magnitude and the best range of the Bayesian thresholding parameter was


found out. The experimental results showed that their method both in visual effectand PSNR were better than many current denoising methods.

Tianfeng [171] in 2014, introduced the denoising performance evaluation stan-dard and imagewavelet threshold denoisingmethod and then provided the quaternionbased on wavelet transform domain hidden markov tree model for image denoising(Q-HMT), non-gaussian distribution model and hybrid statistical model of imagedenoising algorithm. Good results had been achieved on the subjective and objec-tive. On the basis of the introduction of additive model through quaternion wavelettransform, author using improved coefficient classification criterion, the coefficientswere divided into two categories: important coefficient and the important factor, pro-posed to improve the Donoho threshold and the new threshold function, and it dealswith the important coefficient, estimate the excluding of quaternion wavelet trans-form coefficient, and the coherence of SAR image was obtained. Coherent specklenoise suppression experiments of real SAR image, on the objective indicators andvisual effect, the proposed method was superior to the current many methods.

Fang-Fei et al. [62] in 2014, proposed an algorithm by combining the quaternionwavelet transform model with the traditional HMT model. The new algorithm hadthe advantage of good translation without deformation and the advantage of therich phase. The experimental results showed that this algorithm was superior to thetraditional algorithm of denoising in peak signal-to-noise ratio and image effect wasmore superior to the traditional denoising algorithm.

Kadiri et al. [82] in 2014, studied the potential of the quaternionwavelet transformfor the analysis and processing of multispectral images with strong structural infor-mation. They showed an application of this transform in satellite image denoisingand proposed approach relies on the adaptation of thresholding procedures basedon the dependency between magnitude quaternionic coefficients in local neighbor-hoods and phase regularization. In addition, they introduced a non-marginal aspectof multispectral representation. The results obtained indicate the potential of thismultispectral representation with magnitude thresholding and phase smoothing innoise reduction and edge preservation compared with classical wavelet thresholdingmethods that do not use phase or multiband information.

Malleswari and Madhu [112] in 2016, proposed an algorithm of image denoisingbased on Quaternion Wavelet Transform model. The experimental results showedthat this algorithm was superior to the traditional algorithm of denoising in peaksignal to noise ratio and the visual appearance of the image was also better whencompared to the traditional denoising algorithm.

5.9.4 Image Restoration Using Ridgelet Transform

Wavelet is a useful technique to extract piecewise smooth information from a onedimensional signal but does not give satisfactory performance in case on a two dimen-sional signal. Wavelet transform detect point singularities but it fails to capture linesingularities. An image contains singularities along line or curve which cannot be


represented efficiently by wavelet based techniques. Therefore, new image process-ing schemes which are based on true two dimensional transform are expected toimprove the performance over the current wavelet based methods.

The higher dimension issues which cannot he handled by wavelets are put forthby Candes and Donoho [20] in 1999. They discussed line like phenomena in dimen-sion 2 and plane like phenomena in dimension 3. They gave the idea of analysis ofridge functionsψ(u1x1 + · · · + unxn)whose ridge profilesψ are used for performingwavelet analysis in Radon domain. They discussed ridgelet frames, ridgelet orthonor-mal basis, ridges and described a new notion of smoothness naturally attached in theridgelet algorithm.

Do andVetterli [48] in 2000 presented a finite implementation of the ridgelet trans-form. The transform is invertible, non-redundant and achieved via fast algorithms.Furthermore they showed that this transformwas orthogonal, hence it allowed one touse linear approximations for representation of images. Finite ridgelet transformwasconstructed using finite radon transform. The finite radon transform was redundantand non-orthogonal. Radon transform was defined as a line integral of the imageintensity f (x, y), over the line that is at a distance s from the origin and perpendicularto a line passing through origin, at an angle θ to the x-axis.

Do and Vetterli [47] in 2000, proposed a new finite orthonormal image transformbased on the Ridgelet. The Finite ridgelet image transform (FRIT) was shown torepresent effectively images with linear discontinuities. Authors observed that FRIThas potential in restoring images that are smoth away from edges. Finite ridgeletimage transform was expected to work well also for images with smooth edges byapplying on suitable size block.

Cane and Andres [23] in 2004, proposed a new implementation of the ridgelettransform based on discrete analytical 2D line, called discrete analytical ridgelettransform (DART). Discrete radon transform was computed using Fourier strategy.The innovative step given by Cane and Andre was the radial discrete analytical linesin Fourier domain. These discrete analytical lines in Fourier domain were having aparameter called arithmetical thickness, which could be used for specific applicationsuch as denoising. Authors applied DART to each tile for denoising of digital images.But the interesting approach laid by authors was its extendibility to higher dimension.

The thresholded image was regrouped and subjected to Wiener filtering to obtainthe final denoised image. Zhou et al. [204] in 2004were used PSNRandSSIM indicesfor denoising performance evaluation. For Lena and Barbara images corrupted bynoise of standard deviation equal to 10, the corresponding denoised images hadPSNR values of 35.11dB and 33.99dB. The corresponding SSIM values were 0.963and 0.971.

A new denoising method, by integrating the dual-tree complex wavelets intothe ordinary ridgelet has been proposed by Chen et al. [30] in 2007. The dual-tree complex wavelet has shift invariance property and ridgelet transform have highdirectionality. Complex Ridgelet was obtained by applying 1-D dual-tree complexwavelet transform on radon transform coefficients. Hard thresholding was used fordenoising application. PSNR improvement on Lena image was 7.13dB using thistechnique.


Kang andZhang [83] in 2012, proposed amethod to remove the noise inQuickBirdimages based on the ridgelet transform. Experimental results showed that ridgelettransform performs effectively in removing the noise in QuickBird images comparedwith other methods.

Shuyuan et al. [156] in 2013 proposed a ridgelet support vector based imagedenoising algorithm. A multiscale ridgelet support vector filter and Geometric Mul-tiscale Ridgelet Support Vector Transform (GMRSVT) are derived which have shiftinvariant property. This is a dictionary based approach so the performance of algo-rithm is dependent on the extent of learning. Performance of this algorithm is vali-dated on Lena image. Lena image corrupted by Gaussian noise of standard deviation20 is denoised and the PSNR is obtained as 33.34dB.

Krishnanaik et al. [93] in 2013 proposed a new image denoising technique byintegrating the dual-tree complex wavelets into the usual ridgelet transform. In thisprocedure normal hard thresholding of the complex ridgelet coefficient was usedand results obtained showed better than VisuShrink, the ordinary ridgelet imagedenoising, and wiener2 filter, and also Complex ridgelets applied to curvelet imagedenoising.

Kaur andMann [85] in 2013, proposed the ridgelet transformwithM-bandwavelettransform, called M-band ridgelet transform, for medical image segmentation. Theperformance of the proposedmethodwas tested on ultrasound images underGaussiannoise. The results of the proposed method were compared with the ridgelet andcurvelet transform in terms of PSNR, MSE. The results after being investigatedshowed significant improvements compared to the ridgelet and curvelet denoisingalgorithms.

Liu et al. [106] in 2014 proposed finite ridgelet transform based algorithm fordigital image denoising. Patch wise denoising was performed to obtain better PSNRand superior visual quality. The noisy image was first grouped into patches followedby finite ridgelet transform and hard thresholding.

Vetrivelan and Kandaswamy [175] in 2014 proposed ridgelet transform hardthresholding algorithm for image denoising to preserve the details of the image.Ridgelet transform was used as it is concentrated near the edges of the image andit represented one-dimensional singularity in two-dimensional spaces. Wavelet wasgood in representing point singularities. When wavelet was linked with ridgelet,denoised image quality would be improved.

Due to the fusion of the properties of two transforms i.e. Wavelet transform andRadon transform, the Ridgelet Transform possess improved denoising and edge pre-serving capabilities. Kumar and Bhurchandi [94] in 2015, proposed algorithm forimage denoising using Ridgelet transform and cycle spinning. The proposed algo-rithm yielded better PSNR for low and moderate magnitudes of zero mean Gaussiannoise. The algorithm showed remarkable denoising capability of Ridgelet transformin while protecting edges compared to other contemporary algorithms.

Krishnanaik and Someswar [92] in 2016, introduced sliced ridgelet transformfor image de-noising, and to achieved the scalability and accuracy and in a reliablemanner of image processing. Sliced ridgelet transforms ridge function was segre-gated to multiple slices with constant length. Single dimension wavelet transforms


were used to compute the angle values of each slice in sliced ridgelet transform.Ridgelet co-efficient were obtained for the base threshold calculation to implementthe accurate de-noising. The proposed method was based on two operations: one wasthe redundant directional wavelet transform based on the radon transform, and otherwas threshold designing of the ridgelet coefficient. Authors compared the accuracyand scalability of image de-noising with other popular approaches like wavelets,curvlets and some other inter-relevant technologies. Experimental results were prov-ing that the sliced ridgelet approach was having the better performance than the otherpopular techniques.

Huang et al. [77] in 2016, proposed a new multiscale decomposition algorithmcalled adaptive digital ridgelet (ADR) transform. This algorithm could adaptivelydeal with line and curve information in an image by considering its underlying struc-ture.As the key part of the adaptive analysis, the curve parts of an imagewere detectedaccurately by a new curve part detection method. ADR transform was applied toimage denoising experiment and experimental results demonstrate its efficiency forreducing noises as PSNR values could be improved maximally 5dB compared withother methods and MAE values were also considerably improved. A new compar-ison criterion was also proposed and using this criterion, it was shown that ADRtransform can provide a better performance in image denoising.

5.9.5 Image Restoration Using Curvelet Transform

With the introduction of wavelet transform in early 1980, several general and hybriddenoising models have been proposed, in which either mathematical algorithms orheuristic algorithms were used for specific applications. Wavelet has been exploredso much that it has more or less become a household word. Everyday life we comeacross with electronic instruments which have wavelet applications. Not only engi-neers but also non-technical persons know about wavelets. Multiresolution capacitywhich means representation should allow images to be successively approximatedfrom course to fine resolutions. The basis element should be able to localize fre-quency components in multiple directions with very less redundancy. Wavelet doesnot performwell, when the singularities are in higher dimensions. These deficienciesinspired the researchers to extend the wavelet transform. Some of these extensionsare done through geometry of the space and some by preserving properties of thetransform such as invariance, translation, rotation and singularities along certaindirections.

One of the generalizations of the wavelet transform is the directional wavelettransform which in addition to scaling and translation, accounts for rotation. Thismakes the directional wavelet transform better in detecting singularities of higherdimensions.

In the last few years, there have been developments which discovered new multi-scale transforms to overcome wavelets limitations. Those are ridgelets and curvelets.Ridgelet and curvelet performed better even in the early stages of their development.


Ridgelet transform which is an application of 1D wavelet transform is also based onradon transforms. Radon transforms detect singularities along straight line, while theCurvelet transform which may be viewed as directional parabolic dilation transform,in which the parabolic dilation is applied in the frequency domain. It can detect sin-gularities along curves better than the ridgelet and wavelet transform, in addition tosingularities along lines.

The problem of detecting singularities is closely related to image processingbecause in two dimensional images, smooth regions are separated by edges andsmooth cures, which suggest that wavelet transform and ridgelet transform cantbe the best tool for some image processing applications. Till now we have gonethrough wavelet, complex wavelet, quaternion wavelet, ridgelet and to overcome thedrawbacks of wavelet and ridgelet, now we will move on to curvelet transform. Thecurvelet transform represents curves better than wavelet and ridgelet.

Candes and Donoho [21] in 2001 introduced the curvelet transform well-matchedfor objects which are smooth and away from discontinuities across curves. Curvelettransformhas been proved to be effective at processing of images along curves insteadof radial directions. Curvelet can also capture structural information along multiplescales, locations and orientations. Curvelet captures this structural information inthe frequency domain. Even if we subject complex texture structure of CT images,curvelet will reasonability improve quality (texture) of the image.

Starck et al. [161] in 2002 used curvelet transform for representing astronomicalimages. The noisy image was given as an input to the curvelet transform and theresultant imagewas compared with the establishedmethod of thresholded coefficientof wavelet transform. Curvelet transform enhance elongated features and better ringand edges are seen in the astronomical images. The experiments results describedthat curvelet reconstruction did not contain the quantity of disturbing artifacts alongedges that one see in wavelet reconstructions. PSNR of denoised Lenna image was31.95dB which was better than most of the algorithms.

Starck et al. [162, 163] in 2003, were designed the curvelet transform to representedges and other singularities along curves much more efficiently than traditionaltransforms i.e. using many fewer coefficients for a given accuracy of reconstruction.In the curvelet transform, the frame elements are indexed by scale, location anddirection. The curvelets elements obey special scaling law, where the length of thesupport of a frame elements and the width of the support are linked by the relationwidthlength2. All these properties are very attractive and have already led to a widerange of applications such as tomography, astronomy etc.

Ma andPlonka [109] in 2007 presented an almost optimal non-adaptive sparse rep-resentation for curve like features and edges using curvelets. The authors describeda broad range of applications involving image processing, seismic data exploration,image denoising etc. They proposed formula for soft, hard and continuous garrote andtotal variation (TV) constraint curvelet shrinkage for denoising digital images. Theyproved that total variation constraint curvelet shrinkage leads to a promising PSNRat the expense of computation time. PSNR improvement of 4.77dB was obtained forBarbara image using this technique.


Patil and Singhai [132] in 2010, proposed a soft thresholding multiresolutionanalysis technique based on local variance estimation for image denoising. Thisadaptive thresholding with local variance estimation effectively reduced image noiseand also preserved edges. In the proposedmethod, 2D fast discrete curvelet transform(2DFDCT) out performedwavelet based image denoising andPSNRusing 2DFDCTwas found approximately doubled.

Patil et al. [131] in 2010 proposed an approach of reconstruction of SR image usinga sub-pixel shift image registration and Curvelet Transform (CT) for interpolation.The experimental results demonstrated that Curvelet Transform performed better ascompared to StationaryWavelet Transform. Also, it was experimentally verified thatthe computational complexity of the SR algorithm was also reduced by using CT forinterpolation.

Patil and Singhai [133] in 2011 proposed SR reconstruction using a sub-pixelshift image registration and Fast Discrete Curvelet transform (FDCT) for imageinterpolation. Experimentation results showed appropriate improvements in PSNRand MSE and also it was experimentally verified that the computational complexityof the SR algorithm was reduced.

Palakkal and Prabhu [129] in 2012 proposed a denoising algorithm for imagescorrupted by Poisson noise. They had used fast discrete curvelet transform andwavedatom technique along with a variance stabilizing transform for denoising of images.This algorithmwas tested on standard images corrupted bypoissonnoise. ForBarbaraimage having maximum intensity of 120 and input PSNR value of 24.04dB, thedenoised image PSNRwas 29.45dB. The other algorithms that were discussed focuson denoising for Gaussian noisy image whereas this algorithm examines Poissonnoisy images.

A curvelet is a effective spectral transform, which allows sparse representationsof complex data. This spectral technique is based on directional basis functions thatrepresent objects having discontinuities along smooth curves. Oliveira et al. [127] in2012 applied this technique to the removal of Ground Roll, which was an undesiredfeature signal present in seismic data obtained by sounding the geological structuresof the Earth. They decomposed the original seismic data by curvelet transform inscales and angular domains. For each scale the curvelet denoising technique alloweda very efficient separation of the Ground Roll in angle sections. The precise identifi-cation of the Ground Roll pattern allowed an effective erasing of its coefficients. Incontrast to conventional denoising techniques they did not use any artificial attenua-tion factor to decrease the amplitude of the Ground Roll coefficients. They estimatedthat, depending on the scale, around 75 percent of the energy of the strong undesiredsignal is removed.

Kaur et al. [84] in 2012 applied curvelet transform denoising method to noisyimages comparison of images. The results were compared qualitatively (visually)and quantitatively (using quality metrics) and it was proved that values of curveletmethods for all quality metrics were better than the other methods.

Vaghela [173] in 2013 proposed a newmethod for image restoration using curvelettransform. Experimental results showed that curvelet transform coefficient technique


was conceptually simpler, faster and far less redundant than the existing techniquein case of Gaussian noise irrespective of type of images.

Parlewar and Bhurchandi [130] in 2013 developed a modified version of curvelettransform called the 4 quadrant curvelet transform for image denoising. The authorsused curvelet transform by taking radon projections in four quadrants and then aver-aging the result. The 4 quadrant curvelet transformwas applied only to patches whichconsist of edges instead of applying to complete image. PSNR of denoised Barbaraimage corrupted by noise of standard deviation 10 was obtained as 32.7dB.

Denoising of the astronomical images is still a big challenge for astronomers andpeoplewho process astronomical data. Anisimova et al. [7] in 2103 proposed an algo-rithm based onCurvelet and Starlet transform for astronomical image data denoising.The proposed algorithm had been tested on image data from MAIA (Meteor Auto-matic Imager and Analyser) system. Their influence on important photometric datalike stellar magnitude and FWHM (Full Width at Half Maximum) had been studiedand compared with conventional denoising methods.

Wu et al. [187] in 2014 developed a curvelet transform based non-local meansalgorithm for digital image denoising. The noisy image was subjected to curvelettransform followed by inverse curvelet transform. The similarity between curvelettransform reconstructed image pixels and the noisy image pixels was used for denois-ing the image by employing non-local means method.

Bains and Sandhu [13] in 2015, presented a comparative analysis of various imagedenoising techniques using curvelet transforms. A lot of combinations had beenapplied in order to find the best method that could be followed for denoising intensityimages. The experimental results demonstrated that curvelet transform outperformsother transform for denoising all of the above mentioned images. Curvelet transformdenoised the images with more precision as compared to DWT because of its inbornquality of keeping the data intact to a greater extent. PSNR showed an apprehensiveimprovement, if the noisy images were denoised using curvelet transform.

AnjumandBhyri [9] in 2015, used non-linear technique such as curvelet transformand edge detection in image processing for removing of noise present in image. Thebest results were obtained with denoising the test images corrupted by random noise,spackel noise, Gaussian noise and salt and pepper noise in terms of PSNR and itwas noticed that the lowest PSNR gain was obtained for biomedical images whencompared to satellite images.

Raju et al. [140] in 2016 proposed the denoising of remotely sensed imagesbased on Fast Discrete Curvelet Transform (FDCT). The Fast Discrete CurveletTransform had been discussed via Wrapping (WRAP) and Unequally-Spaced FastFourier Transform (USFFT). With its optimal image reconstruction capabilities, thecurvelet outperformed the wavelet technique in terms of both visual quality and PeakSignal toNoiseRatio (PSNR).Mainly the author focused on the analysis of denoisingthe Linear Imaging Self Scanning Sensor III (LISS III) images, Advanced Very HighResolution Radiometer (AVHRR) images from National Oceanic and AtmosphericAdministration 19 (NOAA 19), METOP satellites for the Tirupati region, AndhraPradesh, India. Numerical illustrations demonstrated that this method was highlyeffective for denoising the satellite images.


Zanzad and Rawat [200] in 2016, described a comparison of the discriminatingpower of the various multiresolution based thresholding techniques i.e. wavelet,Curvelet for image denoising. The experimental results showed that the curvelettransform gives better results/performance than wavelet transform method.

Talbi and Cherif [169] in 2017, proposed a new image denoising technique whichcombines two denoising approaches. The first one was a curvelet transform baseddenoising technique and the second onewas a two-stage image denoising by principalcomponent analysis with local pixel grouping (LPG-PCA). This proposed techniqueconsisted at first step in applying the first approach to the noisy image in order toobtain a first estimate of the clean image. The second step consisted in estimatingthe level of noise corrupting the original image. The third step consisted in using thisfirst clean image estimation, the noisy image and this noise level estimate as inputsof the second image denoising system (LPG-PCA based image denoising) in orderto obtain the final denoised image. The proposed image denoising technique wasapplied on a number of noisy images and the obtained results from PSNR and SSIMcomputations show its performance.

5.9.6 Image Restoration Using Contourlet Transform

Contourlet transform is designed to efficiently represent images made of smoothregion and curved boundaries. The contourlet transform has a fast implementa-tion based on laplacian pyramid decomposition followed by directional filter banksapplied on each band pass subbands.

Donoho employed a discrete domain multiresolution and multi-direction expan-sion using non-separable filter banks to construct contourlet transform. Discrete con-tourlet transform used an iterative filter bank that requires N operations for N -pixelimage. The link was developed between the filter banks and associated continu-ous wavelet domain via a directional multiresolution analysis. Donoho [56] in 1998proved that due to directional filter bank, contourlet was able to develop smoothobject boundaries. However, the major drawback was that its basis images were notlocalized in frequency domain. Donoho analyzed and proposed a new contourlet thathad basis which were localized in frequency domain. This algorithm outperformedother contemporary algorithms quantitatively as well as visually.

Do and Vertalli [49] in 2005, constructed a discrete-domain multiresolution andmultidirection expansion using non-separable filter banks, in much the same waythat wavelets were derived from filter banks. This construction resulted in a flexi-ble multiresolution, local, and directional image expansion using contour segments,called contourlet transform. Furthermore, they established a specific link between thedeveloped filter bank and the associated continuous domain contourlet expansion viaa directional multiresolution analysis framework and showed that contourlets, withparabolic scaling and sufficient directional vanishing moments, achieved the optimalapproximation rate for piecewise smooth functions with discontinuities along twicecontinuously differentiable curves. Finally, they provided some numerical experi-


ments demonstrating the potential of contourlets in several image processing appli-cations.

Matalon et al. [113] in 2005 proposed a novel denoising method based on theBasis Pursuit Denoising (BPDN)method. Their method combined the image domainerror with the transform domain dependency structure, resulting in a general objec-tive function, applicable for any wavelet like transform. They focused on the Con-tourlet Transform (CT), a relatively new transform designed to sparsely representimages. The resulting algorithm proved superior to the classic Basis-Pursuit Denois-ing (BPDN), which did not account for these dependencies.

Eslami and Radha [60] in 2005 developed a procedure to obtain a TI version ofa general multi-channel multidimensional subsampled FB. They proposed a gener-alized algorithm a trous, and then applied the derived approach to the contourlettransform. In addition to the proposed TI contourlets, they introduced semi-TI con-tourlets, which was less redundant than the TICT. Furthermore, they employed theirproposed schemes in conjunction with the TIWT to image denoising. Their simu-lation results indicated the potential of the TICT and STICT for image denoising,where one can achieve better visual and PSNR performance at most cases whencompared with the TIWT.

The research of Eslami and Hayder was extended by Alparone et al. [5] in 2006 tomodify contourlet transform for image denoising using cycle spinning. In the clas-sical denoising algorithms, many visual artifacts were produced due to the lack oftranslation invariance. They used cycle spinning based technique to develop transla-tion invariant contourlet denoising scheme. The results demonstrated enhancementon the images corrupted with additive Gaussian noise.

Cunha et al. [43] in 2006, developed the nonsubsampled contourlet transform(NSCT) and study its applications. They constructed NSCT based on a nonsubsam-pled pyramid structure and nonsubsampled directional filter banks. This constructionprovided a flexible multiscale, multidirection, and shift-invariant image decompo-sition that could be efficiently implemented via the trous algorithm. They appliedNSCT in image denoising and enhancement applications and found the better resultsto other existing methods in the literature.

Po and Do [135] in 2006 provided detailed study on the statistics of the contourletcoefficients of natural images: using histograms to estimate the marginal and jointdistributions, and mutual information to measure the dependencies between coeffi-cients. They provided a model for contourlet coefficients using a hiddenMarkov tree(HMT)model with Gaussianmixtures that can capture all inter-scale, inter-direction,and inter-location dependencies. In addition, They presented experimental resultsusing this model in image denoising and texture retrieval applications. In denoising,the contourlet HMT outperformed other wavelet methods in terms of visual quality,especially around edges.

Sun et al. [168] in 2008 adopted multiscale geometry method, distilled the prin-cipal component from the image after Contourlet transform, lowered the dimensionof the high frequency subdomains, eliminated the noise by minimum variance costfunction. The entire arithmetic without estimate noise, compared to Contourlet hard


threshold denoising and wavelet hard threshold denoising, PSNR increased 1dB, thedenoising effect was better than other methods in the experiment.

Sivakumar et al. [158] in 2009, used the new algorithm based on the ContourletTransform. This algorithm was more efficient than the wavelet algorithm in ImageDenoising particularly for the removal of speckle noise. The parameters consideredfor comparing the wavelet and Contourlet Transformswere SNR and IEF. The resultsshowed that the proposed algorithm outperformed the wavelet in terms of SNR, IEFvalues and visual perspective as well.

Guo et al. [71] in 2011, introduced the characteristics of multi-resolution andmulti-direction decomposition about Contourlet transform, by comparing with thewavelet transform, the theoretical basis and advantages of Contourlet transform.Authors strained on analyzing the principle of threshold denoising, and laid forwarda new kindmethod of multi-threshold image denoising. Experimental results showedthat the effect of image denoising method better than wavelet transform image andindividually threshold denoising effect was good, and this method was simple oncalculation with fast speed.

Liu et al. [104] in 2012, presented an image denoising algorithm based on non-subsampled contourlet transform (NSCT). A second order random walk with restartkernel was employed to describe the geometric features like edges and texture. NSCTwas then used to capture these features. This algorithm used an iterative approach forimage denoising. The iterative process continues till the current RMSE is smaller thanthe preceding RMSE. Denoising of Barbara image corrupted by noise of variance0.01, yields RMSE value slightly less than 6.5.

Zhou and Wang [205] in 2012, proposed an image denoising algorithm based onnonsubsampled contourlet transform. It used the Symmetric Normal Inverse Gaus-sian (SNIG) model and models the pixel values as random variables. First the imagewas decomposed using NSCT and the noise was estimated. SNIG model was thenapplied to the noisy image. The algorithm was validated on standard image cor-rupted by noise of different strengths. For Barbara and Lena images corrupted bywhite noise of standard deviation equal to 10, the corresponding denoised images hadPSNR values of 34.08 and 30.77dB. This algorithm was validated only on imagescorrupted by Gaussian noise.

Borate and Nalbalwar [17] in 2012, proposed a technique to recover the super-resolved image from a single observation using contourlet based learning and usefulwhen multiple observations of a scene were not available so one must make the bestuse of a single observation to improve its resolution. Experimental results showedappropriate improvements over conventional interpolation techniques.

Shah et al. [153] in 2013 discussed a novel approach of getting high resolutionimage from a single low resolution image. In their method, the Non Sub-sampledContourlet Transform (NSCT) based learning was used to learn the NSCT coeffi-cients at the finer scale of the unknown high-resolution image from a dataset of highresolution images. The cost function consisting of a data fitting term and a Gaborprior term was optimized using an Iterative Back Projection (IBP). By making useof directional decomposition property of the NSCT and the Gabor filter bank withvarious orientations, the proposed method was capable to reconstruct an image with


less edge artifacts. The experimental results showed the better performance of theproposed method in terms of RMS measures and PSNR measures.

Shen et al. [155] in 2013, proposed an adaptivewindowbased contourlet transformdomainmethod for image denoising.An elliptical shapedwindowwas used instead ofa square window so as to estimate the signal variance. The improvement in denoisingperformance byusing ellipticalwindowwas represented graphically. ForLena image,denoising by this algorithm improved the PSNR from 22.10 to 31.25dB i.e. animprovement of 9.15dB.

Wang et al. [181] in 2013, presented a SVM classification based non-subsampledcontourlet transform (NSCT) domain technique for denoising of images. To distin-guish between noisy pixels and edge pixels, the authors have used SVM to classifyNSCT coefficients into smooth regions and texture regions. A significant increasein PSNR value of Lena image from 22.11 to 31.64dB was obtained using this tech-nique. This algorithm yielded the highest PSNR improvement of 9.53dB in case ofLena image.

Yin et al. [198] in 2013, proposed amodifiedversionof non-subsampled contourlettransform based image denoising algorithm. The derived non-subsampled dual-treecomplex contourlet transform (NSDTCT) was obtained from the merger of dual-treecomplex wavelet transform and the non-subsampled directional filter banks. TheNSDTCT was followed by non-local means filtering to obtain the denoised image.PSNR of denoised Lena image corrupted by noise of standard deviation 10 wasobtained as 35.98dB.

Padmagireeshan et al. [128] in 2013, proposed a medical image denoising algo-rithm using contourlet transformwith directional filter banks and Laplacian pyramid.The performance of the proposed method was analysed with the existing methods ofdenoising using wavelet transform and block DCT. Simulation results showed thatcontourlet transform had better denoising capabilities compared to existing methods.

Sakthivel [149] in 2014, proposed contourlet based image denoising algorithmwhich can restore the original image corrupted by salt and pepper noise, Gaussiannoise, Speckle noise and the Poisson noise. The noisy image was decomposed intosub bands by applying contourlet transform, and then a new thresholding functionwas used to identify and filter the noisy coefficient and take inverse transform toreconstruct the original image. The simulation result of the proposed method wascomparedwith other simulation resultswhich used the various thresholding functionsnamely Bayes Shrink and Visu Shrink. It was observed that the proposed algorithmcan remove Poisson and speckle noises effectively.

Wei et al. [184] in 2015, proposed the image denoising method based on animproved Contourlet to remove the noise of the traction machine and the steel wirerope effectively and protect the details of the image better. The Experimental resultsshowed that the proposed algorithm was improved in the denoising performance andthe visual effects. Meanwhile, the image details were protected better.

Image denoising is a very important step in cryo-transmission electronmicroscopy(cryo-TEM) and the energy filtering TEM images before the 3D tomography recon-struction, as it addresses the problem of high noise in these images, which leads


to a loss of the contained information. High noise levels contribute in particular todifficulties in the alignment required for 3D tomography reconstruction.

Ahmed et al. [4] in 2015 investigated the denoising of TEM images that wereacquired with a very low exposure time, with the primary objectives of enhancingthe quality of these low-exposure time TEM images and improving the alignmentprocess. They proposed denoising structures to combine multiple noisy copies of theTEM images and the structures were based on Bayesian estimation in the transformdomains instead of the spatial domain to build a novel feature preserving imagedenoising structures via wavelet domain, the contourlet transform domain and thecontourlet transform with sharp frequency localization. Numerical image denoisingexperiments demonstrated the performance of the Bayesian approach in the con-tourlet transform domain in terms of improving the signal to noise ratio (SNR) andrecovering fine details that may be hidden in the data. The SNR and the visual qualityof the denoised images were considerably enhanced using these denoising structuresthat combine multiple noisy copies.

Li et al. [101] in 2015 proposed an algorithm for image denoising based onNonsubsampled Contourlet Transform (NSCT) and bilateral filtering in the spa-tial domain is proposed. The noisy image was first decomposed into multi-scaleand multi-directional subbands by NSCT, and direction subbands of each high-passcomponent was processed by the new threshold function which was obtained by theBayes threshold that based on stratified noise estimation. During the reconstruction,the low-pass subband constructed image was further denoised by the bilateral fil-tering in the spatial domain. Experimental results demonstrated that the proposedmethod improved de-noising performance.

Divya and Sasikumar [46] in 2015, proposed a technique of noise removal fromdigital images. In this process, the image was first transformed to the nonsubsampledcontourlet transform (NSCT) domain and then support vector machine (SVM) wasused for classifying noisy pixels from the edge related ones. The proposed methodhad the advantage of achieving a good visual quality with very less quantity ofdisturbing artifacts.

Jannath et al. [78] in 2016 compared the performance of DWT and Contourlettransform for image denoising. They found that Contourlet transform Speckle noisewas better removal of Block Shrink and Poisson noise was work well for the BayesShrink.

Jannath et al. [79] in 2016, denoised Gaussian noises and Speckle noises in MRimages undergo a contourlet domain for decompositionof input images.After decom-position some threshold methods were applied such as Bayes Shrink, Neigh Shrink,andBlock Shirnk. These Thresholdmethodswere used to unfasten the noises. Finallythey analysed the performance of denoised image to find the better result. Perfor-mance of medical image denoising was reckoning by Peak signal to noise ratio(PSNR), structural similarity index (SSIM), image quality index (IQI) and normal-ized cross correlation (NCC).

Kourav and Chandrawat [91] in 2017, proposed non-subsampled contourlet trans-form (NSCT) for image denoising and compared the result with discrete wavelet


transform (DWT). They observed that the performance of NSCT was better than theDWT on the basis of PSNR, RMSE and SSIM.

Oad andBhavana [126] in 2017 proved to the pros of the contourlet transform overwavelet transform. The proposed methodology integrating the wiener and medianfiltering to enhance the performance of the denoising over wavelet transform. Thesimulation was performed on three images Lena, Peppers and Barbara with threeparameters peak signal to noise ratio (PSNR), root mean square error (RMSE) andelapsed time found that the proposed contourlet transformwas better than thewavelettransform based denoising.

5.9.7 Image Restoration Using Shearlet Transform

Removing or reducing noises from image is very important task in image processing.Image Denoising is used to improve and preserve the fine details that may be hiddenin the data. In Image processing, noise is not easily eliminated as well as preservingedges is also difficult. The shearlet representation has emerged in recent years asone of the most effective frameworks for the analysis and processing of multidi-mensional data. Shearlet is the greatest method for preserving the edges. ShearletTransform combines multiscale and multi-directional representation and is very effi-cient to capture intrinsic geometry of the multidimensional image and is optimallysparse in representing image containing edges, which enable them to capture intrin-sic geometric features of image. It can work well in both natural images and medicalimages for identifying the Anistropic features and preserved smooth edges. Shearletis best because it has retained the accurate information. During these advantages Itmotivates and justified to do work in Shearlet transform.

Unlike wavelets, shearlets form a pyramid of well-localized functions definednot only over a range of scales and locations, but also over a range of orientationsand with highly anisotropic supports. As a result, shearlets are much more effectivethan traditional wavelets in handling the geometry of multidimensional data, andthis was exploited in a wide range of applications from image and signal processing.However, despite their desirable properties, the wider applicability of shearlets islimited by the computational complexity of current software implementations. Forexample, denoising a single 512 512 image using a current implementation of theshearlet-based shrinkage algorithm can take between 10s and 2 min, depending onthe number of CPU cores, and much longer processing times are required for videodenoising. On the other hand, due to the parallel nature of the shearlet transform, itis possible to use graphics processing units (GPU) to accelerate its implementation.

Easley et al. [57] in 2008 introduced a new discrete multiscale directional repre-sentation called the discrete shearlet transform. This approach combined the powerof multiscale methods with a unique ability to capture the geometry of multidimen-sional data and was optimally efficient in representing images containing edges.They described two different methods of implementing the shearlet transform. Thenumerical experiments presented in this paper demonstrated that the discrete shearlet


transform was very competitive in denoising applications both in terms of perfor-mance and computational efficiency.

Chen et al. [31] in 2009, proposed a new local adaptive shrinkage threshold denois-ing method based on the shearlet transform by incorporating neighbouring shearletcoefficients of image and noise level. Experimental results showed that the methodoutperformed the several methods.

Li et al. [103] in 2011, presented a new image denoising scheme by combiningthe shearlet shrinkage and improved total variation (TV). According to the artifactsthat appear in the result image after applying shearlet denoising approach, the imagewas further denoised by a TV model, which was improved on the fidelity term.Experiment results showed that the proposed scheme could remove image noise andpreserved the edge texture, removed Gibbs-like artifacts effectively and had lowercomputational complexity.

Deng et al. [44] in 2012, proposed an efficient algorithm for removing noisefrom corrupted image by incorporating a shearlet-based adaptive shrinkage filterwith a non-local means filter. Firstly, an adaptive Bayesian maximum a posterioriestimator, where the normal inverse Gaussian distribution was used as the priormodel of shearlet coefficients, was introduced for removing the Gaussian noise fromcorrupted image. Secondly, the nonlocal means filter was used to suppress unwantednonsmooth artifacts caused by the shearlet transform and shrinkage. Experimentalresults demonstrated that the proposed method can effectively preserve the imagefeatureswhile suppressing noise and unwanted nonsmooth artifacts. It achieved state-of-the-art performance in terms of SSIM and PSNR.

Fan et al. [61] in 2013 proposed an filter algorithm which comprehensive utilizeMulti-Objective Genetic Algorithm (MOGA) and Shearlet transform based on aMulti-scale Geometric Analysis (MGA) theory. Experimental results showed thattheir algorithm was more effective in removing Rician noise, and giving better PeakSignal Noise Ratio (PSNR) gains, without manual intervention in comparison withother traditional filters.

Nonsubsampled shearlet transform (NSST) is an effective multi-scale and multi-direction analysis method, it not only can exactly compute the shearlet coefficientsbased on a multiresolution analysis, but also can provide nearly optimal approx-imation for a piecewise smooth function. Yang et al. [194] in 2014, proposed anew edge/texture-preserving image denoising using twin support vector machines(TSVMs) Based on NSST. In this proposed method, firstly, the noisy image wasdecomposed into different subbands of frequency and orientation responses usingthe NSST, then, the feature vector for a pixel in a noisy image was formed by thespatial geometric regularity in NSST domain, and the TSVMs model was obtainedby training. Next, the NSST detail coefficients were divided into information relatedcoefficients and noise-related ones by TSVMs training model. Finally, the detail sub-bands of NSST coefficients were denoised by using the adaptive threshold. Exper-imental results demonstrated that their method could obtain better performances interms of both subjective and objective evaluations than those state-of-the-art denois-ing techniques. Especially, the proposed method could preserve edges and texturesvery well while removing noise.


Gilbert et al. [69] in 2014 presented an open source stand-alone implementationof the 2D discrete shearlet transform using CUDA C++ as well as GPU-acceleratedMATLAB implementations of the 2D and 3D shearlet transforms. They had instru-mented the code so that they could analyze the running time of each kernel underdifferent GPU hardware. In addition to denoising, authors described a novel appli-cation of shearlets for detecting anomalies in textured images. In that application,computation times could be reduced by a factor of 50 or more, compared tomulticoreCPU implementations.

Image normally has both dots-like and curve structures. But the traditionalwaveletor multidirectional wave (ridgelet, contourlet, curvelet, etc.) could only restore oneof these structures efficiently so that the restoration results for complex images areunsatisfactory. Ding and Zhao [45] in 2015 proposed a combined sparsity regulariza-tion method for image restoration based on the respective advantages of shearlet andwavelet in the sparsity regularization method. This method could efficiently restorethe dots-like and curve structures in images, generating higher SNR of restoredimages. They did not use the traditional soft and hard-threshold algorithm to improveconvergence rate, but adopted semismooth Newton method with super linear conver-gence rate. Numerical results showed the analysis results of Lena image restorationfrom various perspectives and demonstrated that the combined sparsity regulariza-tionmethod could restoremore accurately and efficiently in the way of SNR, residualerrors and local dots or curve structures of images.

To remove noise and preserve detail of image as much as possible, Hu et al. [76] in2015, proposed image filter algorithm which combined the merits of Shearlet trans-formation and particle swarm optimization (PSO) algorithm. Experimental resultshad shown that proposed algorithm eliminates noise effectively and yields good peaksignal noise ratio (PSNR).

Sharma and Chugh [154] in 2015 presented a proficient approach intended forimage denoising based on Shearlet transform and the Bayesian Network. The pro-jected technique used the geometric dependencies in the shearlet domain in thedirection of the Bayesian Network which was next used for predict the noise prob-ability. The Shearlet transform provided improved approximation particularly indifferent scales, and directional discontinuities which make it preferable designedused within support of processing the pixel around the edge. The later result provedthat the future technique better wavelet base method visually and mathematical inconditions of PSNR (peak signal-to-noise ratio).

Satellite images have become universal standard in almost all applications ofimage processing. However, satellite images are mostly degraded due to the inaccu-racy or limitations of the transmission and storage devices. Development of a denois-ing algorithm in satellite images is still a challenging task for many researchers.Mostof the state of the art denoising schemes employ wavelet transform but the main lim-itation of wavelet transform is it can capture only limited information along differentdirections. Hence edges in an image get distorted. Shearlet transformation is a sparse,multiscale and multidimensional alternative to wavelet transform.

Anju and Raj [8] in 2016 presented a novel image denoising algorithm utilizingshearlet transform and Otsu thresholding for denoising the satellite images and it was


found to exhibit superior performance among other state of the art image denoisingalgorithms in terms of peak signal to noise ratio (PSNR) and visual quality.

Ehsaeyan [58] in 2016 presented a new approach for image denoising based onshearlet transform, Wiener filter and NeighShrink SURE model. In this method, thelow frequency sub-band coefficients were denoised by applying the adaptive Wienerfilter. As for the high frequency sub-band coefficients, they were refined accordingto the NeighShrink rule. The visual effect image and detailed measurements showedthat his method was more effective, which was not only better in reducing noise,but also had an advantage in preserving the information of edges. Measured resultsrevealed that his scheme had the best PSNRs in most cases.

Mugambi et al. [121] in 2016 proposed an algorithm for image denoising basedon Shearlet Transform and PCA (Principle Component Analysis). The combinedmethod gave better results both byhuman visual and by PSNR values.

Fathima et al. [63] in 2017 proposed the noise removal shearlet transform by hardthreshold for denoising. The multiscale and multidirectional aspects of the shearlettransform provided a better estimation capability for images exhibiting piecewisesmooth edges, Quantitative performance measure such as MSE, RMS, PSNR wereused to evaluated the denoised image effect. TheShearlet Transformwith hard thresh-old was an efficient technique for improving the quality of the image.

Bharath et al. [16] in 2017 proposed a technique by integrating Wavelet andShearlet transform which effectively removes the noise to the maximum extent andrestored the image by edge detection which can be identified. The simulation wasdone on synthetic image and showed improvement with existing methods.

References

1. Abdullah, A.J.: Denoising of an image using discrete stationarywavelet transform and variousthresholding techniques. J. Signal Inf. Process. 4, 33–41 (2013)

2. Achim, A., Kuruoglu, E.E.: Image denoising using bivariate-stable distributions in the com-plex wavelet domain. IEEE Signal Process. Lett. 12(1) (2005)

3. Achim, A., Tsakalides, P., Aezerianos, A.: SAR image denoising via Bayesianwavelet shrink-age based on Heavy-Tailed modeling. IEEE Trans. Geosci. Remote Sens. 41(8), 1773–1784(2003)

4. Ahmed, S.S., Messali, Z., Ouahabi, A., Trepout, S., Messaoudi, C., Marco, S.: Nonpara-metric denoising methods based on contourlet transform with sharp frequency localization:application to low exposure time electron microscopy images. Entropy 17, 3461–3478 (2015)

5. Alparome, L., Baronti, S., Garzelli, A. Nencini, F.: The curvelet transform for fusion of very-high resolution multi-spectral and panchromatic images. In: Proceedings of the 25th EARSeLSymposium, Porto, Portugal. Millpress Science Publishers, Netherlands (2006)

6. Anbarjafari, G., Demirel, H.: Image super resolution based on interpolation of wavelet domainhigh frequency subbands and the spatial domain input image. ETRI J. 32(3), 390–394 (2010)

7. Anisimova, E., Bednar, J., Pata, P.: Astronomical image denoising using curvelet and starlettransform. In: 23th Conference Radioelektronika, Pardubice, Czech Republic (2013)

8. Anju, T.S., Raj, N.R.N.: Shearlet transform based satellite image denoising using optimizedotsu threshold. Int. J. Latest Trends Eng. Technol. (IJLTET) 7(1), 1572–1575 (2016)

9. Anjum, K., Bhyri, C.: Image denoising using curvelet transform and edge detection in imageprocessing. Proc. NCRIET-2015 Indian. J. Sci. Res. 12(1), 60–63 (2015)


10. Argenti, F., Torricelli, G.: Speckle suppression in ultrasonic images based on undecimatedwavelets. EURASIP J. Appl. Signal Process. (2003)

11. Bahri,M., Ashino, R., Vaillancourt, R.: Two-dimensional quaternionwavelet transform.Appl.Math. Comput. 218(1), 10–21 (2011)

12. Bahri, M.: Construction of quaternion-valued wavelets. Matematika 26(1), 107–114 (2010)13. Bains, A.K., Sandhu, P.: Image denoising using curvelet transform. Int. J. Current Eng. Tech-

nol. 5(1), 490–493 (2015)14. Baraniuk, R.G.: Optimal tree approximation with wavelets. Proc. SPIE Tech. Conf. Wavelet

Appl. Signal Process. VII 3813, 196–207 (1999)15. Belzer, B., Lina, J.M., Villasenor, J.: Complex, linear-phase filters for efficient image coding.

IEEE Trans. Signal Proc. 43, 2425–2427 (1995)16. Bharath,K.S.,Kavyashree, S.,Ananth,V.N.,Kavyashree,C.L.,Gayathri,K.M.: Imagedenois-

ing using wavelet and shearlet transform. Recent Adv. Commun. Electron. Electr. Eng. 5(4),8–14 (2017)

17. Borate, S.K.P., Nalbalwar, S.L.: Learning based single-frame image super-resolution usingcontourlet transform. Int. J. Comput. Sci. Inf. Technol. 3, 4707–4711 (2012)

18. Buccigrossi, R.W., Simoncelli, E.P.: Image compression via joint statistical characterizationin the wavelet domain. IEEE Image Process. 8(12), 1688–1701 (1999)

19. Bulow, T.: Hypercomplex Spectral Signal Representations for the Processing and Analysis ofImages. Ph.D. thesis, Christian Albrechts University, Kiel, Germany (1999)

20. Candes, E.J., Donoho, D.L.: Ridgelets: a key to higher-dimensional intermittency? Philos.Trans. R. Soc. Lond. A. 335, 2495–2509 (1999)

21. Cands, E.J., Donoho, D.L.: Curvelets and curvilinear integrals. J. Approx. Theory 113, 59–90(2001)

22. Candes, E.J.: Ridgelets: Theory and Applications. Stanford University, Stanford (1998)23. Carre, P., Andres, E.: Discrete analytical ridgelet transform. Signal Process. 84, 2165–2173

(2004)24. Chan, W.L., Choi, H., Baraniuk, R.G.: Coherent multiscale image processing using dual-tree

quaternion wavelets. IEEE Trans. Image Process. 17(7), 1069–1082 (2008)25. Chang, S.G., Yu, B., Vetterli, M.: Spatially adaptive wavelet thresholding with context mod-

eling for image denoising. IEEE Trans. Image Process. 9, 1522–1531 (2000)26. Chang, S.G., Yu, B., Vetterli, M.: Adaptive wavelet thresholding for image denoising and

compression. IEEE Trans. Image Process. 9, 15321546 (2000a)27. Chang, S.G., Yu, B., Vetterli, M.:Wavelet thresholding for multiple noisy image copies. IEEE

Trans. Image Process. 9, 1631–1635 (2000b)28. Chan, R.H., Chan, T.F., Shen, L., Shen, Z.: Wavelet algorithms for high-resolution image

reconstruction. Siam J. Sci. Comput. 24(4), 1408–1432 (2003)29. Chaudhari, Y.P., Mahajan, P.M.: Image denoising of various images using wavelet transform

and thresholding techniques. Int. Res. J. Eng. Technol. (IRJET) 4(2), (2017)30. Chen, G.Y., Kgl, B.: Image denoising with complex ridgelets. Pattern Recognit. 40, 578–585

(2007)31. Chen, X., Sun, H., Deng, C.: Image denoising algorithm using adaptive shrinkage threshold

based on shearlet transform. In: International Conference on Frontier of Computer Scienceand Technology, pp. 254–257 (2009)

32. Chipman, H.A., Kolaczyk, E.D., McCulloch, R.E.: Adaptive Bayesian wavelet shrinkage. J.Am. Stat. Assoc. 92(440), 1413–1421 (1997)

33. Chitchian, S., Fiddy, M.A., Fried, N.M.: Denoising during optical coherence tomographyof the prostate nerves via wavelet shrinkage using dual tree complex wavelet transform. J.Biomed. Opt. 14(1) (2009)

34. Cho, D., Bui, T.D.: Multivariate statistical modeling for image denoising using wavelet trans-forms. Signal Processing. 20(1), 77–89 (2005)

35. Choi,H., Baranuik,R.G.:Multiplewavelet basis image denoising using besov ball projections.IEEE Signal Process. Lett. 11(9) (2004)

References 191

36. Choi, H., Baranuik, R.G.: Analysis of wavelet domain wiener filters. In: IEEE InternationalSymposium on Time Frequency and Time-Scale Analysis, Pittsburgh (1998)

37. Chouksey, S., Shrivastava, S., Mishra, A., Dixit, A.: Image restoration using bi-orthogonalwavelet transform. Int. J. Sci. Eng. Res. 6(1), 1594–1600 (2015)

38. Cohen, I., Raz, S., Malah, D.: Translation invariant denoising using the minimum descriptionlength criterion. Signal Process. 75(3), 201–223 (1999)

39. Coifman, R., Donoho, D.: Translation Invariant Denoising. Lecture Notes in Statistics:Wavelets and Statistics, pp. 125–150. Springer, New York (1995)

40. Corrochano, E.B.: Multi-resolution image analysis using the quaternion wavelet transform.Numer. Algorithms 39(13), 35–55 (2005)

41. Corrochano, E.B.: The theory and use of the quaternion wavelet transform. J. Math. ImagingVis. 24(1), 19–35 (2006)

42. Crouse,M.S., Nowak, R.D., Baraniuk, R.G.:Wavelet-based statistical signal processing usingHidden Markov models. IEEE Trans. Signal Process. 46(4), 886–902 (1998)

43. DaCunha, A.L., Zhou, J., Do,M.N.: The nonsubsampled contourlet transform: theory, design,and applications. IEEE Trans. Image Process. 15(10), 3089–3101 (2006)

44. Deng, C., Tian, W., Hu, S., Li, Y., Hu, M., Wang, S.: Shearlet-based image denoising usingadaptive thresholding and non-local means filter 6(20), 333–342 (2012)

45. Ding, L., Zhao, X.: Shearlet-wavelet regularized semismooth newton iteration for imagerestoration. Math. Probl. Eng. 2015, Article ID 647254, 1–12 (2015)

46. Divya, V., kumar, Dr. S.: Noise removal in images in the non subsampled contourlet transformdomain using orthogonal matching pursuit. Int. J. Innov. Res. Comput. Commun. Eng. 3(6)(2015)

47. Do, M.N., Vetterli, M.: Image denoising using orthonormal finite ridgelet transform. In: Pro-ceedings of SPIE Conference on Wavelet Applications in Signal and Image Processing VIII,San Diego, vol. 831 (2000)

48. Do, M.N., Vetterli, M.: Orthonormal finite ridgelet transform for image compression. In:International Conference on Image Processing, Vancouver, BC, Canada, pp. 367–370. IEEE(2000)

49. Do, M.N., Vetterli, M.: The contourlet transform: an efficient directional multiresolutionimage representation. IEEE Trans. Image Process. 14, 2091–2106 (2005)

50. Donoho, D.L., Flesia, A.G.: Digital ridgelet transform based on true ridge functions. Stud.Comput. Math. 10, 1–30 (2003)

51. Donoho, D.L., John Stone, I.M.: Ideal spatial adaptation via wavelet shrinkage. Biometrika81(3), 425–455 (1994)

52. Donoho, D.L., John Stone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. J.Am. Stat. Assoc. 12(90), 1200–1224 (1995)

53. Donoho, D.L.: De-noising by soft-threshold. IEEE Trans. Inf. Theory 41(3), 613–627 (1995)54. Donoho, D.L.: Orthonormal ridgelets and linear singularities. SIAM J. Math. Anal. 31, 1062–

1099 (2000)55. Donoho, D.L.: CART and best-ortho-basis: a connection. Ann. Statist. 1870–1911 (1997)56. Donoho, D.: Digital Ridgelet Transform via Rectopolar Coordinate Transform. Stanford Uni-

versity, Stanford CA, Technical Report 12, 854–865 (1998)57. Easley, G., Labate, D., Lim,W.Q.: Sparse directional image representations using the discrete

shearlet transform. Appl. Comput. Harmon. Anal. 25, 25–46 (2008)58. Ehsaeyan, E.: A new shearlet hybrid method for image denoising. Iranian J. Electron. Electr.

Eng. 12(2), 97–104 (2016)59. Eslami,R., Radha,H.:Wavelet-based contourlet transformand its application to image coding.

In: International Conference on Image Processing. ICIP, pp. 3189–3192. IEEE, Singapore(2004)

60. Eslami, R. Radha, H.: Image Denoising Using Translation-Invariant Contourlet Transform,ICASSP, 557–560 (2005)

61. Fan, Z., Sun, Q., Ruan, F., Gong, Y. Ji, Z.: An improved image denoising algorithm based onshearlet. Int. J. Signal Process. Image Process. Pattern Recognit. 6(4) (2013)


62. Fang-Fei, Y.U., Xia, L., LI-gang, S.: Image denoising based on the quaternion wavelet Q-HMTmodel. In: Proceedings of the 33rd Chinese Control Conference, China, pp. 6279–6282(2014)

63. Fathima, K.M.N.S.A., Shajun, N.S.: Shearlet transform based denoising on normal imagesusing hard threshold. Int. J. Innov. Res. Comput. Commun. Eng. 5(1), 122–129 (2017)

64. Figueiredo, M.A.T., Nowak, R.D.: An EM algorithm for wavelet-based image restoration.IEEE Trans. Image Process. 12(8), 906–916 (2003)

65. Firoiu, I., Isar, A., Isar, D.: A Bayesian approach of wavelet based image denoising in ahyperanalytic multi-wavelet context. WSEAS Trans. Signal Process. 6, 155–164 (2010)

66. Fodor, I.K., Kamath, C.: Denoising throughwavelet shrinkage: an empirical study. J. Electron.Imaging. 12(1), 151–160 (2003)

67. Gadakh, J.S., Thorat, P.R.: A wavelet based image restoration for MR images. Int. J. Eng.Comput. Sci. 4(9), 14111–14117 (2015)

68. Giaouris, D., Finch, J.W.: Denoising using wavelets on electric drive applications. Electric.Power Syst. Res. 78, 559–565 (2008)

69. Gibert, X., Patel, V.M., Labate, D., Chellapa, R.: Discrete shearlet transform on GPU withapplications in anomaly detection and denoising. EURASIP J. Adv. Signal Process. 2014(64),1–14 (2014)

70. Gnanadurai, D., Sadasivam,V.: An efficient adaptive thresholding technique forwavelet basedimage denoising. Int. J. Electr. Comput. Energ. Electron. Commun. Eng. 2(8), 1703–1708(2008)

71. Guo, D., Chen, J.: The application of contourlet transform to image denoising. Proc. Eng. 15,2333–2338 (2011)

72. Guo, H., Odegard, J.E., Lang, M., Gopinath, R.A., Selesnick, I.W., Burrus, C.S.: Waveletbased speckle reduction with application to SAR based ATD/R. Int. Conf. Image Process. 1,75–79 (1994)

73. Gupta, K., Gupta, S.K.: Image denoising techniques- a review. Int. J. Innov. Technol. Explor.Eng. (IJITEE) 2(4), (2013)

74. He, J.X., Yu, B.: Continuous wavelet transforms on the space L2(R,H , dx). Appl. Math. Lett.17(1), 111–121 (2004)

75. Hsung, T.C., Lun, D.P.K., Ho, K.C.: Optimizing the multiwavelet shrinkage denoising. IEEETrans. Signal Process. 53(1), 240–251 (2005)

76. Hu, K., Song, A., Xia, M., Fan, Z., Chen, X., Zhang, R., Zheng, Z.: An image filter based onshearlet transformation and particle swarm optimization algorithm. Math. Probl. Eng. 2015,Article ID 414561, (2015)

77. Huang, Q., Hao, B., Chang, S.: Adaptive digital ridgelet transform and its application in imagedenoising. Digital Signal Process. 52, 45–54 (2016)

78. Jannath, F.P., Latha, R.G.L., Shajun, N.S.: Empirical evaluation of wavelet and contourlettransform for medical image denoising using shrinkage thresholding techniques. Int. J. Innov.Res. Comput. Commun. Eng. 4(5), 9028–9036 (2016)

79. Jannath, F.P. , Shajun, N.S., Sathik, Dr.M.M.: Noise reduction inMRI images using contourlettransform and threshold shrinkages techniques. Int. J. Comput. Sci. Inf. Technol. 7(2), 723–728 (2016)

80. Jain, A.K.: Fundamentals of Digital Image Processing. Prentice-Hall, New Jersey (1989);Digital image processing, Pearson publication (2006)

81. Jansen, M., Bulthel, A.: Empirical bayes approach to improve wavelet thresholding for imagenoise reduction. J. Am. Stat. Assoc. 96(454), 629–639 (2001)

82. Kadiri, M., Djebbouri, M., Carr, P.: Magnitude-phase of the dual-tree quaternionic wavelettransform for multispectral satellite image denoising. EURASIP J. Image Video Process. 41,1–16 (2014)

83. Kang, J., Zhang, W.: Quickbird remote sensing image denoising using ridgelet transform. In:Lei, J., Wang, F.L., Deng, H., Miao, D. (eds.) Emerging Research in Artificial Intelligenceand Computational Intelligence. Communications in Computer and Information Science, vol.315, pp. 386–391. Springer, Berlin (2012)

References 193

84. Kaur, J., Kaur, M., Kaur, J.: Denoising gray scale images using curvelet transform. Int. J.Electron. Commun. Comput. Eng. 3(3), 426–429 (2012)

85. Kaur, R., Mann, P.S.: Image denoising using M-band ridgelet transform. Int. J. Sci. EmergingTechnol. Latest Trends 10(1), 12–15 (2013)

86. Kingsbury, N.G.: The Dual-Tree Complex Wavelet Transform: A New Technique for ShiftInvariance and Directional Filters. In: Proceedings of the 8th IEEE DSP Workshop, BryceCanyon, UT, USA, Paper no. 86 (1998)

87. Kingsbury, N.G.: The dual-tree complex wavelet transform: a new efficient tool for imagerestoration and enhancement. In: Proceedings of the EUSIPCO 98, Rhodes, Greece, pp. 319–322 (1998)

88. Kingsbury, N.G.: A dual-tree complex wavelet transform with improved orthogonality andsymmetry properties. In Proceedings of the IEEE International Conference on Image Proc(ICIP) (2000)

89. Koc, S., Ercelebi, E.: Image restoration by lifting-basedwavelet domain E-median filter. ETRIJ. 28(1), 51–58 (2006)

90. Kongo, R.M., Masmoudi, L., Mohammed, C.: Dual-tree complex wavelet in medical ultra-sounds images restoration. In: International Conference on Multimedia Computing and Sys-tems (ICMCS) (2012)

91. Kourav, D., Chandrawat, U.B.S.: Comparative performance analysis of contourlet transformusing non subsampled algorithm in digital image processing. Int. Adv. Res. J. Sci. Eng.Technol. 4(5), 148–152 (2017)

92. Krishnanaik, V., Someswar, G.M.: Sliced ridgelet transform for image denoising. Int. J. Innov.Res. Comput. Commun. Eng. 4(6), (2016)

93. Krishnanaik, V., Someswar, G.M., Purushotham, K., Rao, R.S.: Image denoising using com-plex ridgelet trasform. Int. J. Eng. Res. Technol. (IJERT). 2(10) (2013)

94. Kumar, P., Bhurchandi, K.: Image denoising using ridgelet shrinkage. In: Proceedings of theSPIE 9443, Sixth International Conference on Graphic and Image Processing (ICGIP 2014),94430X (2015)

95. Kumar, R., Saini, B.S.: Improved image denoising technique using neighboring wavelet coef-ficients of optimal wavelet with adaptive thresholding. Int. J. Comput. Theory Eng. 4(3),395–400 (2012)

96. Kumar, S.P., Reddy, T.S.: Complex wavelet transform based hyper analytical image denoisingby bivariate shrinkage technique 43(3) (2017)

97. Kumar, S.: Image de-noising usingwavelet-like transform. Int. J. Adv. Res. Innov. 5(1), 70–72(2017)

98. Lawton, W.: Applications of complex valued wavelet transforms to sub-band decomposition.IEEE Trans. Signal Proc. 41, 3566–3568 (1993)

99. Lang, M., Guo, H., Odegard, J.E., Burrus, C.S.: Nonlinear Processing of a Shift InvariantDWT for Noise Reduction. SIPE, Mathematical Imaging: Wavelet Applications for Dual Use(1995)

100. Leena, P.V., Remmiya, Devi G., Sowmya, V., Somam, K.P.: Least square based image denois-ing using wavelet filters. Indian. J. Sci. Technol. 9(30), 1–6 (2016)

101. Li, C.J., Yang, H. X., Cai, Y.Y., Song, B.: Image denoising algorithm based on nonsubsam-pled contourlet transform and bilateral filtering. In: International Conference on ComputerInformation Systems and Industrial Applications (CISIA 2015), pp. 666–669 (2015)

102. Li, C., Liu, C., Zou, H.: Wavelet domain image restoration model based on variation regular-ization. Int. J. Signal Process Image Process Pattern Recognit. 7(2), 165–176 (2014)

103. Li, Y., Chen, R.M., Liang, S.: A new image denoising method based on shearlet shrinkage andimproved total variation. In: International Conference on Intelligent Science and IntelligentData Engineering, ISCIDE, pp. 382–388 (2011)

104. Liu, G., Zengand, X., Liu, Y.: Image denoising by random walk with Restart Kernel andnon-sampled contourlet transform. IET Signal Process. 6, 148–158 (2012)

105. Lu, J., Weaver, J.B., Healy, D.M., Xu, Y.: Noise reduction with multiscale edge representationand perceptual criteria. In: Proceedings of the IEEE-SP International Symposium on TimeFrequency and Time-Scale Analysis, Victoria, BC, pp. 555–558 (1992)


106. Liu,Y.X., Law,N.F., Siu,W.C.: Patch based image denoising using the finite ridgelet transformfor less artifacts. J. Vis. Commun. Image Represent 25, 1006–1017 (2014)

107. Liu, Y., Li, L., Li, C.: Elimination Noise by wavelet threshold and subband enhancement. J.Xiamen Univ. (Nat. Sci.) 51(3), 342–347 (2012)

108. Luisier, F., Blu, T., Unser,M.: A new SURE approach to image denoising: interscale orthonor-mal wavelet thresholding. IEEE Trans. Image Process. 16(3), 593–606 (2007)

109. Ma, J., Plonka, G.: Combined curvelet shrinkage and nonlinear anisotropic diffusion. IEEETrans. Image Process. 16, 2198–2206 (2007)

110. Malfait, M., Roose, D., Wavelet based image denoising using a Markov random field a priorimodel. IEEE Trans. Image Process. 6(4) (1997)

111. Mallat, S., Hwang, W.L.: Singularity detection and processing with wavelets. IEEE Trans.Inf. Theory. 38(2), 617–643 (1992)

112. Malleswari, N., Madhu, C.: Image denoising based on quaternion wavelet transform. Int. J.Recent Trends VLSI Embedd. Syst. Signal Process. 2(4), 17–20 (2016)

113. Matalon, B., Elad, M., Zibulevsky, M.: Image denoising with the contourlet transform. In:Proceedings of SPARSE, Rennes, France (2005)

114. Mathew, K., Shibu, S.: Wavelet based technique for super resolution image reconstruction.Int. J. Comput. Appl. 33(7), 11–17 (2011)

115. Mihcak, M.K., Kozintsev, I., Ramchandran, K., Moulin, P.: Low-complexity image denoisingbased on statistical modeling of wavelet coefficients. IEEE Signal Process. Lett. 6(12), (1999)

116. Mitiche, L., Mitiche, A.B.H.A., Naimi, H.: Medical image denoising using dual tree complexthresholding wavelet transform. In: IEEE Jordan Conference on Applied Electrical Engineer-ing and Computing Technologies (2013)

117. Mitrea, M.: Clifford Wavelets, Singular Integrals and Hardy Spaces. Lecture Notes in Math-ematics, vol. 1575. Springer, Berlin (1994)

118. Mohideen,K., Perumal,A.,Krishnan,N., Sathik,M.: Image denoising and enhancement usingmultiwavelet with hard threshold in digital mammographic images. Int. Arab. Je. Technol. 2,49–55 (2011)

119. Motwani, M.C., Gadiya, M.C., Motwani, R.C.: Survey of image denoising techniques. ImageProcess. 19(4), 895–911 (2010)

120. Moulin, P., Liu, J.: Analysis of multiresolution image denoising schemes using generalizedGaussian and complexity priors. IEEE Infor. Theory. 45(3), 909–919 (1999)

121. Mugambi, L.N., Waweru, M., Kimwele, M.: Image denoising using a combined system ofshearlets and principle component analysis algorithm. Int. J. Comput. Technol. 3(1), 1–7(2016)

122. Naik, S., Patel, N.: Single image super resolution in spatial and wavelet domain. Int. J. Mul-timed. Appl. 5(4), 23–32 (2013)

123. Naimi, H., Mitiche, A.B.H.A., Mitiche, L.: Medical image denoising using dual tree complexthresholding wavelet transform and wiener filter. J. King Saud Univ. Comput. Inf. Sci. 27,40–45 (2015)

124. Neha, Khera: S.: Image denoising using DWT-EEMD algorithm. Int. J. Electr. Electron. Res.2(3), 37–43 (2014)

125. Neole, B.A., Lad, B.V., Bhurchandi, K.M.: Denoising of digital images using consolidationof edges and separable wavelet transform. Int. J. Comput. Appl. 168(10), 24–32 (2017)

126. Oad, R.: Bhavana.: Efficient image denoising with enhanced contourlet transform with filter-ing. Int. J. Emerg. Technol. Adv. Eng. 7(5), 145–149 (2017)

127. Oliveira, M.S., Henriques, M.V.C., Leite, F.E.A., Corsoc, G., Lucena, L.S.: Seismic denoisingusing curvelet analysis. Physica A 391, 2106–2110 (2012)

128. Padmagireeshan, S.J., Johnson, R.C., Balakrishnan, A.A., Paul, V., Pillai, A.V., Abdul, R.A.:Performance analysis of magnetic resonance image denoising using contourlet transform.In: Third International Conference on Advances in Computing and Communications, pp.396–399 (2013)

129. Palakkal, S., Prabhu, K.: Poisson image denoising using fast discrete curvelet transform andwave atom. Signal Process. 92, 2002–2017 (2012)

References 195

130. Parlewar, P.K., Bhurchandi, K.M.: A 4-quadrant curvelet transform for denoising digitalimages. Int. J. Autom. Comput. 10, 217–226 (2013)

131. Patil, A.A., Singhai, R., Singhai, J.: Curvelet transform based super-resolution using sub-pixel image registration. In: 2nd Computer Science and Electronic Engineering Conference(CEEC) (2010)

132. Patil, A.A., Singhai, J.: Image denoising using curvelet transform: an approach for edgepreservation. J. Sci. Indust. Res. 69, 34–38 (2010)

133. Patil, A.A., Singhai, J.: Discrete curvelet transform based super-resolution using sub-pixelimage registration. Int. J. Signal Process. Image Process. Pattern Recognit. 4(2), 41–50 (2011)

134. Pizurica, A., Philips, W., Lemahieu, I., Acheroy, M.: A joint inter and intrascale statisticalmodel for Bayesian wavelet based image denoising. IEEE Trans. Image Process. 11(5) (2002)

135. Po, D.D.Y., Do, M.N.: Directional multiscale modeling of images using the contourlet trans-form. IEEE Trans. Image Process. 15, 1610–1620 (2006)

136. Poornachandra, S.: Wavelet-based denoising using subband dependent threshold for ECGsignals. Digital Signal Processing. 18, 49–55 (2008)

137. Portilla, J., Strela, V., Wainwright, M.J., Simoncelli, E.P.: Image denoising using scale mix-tures of gaussians in the wavelet domain. IEEE Trans. Image Process. 12(11) (2003)

138. Rahman, S.M.M., AhmadM.O., Swamy, M.N.S.: Improved image restoration using wavelet-based denoising and fourier-based deconvolution. In: Proceedings of 51st IEEE InternationalMid-West Symposium on Circuits and Systems, Knoxville, TN, USA, pp. 249–252 (2008)

139. Raja, V.N.P., Venkateswarlub, T.: Denoising of medical images using dual tree complexwavelet transform. Proc. Technol. 4, 238–244 (2012)

140. Raju, C., Reddy, T.S., Sivasubramanyam, M.: Denoising of remotely sensed images viacurvelet transform and its relative assessment. Proc. Comput. Sci. 89, 771–777 (2016)

141. Rakesh, U., Lakshmi, T.K., Krishna, V.V., Reddy, P.R.: Super resolution image generationusing wavelet domain inter polation with edge extraction via a sparse representation. Int. J.Innov. Sci. Eng. Technol. 2(8), 38–52 (2015)

142. Rakheja, P., Vig, R.: Image denoising using combination of median filtering and wavelettransform. Int. J. Comput. Appl. 141(9) (2016)

143. Rakvongthai, Y., Vo, A.P.N., Oraintara, S.: Complex Gaussian scale mixtures of complexwavelet coefficients. IEEE Trans. Signal Process. 58(7), 3545–3556 (2010)

144. Ramadhan, J., Mahmood, F., Elci, A.: Image denoising by median filter in wavelet domain.Int. J. Multimed. Appl. (IJMA) 9(1) (2017)

145. Rao, R.M.B., Ramakrishna, P.: Image denoising based on MAP estimation using dual treecomplex wavelet transform. Int. J. Comput. Appl. 110(4), 19–24 (2015)

146. Romberg, J.K., Choi, H., Baraniuk, R.G.: Bayesian wavelet domain image modeling usingHidden Markov models. IEEE Trans. Image Process. 10, 1056–1068 (2001)

147. Rosenbaum, W.L., Atkinsa, M.S., Sarty, G.E.: Classification and performance of denoisingalgorithms for low signal to noise ratio magnetic resonance images. SPIE 3979, 1436–1442(2000)

148. Ruikar, S.D., Doye, D.D.: Wavelet based image denoising technique. Int. J. Adv. Comput.Sci. Appl. 2(3), 49–53 (2011)

149. Sakthivel, K.: Contourlet based image denoising using new threshold function. Int. J. Innov.Res. Comput. Commun. Eng. 2(1) (2014)

150. Sathesh, Manoharan, S.: A dual tree complex wavelet transform construction and its applica-tion to image denoising. Int. J. Image Process. (IJIP) 3(6), 293–300 (2010)

151. Selesnick, I.W., Baraniuk, R.G., Kingsbury, N.C.: The dual-tree complex wavelet transform.IEEE Signal Process Mag. 22, 123–151 (2005)

152. Sender, L., Selesnick, I.W.: Bivariate shrinkage functions forwavelet-based denoising exploit-ing interscale dependency. IEEE Trans. Signal Process. 50(11), (2002)

153. Shah, A.J., Makwana, R., Gupta, S.B.: Single image super-resolution via non sub-samplecontourlet transform based learning and a gabor prior. Int. J. Comput. Appl. 64(18), 32–38(2013)


154. Sharma, A., Chugh, S.: An efficient shearlet Bayesian network based approach for imagedenoising. Int. J. Comput. Appl. 128(10) (2015)

155. Shen, X., Wang, K., Guo, Q.: Local thresholding with adaptive window shrinkage in thecontourlet domain for image denoising. Science China. Inf Sci. 56(19) (2013)

156. Shuyuan,Y.,Wang,M., Linfang, Z., Zhiyi,W.: Image noise reduction via geometricmultiscaleridgelet support vector transform and dictionary learning. IEEE Trans. Image Process. 22,4161–4169 (2013)

157. Simoncelli, E.P., Adelson, E.H.: Noise removal via bayesian wavelet coring. In: Third Inter-national Conference on Image Processing, vol. I, pp. 379–382, Lausanne, IEEE Signal ProcSociety (1996)

158. Sivakumar, R., Balaji, G., Ravikiran, R.S.J., Karikalan, R., Janaki, S.S.: Image denoisingusing contourlet transform. In: Second International Conference on Computer and ElectricalEngineering, pp. 22–25 (2009)

159. Soulard, R., Carre, P.: Quaternionic wavelets for texture classification. Pattern Recognit. Lett.32(13), 1669–1678 (2011)

160. Sowmya, K., Kumari, P.S., Ranga, A.: Single image super resolution with wavelet domaintransformation and sparse representation. Int. J. Innov. Res. Comput. Sci. Technol. 4(1), 1–6(2016)

161. Starck, J.L., Cands, E.J., Donoho, D.L.: The curvelet transform for image denoising. IEEETrans. Image Process. 11, 670–684 (2002)

162. Starck, J.L., Donoho, D.L., Cands, E.J.: Astronomical image representation by the curvelettransform. Astron. Astrophys. Berl. 398, 785–800 (2003)

163. Starck, J.L., Murtagh, F., Cands, E.J., Donoho, D.L.: Gray and color image contrast enhance-ment by the curvelet transform. IEEE Trans. Image Process. 12, 706–717 (2003)

164. Strela, V.: Denoising via block wiener filtering in wavelet domain. In: 3rd European Congressof Mathematics, Barcelona, Birkhuser Verlag (2000)

165. Strela, V., Portilla, J., Simoncelli, E.P.: Image denoising via a local gaussian scale mixturemodel in the wavelet domain. In: Proceedings of the SPIE 45th Annual Meeting, San Diego,CA (2000)

166. Strela, V., Walden, A.T.: Signal and Image Denoising via Wavelet Thresholding: OrthogonalandBiorthogonal, Scalar andMultipleWavelet Transforms. Statistic Section Technical ReportTR-98-01, Department of Mathematics, Imperial College, London (1998)

167. Sudha, S., Suresh, G.R., Sukanesh, R.: Wavelet based image denoising using adaptive thresh-olding. Int. Conf. Comput. Intell. Multimed. Appl. 305, 296–300 (2007)

168. Sun, L.L., Li, Y., Zheng, J.M.: Image denoising with contourlet transform based on PCA. In:International Symposium on Computer Science and Computational Technology, pp. 31–33(2008)

169. Talbi, M., Cherif, A.: A hybrid technique of image denoising using the curvelet transformbased denoising method and two-stage image denoising by PCA with local pixel grouping.Current Med. Imaging Rev. 13(4), (2017)

170. Thangadurai, N.G., Patrick, X.J.: Image restoration using blur invariants in wavelet transform.Int. J. Eng. Develop. Res. 5(1), 309–313 (2017)

171. Tianfeng, H.: The method of quaternions wavelet image denoising. Int. J. Signal Process.Image Process Pattern Recognit. 7(4), 325–334 (2014)

172. Traversoni, L.: Image analysis using quaternion wavelets. In: Geometric Algebra with Appli-cations in Science and Engineering. Corrochano, E.B.C., Sobczyk, G. (eds.), Birkhauser, pp.326–343 (2001)

173. Vaghela, A.B.: Image restoration using curvelet transform. J. Inf. Knowl. Res. Electron.Commun. Eng. 2(2), 783–787 (2013)

174. Varinderjit, K., Singh, A., Dogra, A.K.: A review underwater image enhancement. Int. J. Adv.Res. Comput. Commun. Eng. 3(7) (2014)

175. Vetrivelan, P., Kandaswamy, A.: Medical image denoising using wavelet-based ridgelet trans-form. In: Sridhar V., Sheshadri H., Padma M. (eds) Emerging Research in Electronics, Com-puter Science and Technology. Lecture Notes in Electrical Engineering. vol. 248. Springer,New Delhi (2014)

References 197

176. Vijay, A.K., Mathurakani, M.: Image denoising using dual tree complex wavelet transform.Int. J. Res. Eng. Technol. 3(1), 60–64 (2014)

177. Vohra, R., Tayal, A.: Image restoration using thresholding techniques on wavelet coefficients.Int. J. Comput. Sci. Issues 8(5), 400–404 (2011)

178. Wagadre, K., Singh, M.: Wavelet transform based fusion technique for image restoration. Int.J. Comput. Appl. 153(12), 18–20 (2016)

179. Wang, G., Wang, Z., Liu, J.: A New Image Denoising Method Based on Adaptive MultiscaleMorphological Edge Detection. Mathematical Problems in Engineering. Hindwai (2017)

180. Wang, X.T., Shi, G.M., Niu, Y., Zhang, L.: Robust adaptive directional lifting wavelet trans-form for image denoising. IET Image Process. 5, 249–260 (2011)

181. Wang, X.Y., Yang, H.Y., Zhang, Y., Fu, Z.K.: Image denoising using SVM classification innonsubsampled contourlet transform domain. Inf. Sci. 246, 155–176 (2013)

182. Wang, Y., He, Z., Zi, Y.: Enhancement of signal denoising and multiple fault signaturesdetecting in rotatingmachinery using dual-tree complexwavelet transform.Mech. Syst. SignalProcess. 24, 119–137 (2010)

183. Wang, Z., Bovik, A.C., Sheikh, H.R.: Simoncelli, E.P.: Image quality assessment: from errorvisibility to structural similarity. IEEE Trans. Image Process. 13, 600–612 (2004)

184. Wei, S., Liu. J., Qi. Y.W., Wang, S., Gao, E.Y.: An improved contourlet transform imagedenoising method. In: International Conference on Modelling, Identification and Control, pp.18–20 (2015)

185. Wink, A.M., Roerdink, J.B.T.M.: Denoising functional MR images a comparison of waveletdenoising and Gaussian smoothing. IEEE Trans. Med. Imaging 23(3) (2004)

186. Wood, J.C., Johnson, K.:Wavelet-Denoising of ComplexMagnetic Resonance Images. IEEE,New York (1998)

187. Wu, K., Zhang, X., Ding, M.: Curvelet based nonlocal means algorithm for image denoising.AEU Int. J. Electron. Commun. 68, 37–43 (2014)

188. Wu, P., Xu, H., Xie, P.: Research on ground penetrating radar image denoising using nonsub-sampled contourlet transform and adaptive threshold algorithm. Int. J. Signal Process. ImageProcess Pattern Recognit. 9(5), 219–228 (2016)

189. Xie, J., Zhang, D., Xu, W.: Spatially adaptive wavelet denoising using the minimum descrip-tion length principle. IEEE Trans. Image Process. 13(2) (2004)

190. Xingming, L., Jing, Z.: Study of Image Restoration Using Complex Wavelets. Image andSignal Processing (2009)

191. Xu, Y.H., Zhao, Y.R., Hong, W.X.: Based on quaternion of small amplitude phase represen-tation and block voting strategy approach for face recognition. Appl. Res. Comput. 27(10),3991–3994 (2010)

192. Xu, Y., Weave, B., Healy, D.M.: Wavelet transform domain filters: a spatially selective noisefiltration technique. IEEE Trans. Image Process. 3(6), 217–237 (1994)

193. Yang, H.Y.,Wang, X.Y., Qu, T.X., Fu, Z.K.: Image denoising using bilateral filter and gaussianscalemixtures in shiftable complex directional pyramid domain. Comput. Electr. Eng. 37,656–668 (2011)

194. Yang,H.Y.,Wang,X.Y.,Niua. P.P., Liua,Y.C.: Image denoising using non subsampled shearlettransform and twin support vector machines. Neural Netw. 57, 152–165 (2014)

195. Yang, R.K., Yin, L., Gabbouj, M., Astola, J., Neuvo, Y.: Optimal weighted median filteringunder structural constraints. IEEE Trans. Signal Process. 43(3), (1995)

196. Yaseena, A.S., Pavlovaa, O.N., Pavlova, A.N., Hramov, A.E.: Image denoising with the dual-tree complexwavelet transform. In: Proceedings of the SPIE 9917, Saratov FallMeeting 2015:Third International Symposium on Optics and Biophotonics and Seventh Finnish-RussianPhotonics and Laser Symposium (PALS), 99173K (2016)

197. Yin, M., Liu, W., Shui, J., Wu, J.: Quaternion wavelet analysis and application in imagedenoising 2012, Article ID 493976 (2012)

198. Yin, M., Liu,W., Zhao, X., Guo, Q.W., Bai, R.F.: Image denoising using trivariate prior modelin nonsubsampled dual-tree complex contourlet transform domain and non-local means filterin spatial domain. Opt. Int. J. Light. Electron. Opt. 124, 6896–6904 (2013)


199. Yoon,B.J.,Vaidyanathan, P.P.:Wavelet-based denoising by customized thresholding. In: IEEEInternational Conference on Acoustics, Speech and Signal Processing(ICASSP), pp. 925–928(2004)

200. Zanzad, D.P., Rawat, S.N.: Image denoising using wavelet and curvelet transform. Int. J.Innov. Res. Technol. 3(2) (2016)

201. Zhang, L., Bao, P., Wu, X.: Multiscale LMMSE-based image denoising with optimal waveletselection. IEEE Trans. Circuits Syst. Video Technol. 15(4), 468–481 (2005)

202. Zhang,H., Nosratinia, A.,Wells, R.O.: Image denoising viawavelet-domain spatially adaptiveFIRwiener filtering. In: IEEEProceedings of the International Conference onAcoust, Speech,Signal Processing, Istanbul, Turkey (2000)

203. Zhang, X.: Image denoising using dual-tree complex wavelet transform and wiener filter withmodified thresholding. J. Sci. Indust. Res. 75(11), 687–690 (2016)

204. Zhou, W., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from errorvisibility to structural similarity. IEEE Trans. Image Process. 13, 600–612 (2004)

205. Zhou, Y., Wang, J.: Image denoising based on the symmetric normal inverse Gaussian modeland nonsubsampled contourlet transform. IET Image Process. 6, 1136–1147 (2012)

Chapter 6Image Enhancement

The purpose of image enhancement and image restoration techniques is to perk up aquality and feature of an image that result in improved image than the original one.Unlike the image restoration, image enhancement is the modification of an image toalter impact on the viewer. Generally enhancement distorts the original digital values;therefore enhancement is not done until the restoration processes are completed.In image enhancement the image features are extracted instead of restoration ofdegraded image. Image enhancement is the process in which the degraded image ishandled and the appearance of the image by visual is improved. It is a subjectiveprocess and increases contrast of image but image restoration is a more objectiveprocess than image enhancement. Performance of image restoration can bemeasuredvery precisely, whereas enhancement process is difficult to represent inmathematicalform.

6.1 Overview

Image enhancement is one of the measurement issues in high quality pictures such asdigital camera and HDTV. Since clarity of image is very easily affected by weather,lighting, wrong camera exposure or aperture settings, high dynamic range of scene,etc. These conditionsmake an image suffer from loss of information. Image enhance-ment techniques have been widely used in many applications of image processingwhere the subjective quality of images is important for human interpretation. Imageenhancement aims to improve the visual appearance of an image, without affect-ing the original attributes (i.e.,) image contrast is adjusted and noise is removed toproduce better quality image. Image enhancement improves the interpretability orperception of information in images. Contrast is an important factor in any subjec-tive evaluation of image quality. Contrast is created by the difference in luminance


199

200 6 Image Enhancement

reflected from two adjacent surfaces. In other words, contrast is the difference invisual properties that makes an object distinguishable from other objects and thebackground. Due to low contrast image enhancement becomes challenging and alsoobjects cannot be extracted clearly from dark background. Since color images pro-vide more and richer information for visual perception than that of the gray images,color image enhancement plays an important role in digital image processing. Manyalgorithms for accomplishing contrast enhancement have been developed and appliedto problems in image processing.

In the literature there exist many image enhancement techniques that can enhancea digital image without spoiling its important features. The image enhancementmethods can be divided into the following two categories:

1. Spatial-domain processing and2. Frequency-domain processing.Spatial-domain image enhancement acts on pixels directly. The pixel values are

manipulated to achieve desired enhancement. Spatial domain techniques enhancethe entire image uniformly which at times results in undesirable effects.

Frequency-domain image enhancement is a term used to describe the analysisof mathematical functions or signals with respect to frequency and operate directlyon the image transform coefficients. The image is first transformed from spatial tofrequency domain, and the transformed image is then manipulated. It is, in general,not easy to enhance both low- and high-frequency components at the same time usingthe frequency-domain technique.

These traditional techniques thus do not provide simultaneous spatial and spec-tral resolution.Wavelet Transform is capable of providing both frequency and spatialresolution. Wavelet Transform is based upon small waves with varying frequencyand limited duration called wavelets. Since higher frequencies are better resolvedin time and lower frequencies are better resolved in frequency, the use of waveletstherefore ensure good spatial resolution at higher frequencies and good frequency res-olution at lower frequencies. Hence wavelet-based techniques can solve drawbacksof frequency-domain techniques by providing flexibility in analyzing the signal overthe entire time range.

Newly developed wavelet-based multiscale transforms include ridgelet, curvelet,contourlet and shearlet.

6.2 Spatial Domain Image Enhancement Techniques

In the spatial domain image enhancement technique, transformations are directlyapplied on the pixels. The pixel values are manipulated to achieve desired enhance-ment. Spatial domain processes can be expressed as

g(x, y) = T [ f (x, y)], (6.2.1)

6.2 Spatial Domain Image Enhancement Techniques 201

where f (x, y) is the input image, g(x, y) is the processed image, and T is an operatoron f, defined over the neighborhood of (x, y). Spatial domain techniques like thelogarithmic transforms, power law transforms, histogram equalization are based inthe direct manipulation of the pixels in image. Spatial domain methods can again bedivided into two categories; (1) point processing and (2) spatial filtering operations.Now we briefly summarize various spatial domain techniques.

6.2.1 Gray Level Transformation

Gray level transformation is the simplest image enhancement techniques. The valuesof pixels, before and after processing, will be denoted by r and s, respectively, thesevalues are related by an expression of the form s = T (r),where T is a transformationthatmaps a pixel value r into a pixel value s. Gray level transformations are applied toimprove the contrast of the image. This transformation can be achieved by adjustingthe gray level and dynamic range of the image, which is the deviation betweenminimum and maximum pixel value.

In gray level transformations, three basic types of functions are used frequentlyfor image enhancement: linear, logarithmic and power-low.

1. Image Negative

In this method, reverses the pixel value i.e. each pixel is subtracted from L , where,L is the maximum pixel value of the image. This can be expressed as

s = L − 1 − r, (6.2.2)

where s is the negative image or output image, L−1 is the maximum pixel value andr is the input image. The pixel range for both the input image and negative image isin the range (0, L − 1). This type of processing is particularly used for enhancingwhite or gray detail embedded in dark regions of an image, especially when the blackareas are dominant in size.

2. Log Transformation

The general form of the log transformation is given by

s = c log(1 + r), (6.2.3)

where c is a constant and r ≥ 0. This transformation maps a narrow range of lowgray-level values in the input image into a wider range of output levels. Hossain andAlsharif [27] is used Log transformation to expand the dark pixels and compress the


brighter pixel. This compressed the dynamic range of the image with large variationsin pixel values.

3. Power-Law Transformation

The general form of the power-law transformation is given by

s = c rγ, (6.2.4)

where c and γ are positive constants. Sometimes the Eq. (6.2.4) can be written as

s = c (r + ε)γ (6.2.5)

to account for a measurable output when the input is zero. A variety of devicesused for image capture, printing and display respond according to power law. Thisprocess is also called gamma correction. Gamma correction has become increasinglyimportant in the past few years, as use of digital images for commercial purposesover the internet as increased. In addition, gamma correction are useful for generalpurpose contrast manipulation.

6.2.2 Piecewise-Linear Transformation Functions

A practical implementation of some important transformations can be formulatedonly as piecewise linear functions.

Contrast Stretching

Low-contrast images can result of poor illumination, lack of dynamic range in theimaging sensor or evenwrong camera. One of the simplest piecewise linear functionsis a contrast-stretching transformation. The basic idea behind contrast stretching isto increase the dynamic range of the gray levels in the image being processed. Incontrast stretching, upper and lower threshold are fixed and the contrast is stretchedbetween these thresholds. It is contrast enhancement method based on the intensityvalue as shown

I0(x, y) = f (I (x, y)) (6.2.6)

where, the original image is I (x, y), the output image is I0(x, y) after contrastenhancement. The transformation function T is given by

s = T (r), (6.2.7)

6.2 Spatial Domain Image Enhancement Techniques 203

where s is given as

s =⎧⎨

⎩

l · r, if 0 ≤ r < am · (r − a) + v, if 0 ≤ r < bn · (r − b) + w, if 0 ≤ r < l − 1

where l,m, n are the slopes of the three regions, the s is the modified gray leveland r is the original gray level. a and b are the limits of lower and upper threshold.Contrast stretching tend to make the bright region brighter and vice versa.

6.2.3 Histogram Processing

Intensity transformation functions based on information extracted from image inten-sity histograms play a basic role in image processing.

Histogram Equalization

With L total possible intensity levels in the range [0,G], the histogram of a digitalimage is defined as the discrete function

h(rk) = nk, (6.2.8)

where rk is the kth intensity level in the interval [0,G] and nk is the number of pixels inthe image whose intensity level is rk . it is useful to work with normalized histograms,which is obtained by dividing all elements of h(rk) by the total number of pixels in

the image, which we denote by n, i.e. p(rk) = h(rk)

n= nk

n, for k = 1, 2, . . . , L .

Let Pr (r) denote the probability density function (PDF) of the intensity levels ina given image and assume that intensity levels are continuous quantities normalizedto the range [0, 1], where the subscript is used for differentiating between the PDFsof the input and output images. Then

s = T (r) =∫ r

0Pr (w)dw. (6.2.9)

Gonzalez and Woods [23] in 2002, showed that the probability density function ofthe output levels is uniform; that is,

Ps(s) ={l if 0 ≤ s ≤ 1,0 otherwise.


In other words, the preceding transformation generates an image whose intensitylevels are equally likely, and cover the entire range [0, 1]. The result of this intensity-level equalization process is an image with increased dynamic range, which will tendto have higher contrast. It is clear that the transformation function is the cumulativedistribution function (CDF). When dealing with discrete quantities we work withhistograms and call the preceding technique histogram equalization. In general, thehistogram of the processed image will not be uniform, due to the discrete nature ofthe variables. Let Pr (r j ), j = 1, 2, . . . , L , denote the histogram associated with theintensity levels of a given image and the values in a normalized histogram are approx-imations to the probability of occurrence of each intensity level in the image. Fordiscrete quantities, we work with summations, and the equalization transformationbecomes

sk = T (rk) =k∑

j=1

Pr (r j ) =k∑

j=1

n j

n

for k = 1, 2, . . . , L , where sk is the intensity value in the output (processed) imagecorresponding to value rk in the input image.

Histogram equalization produces a transformation function that is adaptive, inthe sense that it is based on the histogram of a given image. However, once thetransformation function for an image has been computed, it does not change unlessthe histogram of the image changes.

Histogram equalization is used for contrast adjustment using the image his-togramWhenROI is represented by close contrast values, this histogram equalizationenhances the image by increasing the global contrast. As a result, the intensities arewell scattered on the histogram and low contrast region is converted to region withhigher contrast. This is achieved by considering more frequently occurring intensityvalue and spreading it along the histogram. Histogram equalization plays a majorrole in images having both ROI and other region as either darker or brighter. Itsadvantage is, it goes good with images having high color depth. For example imageslike 16-bit gray-scale images or continuous data. This technique is widely used inimages that are over-exposed or under-exposed, scientific images like X-Ray imagesin medical diagnosis, remote sensing, and thermal images. Same way this techniquehas its own defects, like unrealistic illusions in photographs and undesirable effectin low color depth images.

6.2.4 Spatial Filtering

Spatial filtering is used to remove noise whose detail was introduced in this chapter.

6.3 Frequency Domain Image Enhancement Techniques 205

6.3 Frequency Domain Image Enhancement Techniques

Image enhancement in the frequency domain is straightforward. We simply computethe Fourier transform of the image to be enhanced, multiply the result by a filter (lowpass filter, high pass filter and homomorphic filter) rather than convolve in the spatialdomain, and take the inverse transform to produce the enhanced image. The ideaof blurring an image by reducing its high frequency components or sharpening animage by increasing the magnitude of its high frequency components is intuitivelyeasy to understand.

6.3.1 Smoothing Filters

The noises, edges and other sharp transitions in the gray level contribute significantlyto the high frequency. Hence smoothing or blurring is achieved by attenuating aspecified range of high frequency components in the transform of a given image,which can be done using a lowpass filter.

A filter that attenuates high frequencies and retains low frequencies unchangedis called lowpass filter. Since high frequencies are blocked, this results a smoothingfilter in the spatial domain. Three are three types of lowpass filters: Ideal lowpassfilter, Gaussian lowpass filter and Butterworth lowpass filter.

1. Ideal LowPass Filter

The most simple lowpass filter is the ideal lowpass filter (ILPF). It suppresses allfrequencies higher than the cut-off frequency r0 and leaves smaller frequenciesunchanged:

H(u, v) ={l if D(u, v) ≤ r0,0 if D(u, v) > r0,

(6.3.1)

where r0 is called the cutoff frequency (nonnegative quantity), and D(u, v) is thedistance from point (u, v) to the frequency rectangle. If the image is of size M × N ,then

D(u, v) =√(

u − M

2

)

+(

v − N

2

)

The lowpass filters considered here are radially symmetric about the origin. Thedrawback of the IDLF function is a ringing effect that occurs along the edges ofthe filtered image. In fact, ringing behavior is a characteristic of ILPF. As we knowthat multiplication in the Fourier domain corresponds to a convolution in the spatialdomain. Due to the multiple peaks of the ideal filter in the spatial domain, the fil-tered image produces ringing along intensity edges in the spatial domain. The cutoff


frequency r0 of the ILPF determines the amount of frequency components passedby the filter. Smaller the value of r0, more the number of image components elimi-nated by the filter. In general, the value of r0 is chosen such that most componentsof interest are passed through, while most components not of interest are eliminated.Hence, it is clear that ILPF is not very practical.

2. Butterworth LowPass Filter

A commonly used discrete approximation to the Gaussian is the Butterworth filter.Applying this filter in the frequency domain shows a similar result to the Gaussiansmoothing in the spatial domain. The transfer function of a Butterworth lowpassfilter (BLPF) of order n, and with cut-off frequency at a distance r0 from the origin,is defined as

H(u, v) = 1

1 +[D(u,v)r0

]2n . (6.3.2)

Is can be easily seen that, frequency response of the BLPF does not have a sharptransition as in the ideal LPF and as the filter order increases, the transition from thepass band to the stop band gets steeper. This means as the order of BLPF increases,it will exhibit the characteristics of the ILPF. The difference can be clearly seenbetween two images with different orders but the same cutoff frequency.

3. Gaussian LowPass Filter

Gaussian filters are important in many signal processing, image processing and com-munication applications. These filters are characterized by narrow bandwidths, sharpcutoffs, and low overshoots. A key feature of Gaussian filters is that the Fourier trans-form of a Gaussian is also a Gaussian, so the filter has the same response shape inboth the spatial and frequency domains. The form of a Gaussian lowpass filter intwo-dimensions is given by

H(u, v) = e−D2(u,v)/2σ2. (6.3.3)

The parameter σ measures the spread or dispersion of the Gaussian curve. Largerthe value of σ, larger the cutoff frequency and milder the filtering is. Let σ = r0. theEq. (6.3.3) becomes

H(u, v) = e−D2(u,v)/2r20 . (6.3.4)

When D(u, v) = r0, the filter is down to 0.607 of its maximum value of 1.The Gaussian has the same shape in the spatial and Fourier domains and therefore

does not incur the ringing effect in the spatial domain of the filtered image. Thisis an advantage over ILPF and BLPF, especially in some situations where any typeof artifact is not acceptable, such as medical image. In the case where tight control


over transition between low and high frequency needed, Butterworth lowpass filterprovides better choice over Gaussian lowpass filter; however, tradeoff is ringingeffect.

TheButterworth filter is a commonly used discrete approximation to theGaussian.Applying this filter in the frequency domain shows a similar result to the Gaussiansmoothing in the spatial domain. But the difference is that the computational costof the spatial filter increases with the standard deviation, whereas the costs for afrequency filter are independent of the filter function. Hence, the Butterworth filteris a better implementation for wide lowpass filters, while the spatial Gaussian filteris more appropriate for narrow lowpass filters.

6.3.2 Sharpening Filters

Sharpening filters emphasize the edges, or the differences between adjacent light anddark sample points in an image. A highpass filter yields edge enhancement or edgedetection in the spatial domain, because edges contain many high frequencies. Areasof rather constant gray level consist of mainly low frequencies and are thereforesuppressed. A highpass filter function is obtained by inverting the correspondinglowpass filter. An ideal highpass filter blocks all frequencies smaller than r0 andleaves the others unchanged. The transfer function of lowpass filter and highpassfilter can be related as follows:

Hhp(u, v) = 1 − Hlp(u, v), (6.3.5)

where Hhp(u, v) and Hlp(u, v) are the transfer function of highpass and lowpassfilter respectively.

1. Ideal HighPass Filter

The transfer function of an ideal highpass filter with the cutoff frequency r0 is:

H(u, v) ={0 if D(u, v) ≤ r0,1 if D(u, v) > r0.

(6.3.6)

2. Butterworth High Pass Filter

The transfer function of Butterworth highpass filter (BHPF) of order n and withcutoff frequency r0 is given by:


H(u, v) = 1

1 +[

r0D(u,v)

]2n . (6.3.7)

The frequency response does not have a sharp transition as in the IHPF. It can be seenthat BHPF behaves smoother and has less distortion than IHPF. Therefore, BHPF ismore appropriate for image sharpening than the IHPF. Also less ringing is introducedwith small value of the order n of BHPF.

3. Gaussian HighPass Filter

The transfer function of a Gaussian high pass filter (GHPF) with the cutoff frequencyr0 is given by:

H(u, v) = 1 − e−D2(u,v)/2r20 . (6.3.8)

The parameter σ, measures the spread or dispersion of the Gaussian curve. Largerthe value of σ, larger the cutoff frequency and milder the filtering is.

6.3.3 Homomorphic Filtering

An image can be expressed as the product of illumination and reflectance compo-nents:

f (x, y) = i(x, y)r(x, y), (6.3.9)

where i(x, y) and r(x, y) are illumination and reflectance components respectively.f (x, y)must be non zero and finite, the reflectance is bounded by 0 (total absorption)and 1 (total reflectance) and nature of illumination is determined by the illuminationsource, i.e., 0 < i(x, y) < ∞. Since Fourier transform is not distributive overmultiplication, first take natural log both side, we have

z(x, y) = ln f (x, y) = ln i(x, y) + ln r(x, y), (6.3.10)

and then apply Fourier transform:

F{z(x, y)} = F{ln f (x, y)} = F{ln i(x, y)} + F{ln r(x, y)},

orZ(u, v) = Fi (u, v) + Fr (u, v), (6.3.11)

where Fi (u, v) and Fr (u, v) are the Fourier transform of ln i(x, y) and ln r(x, y),respectively. If we process Z(u, v) by means of a filter function H(u, v) then,


Fig. 6.1 Block diagram of homomorphic filtering

S(u, v) = H(u, v)Z(u, v) = H(u, v)Fi (u, v) + H(u, v)Fr (u, v) (6.3.12)

where S(u, v) is the Fourier transform of the result. In the spatial domain,

s(x, y) = F−1{S(u, v)} = F−1{H(u, v)Fi (u, v)} + F−1{H(u, v)Fr (u, v)}(6.3.13)

Finally, as z(x, y)was formed by taking the logarithm of the original image f (x, y),the inverse(exponential) operation yields the desired enhanced image, denoted byg(x, y), i.e.,

g(x, y) = es(x,y) = eF−1{H(u,v)Fi (u,v)}eF

−1{H(u,v)Fr (u,v)} = i0(x, y)r0(x, y), (6.3.14)

where, i0(x, y) = eF−1{H(u,v)Fi (u,v)} and r0(x, y) = eF

−1{H(u,v)Fr (u,v)} are the illumi-nation and reflectance components of the output image. The enhancement approachusing the homomorphic filtering approach is described in Fig. 6.1.

The illumination component of an image generally is characterized by slow spatialvariations, while the reflectance component tends to vary abruptly, particularly atthe junctions of dissimilar objects. These characteristics lead to associating the lowfrequencies of the Fourier transform of the logarithm of an image with illuminationand the high frequencies with reflectance. Although these associations are roughapproximations, they can be used as an advantage in image enhancement.

6.4 Colour Image Enhancement

Color images provide more and richer information for visual perception than thatof the gray images. Color image enhancement plays an important role in DigitalImage Processing. The purpose of image enhancement is to get finer details of animage and highlight the useful information. During poor illumination conditions,the images appear darker or with low contrast. Such low contrast images needs tobe enhanced. In the literature many image enhancement techniques such as gammacorrection, contrast stretching, histogram equalization, andContrast-limited adaptivehistogram equalization (CLAHE) have been discussed. These are all old techniqueswhich will not provide exact enhanced images and gives poor performance in termsof Root Mean Square Error (RMSE), Peak Signal to Noise Ratio (PSNR) and Mean


Absolute Error (MAE). Use of the old enhancement technique will not recover exacttrue color of the images. Recently, Retinex, Homomorphic and Wavelet MultiScaletechniques have been popular for enhancing images. These methods are shown toperform much better than those listed earlier.

6.5 Application of Multiscale Transforms in ImageEnhancement

Image enhancement is one of the measure issues in high-quality pictures from digitalcameras and in high definition television (HDTV). Since clarity of the image iseasily affected by weather, lighting, wrong camera exposures or aperture settings, ahigh dynamic range in the scene, etc., these conditions lead to an image that maysuffer from loss of information. Many techniques have been developed to recoverinformation in an image.

Color image enhancement plays an important role in digital image processingsince color images provide more and richer information for visual perception thangray images. The main purpose of image enhancement is to obtain finer details ofan image and to highlight useful information. The images appear darker or withlow contrast under poor illumination conditions. Such low-contrast images needto be enhanced. Image enhancement is basically improving the interpretability orperception of information in images for human viewers, and providing better inputfor other automated image processing techniques.

Because some features in an image are hardly detectable by eye, we often trans-form images before display. Histogram equalization is one of the most well-knownmethods for contrast enhancement. Such an approach is generally useful for imageswith poor intensity distribution. Histogram equalization (HE) is one of the commonmethods used for improving contrast in digital images. However, this technique isnot very well suited to be implemented in consumer electronics, such as television,because the method tends to introduce unnecessary visual deterioration such as thesaturation effect. One of the solutions to overcome this weakness is by preservingthe mean brightness of the input image inside the output image.

In most cases, Brightness preserving histogram equalization (BPDHE) success-fully enhances the image without severe side effects, and at the same time, maintainsthe mean input brightness. BPDHE preserves the intensity of the input image, itis disadvantageous to highlight the details in areas of low intensity. BPDHE is notsuitable for non-uniform illumination images.

Local histogram equalization (LHE) method tries to eliminate the above prob-lem. It makes use of the local information remarkably. However, LHE (Kim et al.,[39]) demands high computational cost and sometimes causes over enhancementin some portion of the image. Nonetheless, these methods produce an undesirablecheckerboard effects on the enhanced images. It makes use of the local informationremarkably. However, LHE demands high computational cost and sometimes causesover enhancement in some portion of the image. Nonetheless, thesemethods producean undesirable checkerboard effects on the enhanced images.

6.5 Application of Multiscale Transforms in Image Enhancement 211

The brightness preserving Bi-HE (BBHE) (Kim, [41]) method decomposes theoriginal image into two sub-images, by using the image mean gray level, and thenapplies the HE method on each of the sub images independently. At some extentBBHE preserves brightness of image; however generated image might not have anatural appearance. Dualistic sub-image histogram equalization (DSIHE) is similarto BBHE but DSIHE uses median value as separation intensity to divide the his-togram into two sub-histogram. The algorithm enhances only the image visual infor-mation effectively, but does not preserve the details and naturalness. The essenceof the named brightness preserving histogram equalization with maximum entropy(BPHEME) [13] tried to find the mean brightness was fixed, then transformed theoriginal histogram to that target one using histogram specification. In the consumerelectronics such as TV, the preservation of brightness is highly demanded.

Minimum mean brightness error bi-histogram equalization (MMBEBHE) wasproposed to preserve the brightness optimally. MMBEBHE is an extension of theBBHE method and in this the separation intensity is minimum mean brightnesserror between input image and output image. The RMSHE method proposed forperforming image decomposition recursively, up to a scalar r , generating 2r sub-image.

Multi histogram equalization(MHE) techniques had been proposed to furtherimprove the mean image brightness preserving capability. MHE proposed a tech-nique for image enhancement based on curvelet transform and perception network.

Since edges play a fundamental role in image understanding, one good way toenhance the contrast is to enhance the edges. Multiscale-edge enhancement (Toet[77]) can be seen as a generalization of this approach, taking all resolution levelsinto account.

In color images, objects can exhibit variations in color saturation with little orno correspondence in luminance variation. Several methods have been proposedin the past for color image enhancement. Many image enhancement techniques,such as gamma correction, contrast stretching, histogram equalization, and contrastlimited adaptive histogram equalization (CLAHE) have been discussed. These areall old techniques that will not provide exact, enhanced images, and that give poorperformance in terms of root mean square error (RMSE), peak signal-to-noise ratio(PSNR) and mean absolute error (MAE). Use of the old enhancement technique willnot recover an exact true color in the image. Recently, retinex, single and multiscaleretinex, and homomorphic and wavelet multiscale techniques have become popularfor enhancing images. These methods are shown to perform much better than thoselisted earlier by Hanumantharaju et al. [24].

The retinex concept was introduced by Land in 1986 as a model for human colorconstancy. The single scale Retinex (SSR) method [31] consists of applying thefollowing transform to each band of the color image:

Ri (x, y) = log(Ii (x, y)) − log(F(x, y) ∗ Ii (x, y))

where Ri (x, y) is the retinex output, Ii (x, y) is the image distribution in the i th

spectral band, F is a Gaussian function, and ∗ is convolution. The retinex method is


efficient for dynamic range compression, but does not provide good tonal rendition[62]. Multiscale Retinex (MSR) combines several SSR outputs to produce a singleoutput image that has both good dynamic range compression and color constancy(color constancy may be defined as the independence of the perceived color fromthe color of the light source [49, 53]), and good tonal rendition [31]. Multiscaleretinex leads the concept of multiresolution for contrast enhancement. It implementsdynamic range compression and can be used for different image processing goals.

Barnard and Funt [10] in 1999, presented improvements to the algorithm, leadingto better color fidelity. MSR softens the strongest edges and leaves the faint edgesalmost untouched.

Using the wavelet transform, the opposite approach was proposed by Velde [81]in 1999 for enhancing the faintest edges and keeping the strongest untouched. Thestrategies are different, but bothmethods allow the user to see details that were hardlydistinguishable in the original image, by reducing the ratio of strong features to faintfeatures.

The review concludes that histogram equalization cannot preserve the brightnessand color of the original image, and a homomorphic filtering technique has a problemwith bleaching of the image. Modern technique retinex (SSR and MSR) performsmuch better than those listed above, because it is based on the color constancy theory,but it still suffers from color violation and the unnatural color rendition problem, asthewavelet transform is a very good technique for image enhancement and denoising,and input images always face noise during image processing.

Wavelet analysis [40] has proven to be a powerful image processing tool in recentyears.When images are to be viewed or processed atmultiple resolutions, thewavelettransform is the mathematical tool of choice. In addition to being an efficient, highlyintuitive framework for the representation and storage of multiresolution images, theWT provides powerful insight into an images spatial and frequency characteristics.The image detail parts are stored in the high-frequency parts of the image transformedby the wavelet, and the imagery constant part is stored in the low-frequency part.Because the imagery constant part determines the dynamic range of the image, thelow frequency part determines the dynamic range of the image. We attenuate thelow-frequency part in order to compress the dynamic range. But details must belost when the low frequency part is attenuated (Xiao et al., [85]). As some detailsare stored in the high-frequency parts very well, the image reconstructed by inversewavelet transform has more detail. The wavelet framework was selected instead ofthe Fourier, because wavelets provide an intrinsically local frequency description ofa signal directly related to local contrast, while the Fourier transform provides onlyglobal frequency information.

6.5.1 Image Enhancement Using Fourier Transform

The Fourier Transform is used in a wide range of applications in image processing,such as image analysis, image filtering, image reconstruction, image enhancementand image compression. Fast Fourier Transform (FFT) is an efficient implementation


of DFT and it is an important tool used in image processing. The main disadvantagesof using DFT, the speed is very slow when compared with FFT. The number ofcomputations for a DFT is on the order of N Squared.

Fast Fourier Transform is applied to convert an image from the image (spatial)domain to the frequency domain. Applying filters to images in frequency domain iscomputationally faster than to do the same in the image domain. An inverse transformis then applied in the frequency domain to get the result of the convolution. Theinteresting property of the FFT is that the transform of N points can be rewrittenas the sum of two N/2 transforms (divide and conquer). This is important becausesome of the computations can be reused thus eliminating expensive operations. TheFFT-based convolution method is most often used for large inputs. For small inputsit is generally faster to use im-filter.

A discrete cosine transform (DCT) expresses a finite sequence of data points intermsof a sumof cosine functionoscillating at different frequencies. Theuseof cosinerather than sine functions is critical in those applications, for compression it turnsout that cosine functions are much more efficient, whereas for differential equationsthe cosines express a particular choice of boundary conditions. The Discrete CosineTransform (DCT) is used in many applications by the scientific, engineering andresearch communities and in data compression in particular.

Kanagalakshmi and Chandra [33] in 2012, proposed and implemented a Fre-quency domain enhancement algorithm based on Log-Gabor and FFT. They foundthat the maximum variations between original and enhanced images; and also theincreased number of terminations and decreased number of bifurcations due to theun-smoothing and noisiness. The results proved that the proposed algorithm can bea better one for the frequency domain enhancement.

Tarar and Kumar [74] in 2013, designed a fingerprint enhancement algorithmwhich can increase the degree of clarity of ridges and valleys. Since high qual-ity fingerprint image acquired by using an adaptive fingerprint image enhancementmethod was critical to the quality of any fingerprint identification system, a finger-print enhancement method based on iterative Fast Fourier Transformation had beendesigned by tarar et al. and comparative analysis with the existing method, i.e., FastFourier Transformation had been shown with the help of graph. The performance ofthe algorithm was evaluated using the goodness index of minutia extraction process.Experiments on a public domain fingerprint database (i.e., FVC 2006) demonstratesthat the use of minutia descriptors leads to an order of magnitude reduction in thefalse accept rate without significantly affecting the genuine accept rate. Based onthe observation of good quality rolled images, the ridge and valleys intervals of eachimage are considered in order to select the Region of Interest (ROI) for effectiveenhancement. Experimental results showed that our algorithm improved the good-ness index aswell asmatching performance of the FIS.Also their algorithmdealtwiththe broken ridges/false minutia problem and removed them from further processing.

Steganography is used to send the data secretly in the carrier. While sendingthis information, noise may get added and it will distort the message which is sent.For the removal of noise we require the features of image enhancement. Hence


Singh et al. [69] worked on these two important fields and were combined togetherso that the receiver could get the image which was noise free and the message hadbeen delivered.

Li [46] in 2013 proposed a novel method of image enhancement with respectto the fractional Fourier transform. This method filtered image in the fractionalFourier domain instead of the Fourier domain which was usually applied to theclassical image enhancement. The fractional Fourier transform had a rotation angle a,characters of image thus change in different transform domain. In a proper fractionalFourier domain with angle a, ideal low-pass filter made image smoother and idealhigh-pass filter loses less information of image than in the traditional Fourier domain,which provided an alternative way to enhance image with proper filter designing.

Umamageswari et al. [79] in 2014, proposed a naturalness preserved enhancementmethod for non-uniform illumination images using transformation techniques. Theirproposed method did not only enhances the details of the image but also preservedthe naturalness.

With the advancement of technologies, the images are createdwithmore andmoreenhancement. Yeleshetty et al. [86] in 2015, proposed a generalized equalizationmodel for image enhancement and further improved the same using Fast FourierTransform and Bilog Transformation. Here, we analysed the relationship betweenimage histogram and contrast enhancement and white balancing. In the proposedsystem, they enhance not only images, but also videos both live and recorded. Theoriginal image was stored in RAW format which was too big for normal displays.

Arunachalam et al. [6] in 2015 implemented two-dimensional Fast Fourier Trans-form (FFT) and Vedic algorithm based on Urdhva Tiryakbhyam sutra. The algorithmwas presented using MATLAB program. The input image was divided into blocksand two-dimensional FFT was applied to enhance or filter the image. The proposedtwo-dimensional FFT design was based on using Urdhva Tiryakbhyam sutra. FFTcomputations using Vedic multiplication sutra gave a significant performance ascompared to the conventional FFT.

Ramiz andQuazi [63] in 2017 proposed a hybridmethodwhichwas very effectivein enhancing the images. Initially, frequency domain analysis was done followed byspatial domain procedures. The performance of the proposed method was assessedon the basis of two parameters i.e. Mean Square Error (MSE) and Peak Signal tonoise ratio (PSNR). Their proposed algorithm provided better PSNR and MSE.

6.5.2 Image Enhancement Using Wavelet Transform

Image enhancement is applied in every fieldwhere images are ought to be understoodand analyzed. It offers a wide variety of approaches for modifying images to achievevisually acceptable images. The choice of techniques is a function of the specific task,image content, observer characteristics, and viewing conditions.Wavelet transform iscapable of providing the time and frequency information of a signal simultaneously.But sometimes we cannot know what spectral component exists at any given time


instant. The best we can do is to investigate that what spectral components exist atany given interval of time. DWT is a linear transformation that operates on a datavector whose length is an integer power of two, transforming it into a numericallydifferent vector of the same length.

An advancement of wavelet theory has taken the interest of researchers in itsapplication to image enhancement. Discrete wavelet transform (DWT) is one ofthe latest wavelet transform. It is simple mathematical tool for image processing.DWT composes filter bank (High pass filter and low pass filter). DWT decomposesimage into four sub-bands. Image resolution enhancement in the wavelet domain is arelatively new research topic and recently many new algorithms have been proposed.Another recent wavelet transform, named stationary wavelet transform (SWT), hasbeen used in several image processing applications. SWT is similar to DWT but itdoes not use down-sampling, hence the sub-bands will have the same size as the inputimage. The SWT is an inherently redundant scheme as the output of each level ofSWT contains the same number of samples as the input. So for a decomposition of Nlevels there is a redundancy of N in the wavelet coefficients. Translation invarianceproperty of DWT is overcome in SWT.

In (Mallat and Hwang [48]) the enhancement was done by noise removing andedge enhancement. The algorithm relied on a multiscale edge representation wherethe noise is connected to the singularities.

Donoho [16] in 1993, proposed the nonlinear wavelet shrinkage algorithm whichreduced wavelet coefficients toward zero, based on a level-dependent threshold. Heprovided a detailed mathematical analysis of a directional wavelet transform andrevealed its connection with the edge enhancement. In addition, he discussed a singlelevel edge sharpening, followed by its refinement to a multiscale sharpening.

Directional wavelet transform decomposes an image into four-dimensional spacewhich augments the image by the scale and directional information. Heric and Potoc-nik [26], in 2006, proposed a novel image enhancement technique by using direc-tional wavelet transform. They showed that the directional information significantlyimproved image enhancement in noisy images in comparison with the classical tech-niques. Image enhancement was based on the multiscale singularity detection withan adaptive threshold whose value was calculated via maximum entropy measure.The proposed technique was tested on synthetic images at different signal-to-noiseratios and clinical images and showed that proposed image enhancement techniquewas robust, accurate, and effective in noisy images too.

Al-Samaraie et al. [4] in 2011, proposed a new method to enhance the satelliteimage which using intelligent aspect of filtering and describe multi-threshold tech-nique with an additional step in order to obtain the perceived image. In the proposedscheme, the edge detected guided smoothing filters succeeded in enhancing lowsatellite images. This was done by accurately detecting the positions of the edgesthrough threshold decomposition and the detected edges were then sharpened byapplying smoothing filter. By utilizing the detected edges, the scheme was capa-ble to effectively sharpening fine details whilst retaining image integrity. The visualexamples shown have demonstrated that the proposed method was significantly bet-


ter than many other well-known sharpener type filters in respect of edge and finedetail restoration.

Kakkar et al. [32] in 2011, proposed a enhancement algorithm using gabor filterin wavelet domain. Present algorithm effectively improved the quality of fingerprintimage. A refined gabor filter was applied in fingerprint image processing, then a goodquality of fingerprint image was achieved, and the performance of the fingerprintrecognition system had been improved.

Prasad et al. [59] in 2012 developed a method to enhance the quality of the givenimage. The enhancement is done both with respect to resolution and contrast. Theproposed technique uses DWT and SVD. To increase the resolution, the proposedmethod uses DWT and SWT. These transforms decompose the given image into foursub-bands, out of which one is of low frequency and the rest are of high frequency.The HF components are interpolated using conventional interpolation techniques.Then we use IDWT to combine the interpolated high frequency and low frequencycomponents. To increase the contrast, we use SVD and DWT. The experimentalresults show that proposed technique gives good results over conventional methods.

Neena et al. [55] proposed a image enhancement technique with respect to res-olution and contrast based on bi cubic interpolation, stationary wavelet transform,discrete wavelet transform and singular value decomposition. They tested proposedtechnique on different satellite images and experimental results showed the proposedmethod provided good results over conventional methods.

Narayana and Nirmala [54] in 2012 proposed an image resolution enhancementtechnique based on the interpolation of the high frequency subbands obtained byDWT and SWT. The proposed technique had been tested on well-known benchmarkimages, where their PSNR,Mean Square Error and Entropy results showed the supe-riority of proposed technique over the conventional and state-of-art image resolutionenhancement techniques.

Saravanan et al. [65] in 2013, proposed that a new image enhancement schemeusing wavelet transform, smooth and sharp approximation of a piecewise nonlinearfilter technique after converting the RGB values of each pixel of the original image toHSV.Themethod had effectively achieved a successful enhancement of color images.The experimental result vividly displays the proposed algorithmwas efficient enoughto remove the noise resulting good enhancement.

Karunakar et al. [34] in 2013 proposed a new resolution enhancement techniquebased on the interpolation of the high-frequency sub band images obtained by DWTand the input image. The proposed technique had been tested on well-known bench-mark images, where their PSNR and RMSE and visual results show the superiorityof the proposed technique over the conventional and state-of-art image resolutionenhancement techniques. The PSNR improvement of the proposed technique wasup to 7.19 dB compared with the standard bicubic interpolation.

Bagawade and Patil [9] in 2013 used SWT (Stationary Wavelet Transform) andDWT (Discrete Wavelet Transform) to enhance image resolution and then interme-diate subbands of image produced by SWT and DWT were interpolated by usingLanczos interpolation. Finally they combined all subbands by using IDWT (Inverse


Discrete Wavelet Transform). This approach provided better result in comparison toother method. For Baboon image they got PSNR value 27.0758dB.

Miaindargi V.C. and Mane A.P. [52] in 2013 proposed an image resolutionenhancement technique based on interpolation of the high frequency subband imagesobtained by discrete wavelet transform (DWT) and the input image. The image edgeswere enhanced by introducing an intermediate stage by using stationary wavelettransform (SWT). DWT was applied in order to decompose an input image into dif-ferent subbands. The quantitative and visual results showed the superiority proposeddecimated resolution technique over the conventional system and state-of-art imageresolution enhancement techniques.

Panwar and Kulkarni [58] in 2014 provided a technique of image resolutionenhancement based on SWT and DWT. The proposed technique was compared withconventional and state-of-art image resolution enhancement techniques. They havealso provided subjective and objective comparison of resultant images and PSNRtable showed the superiority of the proposed method over conventional methods.

Provenzi and Caselles [61] in 2014, proposed a variational model of perceptuallyinspired color correction based on the wavelet representation of a digital image.Qualitative and quantitative tests about the wavelet algorithm had shown that it wasable to enhance both under and over exposed images and to remove color cast.Moreover, in the quantitative test of color normalization property, i.e. the abilityto reduce the difference between images of the same scene taken under diverseillumination conditions, the wavelet algorithm had performed even better than thatof existing methods.

Pai et al. [56] in 2015, provided a method to obtain the sharpened image mainlyfor medical image enhancement by using the wavelet transforms using Haar waveletfollowed by the Laplacian operator. First, a medical image was decomposed withwavelet transform. Secondly, all highfrequency sub-images were decomposed withHaar transform. The contrast of the image was adjusted by linear contrast enhance-ment approaches. Filters were applied to identify the edges. Finally, the enhancedimage was obtained by subtracting resulting image from the original image. Exper-iments showed that this method can not only enhance an images details but can alsopreserve its edge features effectively.

Khatkara and Kumar [38] in 2015 presented a method to enhance the biomedicalimages using combination of wavelets, as image enhancement is the main issue forbiomedical image diagnosis. The results of the proposed method had been comparedwith other wavelets on the basis of different metrics like PSNR (Peak signal to noiseratio) and Beta coefficient and it had been found that the proposed method providesbetter results than the other methods.

Brindha [12] in 2015, proposed a satellite image contrast enhancement techniquebased onDWTand SVD. The experimental result showed that the better performanceand high accuracy when compared with other methods.

Thorat and Katariya [76] in 2015, proposed an image resolution enhancementtechnique based on the interpolation of the high frequency subbands obtained byDWT, correcting the high frequency subband estimation by using SWT high fre-quency subbands and the input image. The proposed technique had been tested on


well-known benchmark images, where their PSNR and visual results showed thesuperiority of proposed technique over the conventional and state-of-art image res-olution enhancement techniques.

Kole and Patil [42] in 2016 proposed a new resolution enhancement techniquebased on the interpolation of the high-frequency sub band images obtained by DWT.The proposed technique had been tested on well-known benchmark images, wheretheir PSNR and RMSE and visual results show the superiority of the proposed tech-nique over the conventional and state-of-art image resolution enhancement tech-niques. This procedure was successful in obtaining the enhanced images to obtaineven the minute details when related to satellite images. The PSNR improvement ofthis technique was up to 7.19 dB compared with the standard bicubic interpolation.

Sumathi and Murthi [73] in 2016, proposed a new satellite image resolutionenhancement technique based on the interpolation of the high frequency sub bandsobtained by discrete wavelet transform (DWT) and the input image. The proposedtechnique had been tested on satellite benchmark images. The quantitative and visualresults showed the superiority of the proposed technique over the conventional andstate of art image resolution enhancement techniques.

Arya and Sreeletha [7] in 2016 provided image resolution enhancement meth-ods using multi-wavelet and interpolation in wavelet domain. They discussed aboutimprovement in the resolution of satellite images based on the multi-wavelet trans-form using interpolation techniques. The quantitative metrics (PSNR, MSE) of theimage calculated showed the superiority of DWT-SWT technique.

Shanida et al. [66] in 2016, proposed a denoising and resolution enhancementtechnique for dental radiography images using wavelet decomposition and recon-struction. Salt and pepper noise present in image was removed by windowing thenoisy image with a median filter before performing the enhancement process. Thebetter performance was achieved using the proposed technique than the conventionaltechniques. Quantitative assessment of the image quality was performed by meansof peak signal to noise (PSNR) calculation.

Light scattering and color change are twomajor sources of distortion for underwa-ter photography. Light scattering is caused by light incident on objects reflected anddeflected multiple times by particles present in the water before reaching the camera.This in turn lowers the visibility and contrast of the image captured. Color changecorresponds to the varying degrees of attenuation encountered by light traveling inthe water with different wavelengths, rendering ambient underwater environmentsdominated by a bluish tone. Any underwater image will have one or more combi-nations of the Inadequate range visibility, Non uniform illumination, Poor contrast,Haziness, Inherent noise, Dull color, Motion blur effect due to turbulence in theflow of water, Scattering of light from different particles of various sizes, Dimin-ished intensity and change in color level due to poor visibility conditions, Suspendedmoving particles and so on.

Badgujar and Singh [8] in 2017 proposed an efficient systematic approachto enhance underwater images using generalized histogram equalization, discretewavelet transform and KL transform. The proposed system provided properly


enhanced underwater image output and the quality of the image was up to the markregarding contrast and resolution. The PSNR value of the image was higher thanother methods like DWT-KLT, DWT-SVD and GHE.

6.5.3 Image Enhancement Using Complex WaveletTransform

Image-resolution enhancement in the wavelet domain is a relatively new researchtopic, and, recently, many new algorithms have been proposed. Complex wavelettransform based approach of image enhancement is one of the recent approachesused in image processing and also an improvement technique of discrete wavelettransform. A one-level CWT of an image produces two complex-valued low-frequency subband images and six complex-valued high-frequency subband images.The high-frequency subband images are the result of direction-selective filters.They show peak magnitude responses in the presence of image features orientedat +75,+45,+15, 15, 45, and 75. Resolution enhancement is achieved by usingdirectional selectivity provided by CWT. The high frequency subband in 6 differentdirections contributes to the sharpness of high frequency details such as edges. TheDT-CWThas good directional selectivity and has the advantage over discrete wavelettransform (DWT). It also has limited redundancy. The DT-CWT is approximatelyshift invariant, unlike the critically sampled DWT. The redundancy and shift invari-ance of the DT-CWT mean that DT-CWT coefficients are inherently interpolable.

Demirel and Anbarjafari [15] in 2010 proposed a satellite image resolutionenhancement technique based on the interpolation of the high-frequency subbandimages obtained by DT-CWT and the input image. The proposed technique usedDT-CWT to decompose an image into different subband images, and then the high-frequency subband images were interpolated. An original image was interpolatedwith half of the interpolation factor used for interpolation of the high-frequency sub-band images.Afterward, all these imageswere combined using IDT-CWT to generatea super-resolved image. The proposed technique had been tested on several satelliteimages, where their PSNR and visual results show the superiority of the proposedtechnique over the conventional and state-of-the-art image resolution enhancementtechniques.

Tamaz et al. [75] in 2012 proposed a satellite image enhancement system con-sisting of image denoising and illumination enhancement technique based on dualtree complex wavelet transform (DT-CWT). The technique firstly decomposed thenoisy input image into different frequency subbands by using DT-CWT and denoisedthese subbands by using local adaptive bivariate shrinkage function (LA-BSF) whichassumed the dependency of subband detail coefficients. In LA-BSF, model param-eters were estimated in a local neighborhood which results in improved denoisingperformance. Further, the denoised image once more was decomposed into the dif-ferent frequency subbands by using DT-CWT. The highest singular value of the low


frequency subbands were used in order to enhance the illumination of the denoisedimage. Finally the image was reconstructed by applying the inverse DT-CWT (IDT-CWT). The experimental results showed the superiority of the proposed method overthe conventional and the state-of-art techniques.

Iqbal et al. [28] in 2013 proposed wavelet-domain approach for RE of the satel-lite images based on dual-tree complex wavelet transform (DT-CWT) and nonlocalmeans (NLM).Objective and subjective analyses revealed superiority of the proposedtechnique over the conventional and state-of-the-art RE techniques.

Bhakiya and Sasikala [11] in 2014 proposed a new satellite image resolutionenhancement technique based on the interpolation of the high-frequency sub bandsobtained by dual tree complex wavelet (DTCWT) transform and the input image.The proposed technique had been tested on satellite benchmark images. The quanti-tative peak signal-to-noise ratio, root mean square error and visual results showed thesuperiority of the proposed technique over the conventional and state-of-art imageresolution enhancement techniques. The PSNR improvement of the proposed tech-nique was up to 19.79 dB.

Mahesh et al. [47] in 2014 proposed a wavelet-domain approach based on doubledensity dual-tree complex wavelet transform (DDDT-CWT) for RE of the satelliteimages and comparedwith dual-tree complexwavelet transform(DT-CWT). Firstly, asatellite image was decomposed by DDDT-CWT to obtain highfrequency subbands.Then the high frequency subbands and the low-resolution (LR) input image wereinterpolated and the high frequency subbands were passed through a low pass filter.Finally, the filtered high-frequency subbands and the LR input image were combinedusing inverse DT-CWT to obtain a resolution-enhanced image. Objective and sub-jective analyses revealed superiority of the proposed technique over the conventionaland state-of-the-art RE techniques.

Kumar and Kumar [44] in 2015 presented Multi scale decomposition for SRE ofthe satellite images based on dual-tree complex wavelet transform and edge preser-vation. In their proposed method, a satellite input image was decomposed by DT-CWT to obtain high-frequency sub bands. The high-frequency sub band and the lowresolution (LR) input image were interpolated using the bi-cubic interpolator. Thesimulated results showed that technique used in this process provided better accuracyrather than prior methods.

Sharma and Chopra [68] in 2015 proposed A method based on dual tree com-plex wavelet transform (DTCWT), contrast limited adaptive histogram equalization(CLAHE) and Wiener filter for enhancing the visual quality of the X-Ray images.Quantitative analysis of proposed algorithm was done by evaluating MSE, PSNR,SNR andContrast Ratio (CR). Their proposed algorithm showed that it outperformedother conventional method for improving visual quality of the X-Ray image. Wienerfilter took less time as compared toNLMfilter,whichwas the advantage in emergencysituations.

Sharma and Mishra [67] in 2015 presented multi scale decomposition based ondual-tree complex wavelet transform and edge preservation for SRE of the satelliteimages. DTCWT decomposed the low resolution input image into high frequencysubbands and low frequency subbands, since DTCWT is nearly shift invariant. High


frequency subbands were interpolated with factor 2(α) and input image is inter-polated with factor 1(α/2). Nonlocal means filter diminished the remaining arti-facts. The PSNR and the RMSE values had been calculated and compared with con-ventional methods which showed the effectiveness and the superiority of proposedmethod.

Arulmozhi and Keerthikumar [5] in 2015 presented that noise reduction ofenhanced images usingDual tree complexwavelet transform andBivariate shrinkagefilter. In their method, initially noisy image pixels were decorrelated to obtain coarserand finer components and more noise details were contaminated in high frequencysubbands. In order to reduce the spatial distortion during filtering, bivariate shrinkagefunction were used in the DTCWT domain. Experimental results showed that theresultant algorithms produced images with better visual quality and evaluation wascarried out in terms of various parameters such as Peak Signal to Noise Ratio, meanStructural Similarity and Coefficient of Correlation.

Firake and Mahajan [21] in 2016 proposed based on dual-tree complex wavelettransform (DT-CWT) and nonlocal means (NLM) for satellite images of RE. Simu-lation results showed the superior performance of proposed techniques.

Deepak and Jain [14] in 2016 improved the medical image such as CT imagequality which was degraded through the Gaussian noise using dual-tree complexwavelet transform. To improve the image quality, noise reduction techniques wereused over lower dose images and noise was reduced with preserving all clinicallyrelevant structures. The proposed scheme was tested on various test images and theobtained results showed the effectiveness of the proposed scheme.

Kaur and Vashist [35] in 2017 proposed a hybrid approach algorithm usingDTCWT, NLM filter and SVD for Medical Image Enhancement and had been testedon a set of medical images. In their method, firstly, The medical input image wasdecomposed using DTCWT. Less artifacts were generated with the help of DTCWTcompared to that ofDWTbecause of nearly shift invariance characteristic ofDTCWT.Further imagequalitywas improvedusingNLMfiltering approach andSVDwasusedfor to get originality of image and obtain a better quality image both quantitativelyand qualitatively. Simulation results showed that proposed technique outperformsother conventional techniques for improving visual quality of medical images forproper manual interpretation and computer based diagnosis.

HemaLatha and Vardarajan [25] in 2017 presented a image resolution enhance-ment of LR image using the dualtree complex wavelet transform. In their method,dual tree complex wavelet transform was applied to low resolution (LR) satelliteimage. Further, the high resolution (HR) image was reconstructed from the low reso-lution image, together with a set of wavelet coefficients, using the inverse DT-CWT.Finally, the inverse dual tree complex transform was taken. Output was high resolu-tion image and the DT-CWT had better performance in terms of PSNR, RMSE, CCand SSIM compared to DWT technique.


6.5.4 Image Enhancement Using Curvelet Transform

Wavelet bases present some limitations, because they are not well adapted to thedetection of highly anisotropic elements, such as alignments in an image, or sheetsin a cube. Recently, other multiscale systems have been developed, which includein particular ridgelets and curvelets and these are very different from wavelet-likesystems. Curvelets and ridgelets take the form of basis elements which exhibit veryhigh directional sensitivity and are highly anisotropic. The curvelet transform usesthe ridgelet transform in its digital implementation. The Curvelet Transform is based,on decomposing the image into different scales, then partitioning into squares whosesizes based on the corresponding scale.

Since edges play a fundamental role in image understanding, one good way toenhance the contrast is to enhance the edges. The curvelet transform representsedges better than wavelets, and is therefore well-suited for multiscale edge enhance-ment. Enhancement of clinical image research based on Curvelet has been developedrapidly in recent years.

Starck et al. [72] in 2003 provided a new method for contrast enhancement basedon the curvelet transform and compared this approach with enhancement based onthe wavelet transform, and the Multiscale Retinex. They found that curvelet basedenhancement out-performed other enhancement methods on noisy images, but onnoiseless or near noiseless images curvelet based enhancement was not remarkablybetter than wavelet based enhancement.

Ren and Yang [64] in 2012 proposed a new method for color microscopic imageenhancement based on curvelet transformvia soft shrinkage and the saturation adjust-ment. They presented a new curvelet soft thresholdingmethod calledmodulus squarefunction, which modifies the high frequency curvelet coefficients of the V compo-nent. The experimental results showed that the method consistently produce thesatisfactory result for the V component degraded by Random noise, Gaussian noise,Speckle noise, and Poisson noise, and the S component was adjusted to render themicroscopic image colorfulness by adaptive histogram equalization. Hence, the pro-posed method was an efficient and reliable method for hue preserving and colormicroscopic image enhancement.

Kumar [43] in 2015 proposed a new method to enhance the colour image basedon Discrete Curvelet Transform (DCT) and multi structure decomposition. Experi-mental results showed that this method provided better qualitative and quantitativeresults.

Abdullah et al. [1] in 2016 proposed an efficient method to enhance low contrastin gray image based on fast discrete curvelet transform via unequally spaced fastFourier transform (FDCT-USFFT). Results showed that the proposed technique wascomputationally efficient, with the same level of the contrast enhancement perfor-mance and proposed technique was better than histogram equalization and wavelettransform in image quality.

Abdullah et al. [2] in 2017 proposed a newmethod for contrast enhancement grayimages based on Fast Discrete Curvelet Transform via Wrapping (FDCT-Wrap).


Experimental results showed that the proposed technique gave very good resultsin comparison to the histogram equalization and wavelet transform based contrastenhancement method.

Farzam and Rastgarpour [18] in 2017 presented a method for image contrastenhancement for cone beam CT (CBCT) images based on fast discrete curvelettransforms (FDCT) that work through Unequally Spaced Fast Fourier Transform(USFFT). Their proposed method first used a two-dimensional mathematical trans-form, namely the FDCT through unequal-space fast Fourier transform on input imageand then applied thresholding on coefficients of Curvelet to enhance the CBCTimages. Consequently, applying unequal-space fast Fourier Transform lead to anaccurate reconstruction of the image with high resolution. The experimental resultsindicated the performance of the proposedmethodwas superior to the existing ones interms of Peak Signal to Noise Ratio (PSNR) and Effective Measure of Enhancement(EME).

6.5.5 Image Enhancement Using Contourlet Transform

The wavelet transform may not be the best choice for the contrast enhancementof natural images. This observation is based on the fact that wavelets are blind tothe smoothness along the edges commonly found in images. Thus, there must bea new multiresolution approach which is more flexible and efficient in capturingthe smoothness over the edges of the images should be used in image enhancementapplications.

The enhancement approach which is proposed by Starck et al. [72] in 2003, basedon the curvelet transform domain, is a modified version of the Veldes [81] algorithm.This approach successfully removes the noise, as the noise are not parts of structuralinformation of the image, and the curvelet transformwill not generate coefficients forthe noise. But the curvelet transform is defined in the polar coordinate which makesit difficult to translate it back to the Cartesian coordinate. Analyzing these problemsDo and Vetterli in 2005 proposed another method called the contourlet transform.

The contourlet framework provides an opportunity to achieve the tasks of cap-turing important features of the image and is defined in Cartesian coordinates. Itprovides multiple resolution representations of an image, each of which highlightsscale-specific image features. Since features in those contourlet transformed compo-nents remain localized in space, many spatial domain image enhancement techniquescan be adopted for the contourlet domain. For high dynamic range and low contrastimages, there is a large improvement by using contourlet transform-based imageenhancement since it can detect the contours and edges quite adequately.

Literature dictates that contourlet transformhas better performance in representingthe image salient features such as edges, lines, curves, and contours than wavelets forits anisotropy and directionality and is thereforewell suited formultiscale edge-basedimage enhancement.


The evaluation of retinal images is widely used to help doctors diagnose manydiseases, such as diabetes or hypertension. Due to the acquisition process, retinalimages often have low grey level contrast and dynamic range. This problem mayseriously affect the diagnostic procedure and its results.

Feng et al. [20] in 2007 presented a newmulti-scale method for retinal image con-trast enhancement based on the Contourlet transform. Firstly they modify the Con-tourlet coefficients in corresponding subbands via a nonlinear function and took thenoise into account for more precise reconstruction and better visualization. Authorscompared this approach with enhancement based on the Wavelet transform, His-togram Equalization, Local Normalization and Linear Unsharp Masking and foundthat the proposed approach outperformed other enhancement methods on low con-trast and dynamic range images, with an encouraging improvement, and might behelpful for vessel segmentation.

Melkamu et al. [50] in 2010 provided a new algorithm for image enhancementby fusion based on Contourlet Transform. The experimental results showed that thefusion algorithm gives encouraging results for both multi modal and multi focalimages.

AlSaif and Abdullah [3] in 2013 proposed a new approach for enhancing contrastof color image based on contourlet transform and saturation components. In theirmethod, the color image was converted to HSI (hue, saturation, intensity) values.The S, which represented the Saturation value of color image, decomposed to itscoefficients by non-sampling contourlet transform, then applying grey-level contrastenhancement technique on some of the coefficients. Then, inverse contourlet trans-form was performed to reconstruct the enhanced S component. The I componentwas enhanced by histogram equalization while the H component did not changed toavoid degradation color balance between the HSI components. Finally the enhancedS and I together with H were converted back to its original color system. The algo-rithm effectively enhanced the Contrast images especially the fuzzy ones with lowContrast. At the same time, this method was easy one and a new approach to achievethe later transformation on contrast enhancement.

Song [71] in 2013 proposed a useful image enhancement scheme based on non-subsampled contourlet transform. Experimental results showed that the proposedenhancement scheme was able to enhance the detail and increase the contrast of theenhanced image at the same time.

Kaur and Singh [37] in 2014 proposed a new algorithm based on the firefly algo-rithm and the contourlet transform for sharpening of ultra sound images. To improvethe results of contourlet transform, authors implemented a new approach based onfirefly and it was providing very high percentage of image quality for ultra soundimages. The results demonstrated the improvement in the quality of the ultra soundimages to find the optimal solution and parameters were calculated which showedthat the proposed approach was performing better than the existing solutions.

Melkamu et al. [51] in 2015 presented a new image enhancement algorithm usingthe important features of the contourlet transform. The results obtained are comparedwith other enhancement algorithms based on wavelet transform, curvelet transform,bandlet transform, histogram equalization (HE), and contrast limited adaptive his-


togram equalization. The performance of the enhancement based on the contourlettransform method was superior than the other methods.

Kaur and Aashima [36] in 2015 proposed a algorithm based on Contourlet trans-formation and BFOA forMedical image enhancement. Firstly, the contourlet utilizedLP and FB decomposition filter to divide the image into different segments and thesesegments had been enhanced independently using different contourlet transformationequations. After the implementation of contourlet transformation the BFO algorithmwas implemented for optimization process. This is a nature inspired approach thathave been used or optimization on the basis of bacterias. After these steps variousparameters had been analyzed for validation of purposed work and different parame-ters were introduced and on the basis of these parameters their system provided thembetter results.

6.5.6 Image Enhancement Using Shearlet Transform

About 20 years ago, the emergence of wavelets represented a milestone in the devel-opment of efficient encoding of piecewise regular signals. The wavelet basis functionis the optimal basis which represents a point singularity objective function, and hasgood time-frequency locality, multi-resolution and sparse representation for a pointsingularity piecewise smooth function. In the two-dimensional image, high dimen-sional singular curves such as edges, contours and textures, contain most of theimage information. Nonetheless, a two-dimensional wavelet frame spanned by twoone dimensional wavelets can just describe the location of singular points in theimage. In the support region, the wavelet basis function only has horizontal, verticaland diagonal direction, and its shape is isotropic square. In fact, the wavelet frame isoptimal for approximating data with point-wise singularities only and cannot equallywell handle distributed singularities such as singularities along curves. Therefore,the wavelet frame, lacking of direction and anisotropy, is hard to sparsely representhigh dimensional singular characteristics like edges and textures.

In order to overcome the classicalwavelet framedefects, scholars propose a varietyof multi-scale geometric analysis methods, in light of the characteristics of visualcortex receiving outside scene information. Notable methods include the curveletand the contourlet. The curvelet is the first and so far the only construction providingan essentially optimal approximation property for 2D piecewise smooth functionswith discontinuities alongC2 curves. However, the curvelet is not generated from theaction of a finite family of operators on a single function, as is the case with wavelets.Thismeans its construction is not associatedwith amultiresolution analysis. This andother issues make the discrete implementation of the curvelet very challenging, as isevident by the fact that two different implementations of it have been suggested by theoriginators. In an attempt to provide a better discrete implementation of the curvelet,the contourlet representation has been recently introduced. This is a discrete time-domain construction, which is designed to achieve essentially the same frequencytiling as the curvelet representation. But the directional sub-bands of contourlet exists


spectrum aliasing. This leads to high similarity between the high frequency sub-bandcoefficients. However, both of them do not exhibit the same advantages the wavelethas, namely a unified treatment of the continuum and digital situation, a simplestructure such as certain operators applied to very few generating functions, and atheory for compactly supported systems to guarantee high spatial localization.

The shearlet frame was described in 2005 by Labate, Lim, Kutnyiok and Weisswith the goal of constructing systems of basis-functions nicely suited for representinganisotropic features (e.g. curvilinear singularities). Many scholars put forward imageenhancement algorithms based on the shearlet frame for different types of images,such as medical, cultural relic, remote sensing, infrared and ultrasound images. Thecommon idea is decomposing the original image into low frequency coefficients andhigh frequency sub-band coefficients of various scales and directions in the shearletdomain, and then enlarging or reducing these two components according to the aimof enhancement. These methods enlarge high frequency coefficients include hardthresholding and soft thresholding. Meanwhile, there is fuzzy contrast enhancementto deal with low frequency coefficients.

The advantage of the Shearlet frame, in particular, is to provide a unique abilityto control the geometric information associated with multidimensional data. Thus,the Shearlet appears to be particularly promising as a tool, enhancing the componentof the 2D data associated with the weak edges.

Wang and Zhu [83] in 2014 proposed a single image dehazing algorithms toimprove the contrast of the foggy images based on shearlet Transform. Firstly, algo-rithm executed shearlet transforms for foggy images, got low frequency coefficientsand high frequency of in all directions and scale factor, then executed fuzzy contrastenhancement for Low-frequency coefficients, then executed fuzzy enhancement forhigh-frequency coefficients of different scales in different directions. Finally, appliedshearlet inverse transform for low-frequency coefficients and high frequency coef-ficients of treated. Experimental results showed that this algorithm can effectivelyimprove the visual effect of the foggy images, and enhance the contrast of the foggyimages.

Premkumar and Parthasarathi [60] in 2014 proposed an efficient approach forcolour image enhancement using discrete shearlet transform. They proposed a novelmethod for image contrast enhancement based onDiscrete Shearlet Transform (DST)for colour images. In order to obtain high contrast enhancement image, the RGBimage was first converted into HSV (Hue, Saturation and Value) color space. Theconverted hue color channel was only taken into the account for DST decomposition.After that higher sub bands of hue component were eradicated and lower sub bandswere only considered for reconstruction. Finally, high contrast image was obtainedby using reconstructed Hue for HSV color space and then it was converted to RGBcolor space.

Wubuli et al. [84] in 2014 proposed a novel enhancement algorithm for med-ical image processing based on Shearlet transform and unsharp masking. In theirmethod, Firstly, histogram equalization was applied to the medical image, then, themedical image was decomposed into low frequency component and high frequencycomponent using shearlet transform. Next, the adaptive threshold denoising and


linear enhancement was applied to the high frequency components while the lowfrequency components were not processed. Finally, the coefficients were increasedthrough unsharp masking algorithm behind the Shearlet inverse transform process.The benchmark results for this algorithms showed that the proposed method couldsignificantly enhance the medical images and thus improve the image qualities.

Sivasankari et al. [70] in 2014 introduced the effective speckle reduction of SARimages based on a new approach of Discrete Shearlet Transform with Bayes Shrink-age Thresholding. The combined effect of soft thresholding in shearlet transformworked better when compared to the other spatial domain filter and transforms andit also performed better in the curvilinear features of SAR images.

Wang et al. [82] in 2015 proposed an image enhancement algorithm based onShearlet transform. In their method, the image was decomposed into low frequencycomponents and high frequency components by Shearlet transform. Firstly, Multi-Scale Retinex (MSR) was used to enhance the low frequency components of Shear-let decomposition to remove the effect of illumination on image then the thresholddenoising was used to suppress noise at high frequency coefficients of each scale.Finally, the fuzzy contrast enhancementmethodwas used to the reconstruction imageto improve the overall contrast of image. The experimental results showed that pro-posed algorithm provided significantly improvement in the image visual effect, andit had more image texture details and anti-noise capabilities.

Pawade andGaikwad [57] in 2016 implemented a novelmethod for image contrastenhancement which includes enhancement in both Intensity/value using discretecosine transform and hue components using discrete Shearlet transform of HSVcolour image simultaneously. The results showed that the perceptibility of an imagewas increased.

Fan et al. [17] in 2016 proposed a novel infrared image enhancement algorithmbased on the shearlet transform domain to improve the image contrast and adaptivelyenhance image structures, such as edges and details. Experimental results showed thatthe proposed algorithm could enhance the infrared image details well and producedfew noise regions, which was very helpful for target detection and recognition.

Tong and Chen [78] in 2017 presented a multi-scale image adaptive enhancementalgorithm for image sensors in wireless sensor networks based on non-subsampledshearlet transform. The performance of the proposed algorithm was evaluated bothobjectively and subjectively and the results showed that the visibility of the imageswas enhanced significantly.

Favorskayaa and Savchinaa [19] in 2017 investigated a process of dental imagewatermarking based on discrete shearlet transform.The proposedwatermarking tech-nique was tested on 40 dental gray scale images with various resolution. The exper-iments showed the highest robustness to the rotations and proportional scaling andthe medium robustness to the translations and JPEG compression. The SSIM estima-tors were found high for the rotation and scaling distortions that showed good HVSproperties.


References

1. Abdullah, M.H., Hazem, M.E., Reham, R.M.: Image contrast enhancement using fast discretecurvelet transform via unequally spaced fast fourier transform (FDCT-USFFT). Int. J. Electron.Commun. Comput. Eng. 7(2), 88–92 (2016)

2. Abdullah, M.H., Hazem, M.E., Reham, R.M.: Image contrast enhancement using fast discretecurvelet transform via wrapping (FDCT-Wrap). Int. J. Adv. Res. Comput. Sci. Technol. 5(2),10–16 (2017)

3. AlSaif, K.I., Abdullah, A.S.: Image contrast enhancement of HSI color space based on con-tourlet transform. Int. J. Inf. Technol. Bus. Manag. 15(1), 60–66 (2013)

4. Al, S., Muna, F., Al, S., Nedhal, A.M.: Colored satellites image enhancement using waveletand threshold decomposition. IJCSI Int. J. Comput. Sci. Issues 8(5), 33–41 (2011)

5. Arulmozhi, S., Keerthikumar, D.N.: Noise reduction of enhanced images using dual tree com-plex wavelet transform and shrinkage filter. Int. J. Adv. Res. Electr. Electron. Instrum. Eng.4(5), 4689–4694 (2015)

6. Arunachalam, S., Khairnar, S.M., Desale, B.S.: Implementation of fast fourier transform andvedic algorithm for image enhancement. Appl. Math. Sci. 9(45), 2221–2234 (2015)

7. Arya, A.R., Sreeletha, S.H.: Resolution enhancement of images using multi-wavelet and inter-polation techniques. Int. J. Adv. Res. Comput. Commun. Eng. 5(7), 228–231 (2016)

8. Badgujar, P.N., Singh, J.K.: Underwater image enhancement using generalized histogramequalization, discrete wavelet transform and KL-transform. Int. J. Innov. Res. Sci. Eng. Tech-nol. 6(6), 11834–11840 (2017)

9. Bagawade, R., Patil, P.: Image resolution enhancement by using wavelet transform. Int. J.Comput. Eng. Technol. 4(4), 390–399 (2013)

10. Barnard, K., Funt, B.: Investigations into Multiscale Retinex, Color Imaging: Vision and Tech-nology, pp. 9–17. Wiley, New York (1999)

11. Bhakiya,M.M., Sasikala,N.T.: Satellite image resolution enhancement usingdual-tree complexwavelet transform and adaptive histogram equalization. Int. J. Adv. Res. Electron. Commun.Eng. 3(4), 467–470 (2014)

12. Brindha, S.: Satellite image enhancement using DWTSVD and segmentation usingMRRMRFmodel. J. Netw. Commun. Emerg. Technol. 1(1), 6–10 (2015)

13. Chanda, B., Majumder, D.D.: Digital Image Processing and Analysis. Prentice-Hall of India(2002)

14. Deepak., Jain, A.: Medical image denoising using dual tree complex wavelet transforms. Int.J. Adv. Res. Comput. Sci. Softw. Eng. 6(11), 252–256 (2016)

15. Demirel, H., Anbarjafari, G.: Satellite image resolution enhancement using complex wavelettransform. IEEE GeoSci. Remote Sens. Lett. 7(1), 123–126 (2010)

16. Donoho, D.L.: Nonlinear wavelet methods for recovery of signals, densities and spectra fromindirect and noisy data. Proc. Symposia Appl. Math. 47, 173–205 (1993)

17. Fan, Z., Bi, D., Gao, S., He, L., Ding, W.: Adaptive enhancement for infrared image usingshearlet frame. J. Opt. 18, 1–11 (2016)

18. Farzam, S., Rastgarpour, M.: An image enhancement method based on curvelet transform forCBCT-images. Int. J. Math. Comput. Phys. Electr. Comput. Eng. 11(6), 200–206 (2017)

19. Favorskayaa, M.N., Savchinaa, E.I.: Content preserving watermarking for medical imagesusing shearlet transform and SVD. The International Archives of the Photogrammetry, RemoteSensing and Spatial Information Sciences, vol. XLII-2/W4, pp. 101–108 (2017)

20. Feng, P., Pan, Y., Wei, B., Jin, W., Mi, D.: Enhancing retinal image by the contourlet transform.Pattern Recognit. Lett. 28, 516–522 (2007)

21. Firake, D.G., Mahajan, P.M.: Satellite image resolution enhancement using dual-tree complexwavelet transform and nonlocal means. Int. J. Eng. Sci. Comput. 6(10), 2810–2815 (2016)

22. Garg, R., Mittal, B., Garg, S.: Histogram equalization techniques for image enhancement. Int.J. Current Eng. Technol. 2(1) (2011)

23. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 2nd edn. Fourth Indian Reprint, Pear-son Education (2003)

References 229

24. Hanumantharaju, M.C., Ravishankar, M., Rameshbabu, D.R., Ramchandran, S.: Color imageenhancement using multiscale retinex with modified color restoration technique. In: SecondInternational Conference on Emerging Applications of Information Technology, pp. 93–97(2011)

25. HemaLatha, M., Vardarajan, S.: Resolution enhancement of low resolution satellite imagesusing dual-tree complex wavelet transform. Int. J. Sci. Eng. Res. 8(5), 1361–1364 (2017)

26. Heric, D., Potocnik, B.: Image enhancement by using directional wavelet transform. J. Comput.Inf. Technol. 2, 299–305 (2006)

27. Hossain, F., Alsharif, M.R.: Image Enhancement based on lograthmic transform co-efficientand adaptive histogram equalization. In: International Conference on Convergence InformationTechnology, pp. 1439–1444 (2007)

28. Iqbal, M.Z., Ghafoor, A., Siddiqui, A.M.: Satellite image resolution enhancement using dual-tree complex wavelet transform and nonlocal means. IEEE GeoSci. Remote Sens. Lett. 103,451–455 (2013)

29. Jain, A.K.: Fundamentals of Digital Image Processing. Pearson Education, Fourth IndianReprint (2005)

30. Jobson, D.J., Rahman, Z., Woodell, G.A.: A multi-scale retinex for bridging the gap betweencolor images and the human observation of scenes. IEEE Trans. Image Process. 6, 965–976(1997)

31. Jobson, D.J., Rahman, Z., Woodell, G.A.: Properties and performance of a center/surroundretinex. IEEE Trans. Image Process. 6, 451–462 (1997)

32. Kakkar,V., Sharma,A.,Mangalam, T.K.,Kar, P.: Fingerprint image enhancement usingwavelettransform and gabor filtering. Acta Technica Napocensis Electron. Telecommun. 524, 17–25(2011)

33. Kanagalakshmi, K., Chandra, E.: Frequency domain enhancement algorithm based on log-gabor filter in FFT domain. Glob. J. Comput. Sci. Technol. 12(7), 15–22 (2012)

34. Karunakar, P., Praveen, V., Kumar, O.R.: Discrete wavelet transform-based satellite imageresolution enhancement. Adv. Electron. Electr. Eng. 3(4), 405–412 (2013)

35. Kaur, G., Vashist, S.: A robust approach for medical image enhancement using DTCWT. Int.J. Comput. Appl. 167(6), 26–29 (2017)

36. Kaur, G.: Aashima: bio-medical image enhancement using counter let transformation andbacterial foraging optimization. Int. J. Adv. Res. Comput. Commun. Eng. 4(9), 397–401 (2015)

37. Kaur, G., Singh, R.: Sharpening enhancement of ultra sound images using firefly algorithm.Int. J. Adv. Res. Comput. Sci. Softw. Eng. 4(8), 1039–1044 (2014)

38. Khatkara, K., Kumar, D.: Biomedical image enhancement using wavelets. In: InternationalConference on Intelligent Computing, Communication and Convergence. Procedia ComputerScience, vol. 48, pp. 513–517 (2015)

39. Kim, J.Y., Kim, L.S., Hwang, S.H.: An advanced contrast enhancement using partially over-lapped sub-block histogram equalization. IEEE Trans. Circuits Syst. Video Technol. 11, 475–484 (2001)

40. Kim, S., Kang, W., Lee, E., Paik, J.: Wavelet-domain color image enhancement using filtereddirectional bases and frequency-adaptive shrinkage. IEEE Trans. Consum. Electron. 56(2)(2010)

41. Kim, Y.T.: Contrast enhancement using brightness preserving bi-histogram equalization. IEEETrans. Consum. Electron. 43, 1–8 (1997)

42. Kole, P.S., Patil, S.N.: Satellite image resolution enhancement using discretewavelet transform.Int. J. Eng. Sci. Comput. 6(4), 3719–3721 (2016)

43. Kumar, B.P.S.: Image enhancement using discrete curvelet transform. Int. Res. J. Eng. Technol.2(8), 1252–1259 (2015)

44. kumar, D., kumar, D.V.S.N.: Image de-noising using dual-tree complex wavelet transform forsatellite applications. Int. Res. J. Eng. Technol. 2(6), 1125–1132 (2015)

45. Land, E.: Recent advances in retinex theory. Vis. Res. 26(1), 7–21 (1986)46. Li, X.M.: Image enhancement in the fractional fourier domain. In: 6th International Congress

on Image and Signal Processing, pp. 299–303 (2013)


47. Mahesh, A., Sreedhar, Ch., Zeeshan, A.: Double density dual-tree complex wavelet transformbased satellite image resolution enhancement. Int. J. Eng. Sci. Res. 4(8), 595–601 (2014)

48. Mallat, S., Hwang, W.L.: Singularity detection and processing with wavelets. IEEE Trans. Inf.Theory 38(2), 617–643 (1992)

49. Mallot, H.A.: Computational Vision: Information Processing in Perception and Visual Behav-ior. MIT Press, Cambridge, MA (2000)

50. Melkamu, H.A., Vijanth, S.A., Lila, I., Ahmad, F.M.H.: Image enhancement by fusion incontourlet transform. Int. J. Electr. Eng. Inform. 2(1), 29–42 (2010)

51. Melkamu, H.A., Vijanth, S.A., Ahmad, F.M.H.: Image enhancement based on contourlet trans-form. SIViP 9, 1679–1690 (2015)

52. Miaindargi, V.C., Mane, A.P.: Decimated and un-decimated wavelet transforms based imageenhancement. Int. J. Ind. Electr. Electron. Control Robot. 3(5), 26–30 (2013)

53. Munteanu,C., Rosa,A.: Color image enhancement using evolutionary principles and the retinextheory of color constancy. In: Proceedings of IEEE Signal Processing Society Workshop onNeural Networks for Signal Processing XI, pp. 393–402 (2001)

54. Narayana, B.R.V.S., Nirmala, K.: Image resolution enhancement by using stationary and dis-crete wavelet decomposition. Int. J. Image Graphics Signal Process. 11, 41–46 (2012)

55. Neena, K.A., Raj, A., Roy, R.C.: Image enhancement based on stationary wavelet transform,integer wavelet transform and singular value decomposition. Int. J. Comput. Appl. 58(11),30–35 (2012)

56. Pai, R.D., Halvi, S., Hiremath, B.: Medical color image enhancement using wavelet transformand contrast stretching technique. Int. J. Sci. Res. Publ. 5(7), 1–7 (2015)

57. Pawade, P.V., Gaikwad, M.A.: Contrast and luminance enhancement using discrete shearlettransform, discrete cosine transform and filtering technique. Int. J. Sci. Technol. Eng. 3(1),342–347 (2016)

58. Pawar, M.M., Kulkarni, N.P.: Image resolution enhancement using multi-wavelet transformswith interpolation technique. IOSR J. Electr. Electron. Eng. 9(3), 9–13 (2014)

59. Prasad, V., Ganesh, N.S., Khan, Habibullah, B.K., Muralidhar, C., Kiran, C, Tulasi, S.: Imageenhancement using wavelet transforms and SVD. Int. J. Eng. Sci. Technol. 4(3), 1080–1087(2012)

60. Premkumar, S., Parthasarathi, K.A.: An efficient approach for colour an efficient approach forcolor image enhancement using discrete shearlet transform. In: International Conference oncurrent Trends in Engineering and Technology, ICCTET (2014)

61. Provenzi, E., Caselles, V.: A wavelet perceptive on variational perceptually-inspired colorenhancement. Int. J. Comput. Vis. 106, 153–171 (2014)

62. Rahman, Z., Jobson, D.J., Woodell, G.A.: Multi-scale retinex for color image enhancement.In: IEEE International Conference on Image Processing (1996)

63. Ramiz, M.A., Quazi, R.: Design of an efficient image enhancement algorithms using hybridtechnique. Int. J. Recent Innov. Trends Comput. Commun. 5(6), 710–713 (2017)

64. Ren, C.C., Yang, J.: A novel color microscope image enhancement method based on HSVcolor space and curvelet transform. Int. J. Comput. Sci. Issues 9(6), 272–277 (2012)

65. Saravanan, G., Yamuna, G., Vivek, R.: A color image enhancement based on discrete wavelettransform. Int. J. Comput.Appl. Proc.Natl. Conf. Emerg. Trends Inf. Commun. Technol. (2013)

66. Shanida,K., Shayini,R., Sindhu,C.S.:Dental image enhancement usingwavelet decompositionand reconstruction. Int. J. Recent Adv. Eng. Technol. 4(7), 73–77 (2016)

67. Sharma,B.K.,Mishra,R.:Application of dual-tree complexwavelet transform in satellite imageresolution enhancement using nonlocal means. Int. J. Sci. Eng. Technol. Res. 4, 1725–1728(2015)

68. Sharma, R., Chopra, P.K.: Enhancing X-ray images based on dual tree complex wavelet trans-form. Int. J. Sci. Eng. Technol. Res. 4(6), 2179–2184 (2015)

69. Singh, I., Khullar, S., Laroiya, S.C.: DFT based image enhancement and steganography. Int. J.Comput. Sci. Commun. Eng. 2(1), 5–7 (2013)

70. Sivasankari, J., Ashlin, M.M., Janani, T., Shahin, D.F.: Despeckling of SAR images based onbayes shrinkage thresholding in shearlet domain. Int. J. Innov. Res. Sci. Eng. Technol. 3(3),1175–1181 (2014)

References 231

71. Song, H.: Image enhancement scheme based on nonsubsampled contourlet transform. In: Pro-ceedings of the SPIE 8878, Fifth International Conference on Digital Image Processing (ICDIP2013), pp. 887804 (2013)

72. Starck, J.L., Murtagh, F., Cands, E.J., Donoho, D.L.: Gray and color image contrast enhance-ment by the curvelet transform. IEEE Trans. Image Process. 12(6), 706–717 (2003)

73. Sumathi, M., Murthi, V.K.: Image enhancement based on discrete wavelet transform. IIOABJ7(10), 12–15 (2016)

74. Tarar, S., Kumar, E.: Fingerprint image enhancement: iterative fast fourier transform algorithmand performance evaluation. Int. J. Hybrid Inf. Technol. 6(4), 11–20 (2013)

75. Tamaz, H., Demirel, H., Anbarjafari, G.: Satellite image enhancement by using dual tree com-plex wavelet transform: denoising and illumination enhancement. In: Signal Processing andCommunications Applications Conference (SIU) (2012)

76. Thorat,A.A.,Katariya, S.S.: Image resolution improvement byusingDWTandSWT transform.I.J.A.R.I.I.E. 1(4), 490–495 (2015)

77. Toet, A.: Multiscale color image enhancement. In: Proceedings of SPIE International Confer-ence on Image Processing and Its Applications, pp. 583–585 (1992)

78. Tong, Y., Chen, J.: Non-linear adaptive image enhancement in wireless sensor networks basedon nonsubsampled shearlet transform. EURASIP J. Wirel. Commun. Netw. 46 (2017)

79. Umamageswari, D., Sivasakthi, L., Vani, R.: Quality analysis and image enhancement usingtransformation techniques. Int. J. Adv. Res. Electr. Electron. Instrum. Eng. 3(3), 298–303(2014)

80. Unaldi, N., Temel, S., Demirci, S.: Undecimated wavelet transform based contrast enhance-ment. Int. J. Comput. Electr. Autom. Control Inf. Eng. 7(9), 1215–1218 (2013)

81. Velde, K.V.: Multi-scale color image enhancement. Proc. Int. Conf. Image Process. 3, 584–587(1999)

82. Wang, J., Jia, Z., Qin, X.: Remote sensing image enhancement algorithm based on shearlettransform and multi-scale retinex. J. Comput. Appl. 35(9), 202–205 (2015)

83. Wang, Y., Zhu, R.: Study of single image dehazing algorithms based on shearlet transform.Appl. Math. Sci. 8(100), 4985–4994 (2014)

84. Wubuli, A., Zhen-Hong, J., Xi-Zhong, Q., Jie, Y., Kasabov, N.: Medical image enhancementbased on shearlet transform and unsharp masking. J. Med. Imaging Health. Inform. 4(5), 814–818 (2014)

85. Xiao, F., Zhou, M., Geng, G.: Detail enhancement and noise reduction with color image detec-tion based on wavelet multi-scale. In: Proceedings of the 2nd International Conference onArtificial Intelligence, Management Science and Electronic Commerce. Zhengzhou, China,pp. 1061–1064. IEEE Press, New York (2011)

86. Yeleshetty, D., Shilpa, A., Sherine, M.R.: Intensification of dark mode images using FFT andbilog transformation. Int. J. Res. Comput. Commun. Technol. 4(3), 186–191 (2015)

Appendix AReal and Complex Number System

The set of natural numbers {1, 2, 3, 4, . . .} will be denoted by N, the set of integers{. . . ,−3,−2,−1, 0, 1, 2, 3, . . .} by Z, the set of rational numbers { xy ; y �= 0} byQ, the set of irrational numbers by Q

′, and union of rational & irrational numbersshall be denoted by R. The set of complex numbers {a + ib, a, b ∈ R, i = √−1}will be denoted by C.

Field. A pair (F,+, ·), where F is a set with operations + (addition) and · (multipli-cation), is called a field if it satisfies the following properties:

For Addition:

1. (Closure) x + y ∈ F, for all x, y ∈ F.

2. (Commutativity) x + y = y + x, for all x, y ∈ F.

3. (Associativity) x + (y + z) = (x + y) + z, for all x, y, z ∈ F.

4. (Additive identity) There exists an element 0, in F such that x + 0 =x, for all x ∈ F.

5. (Additive Inverse) There exists an element −x , in F such that x + (−x) =0, for all x ∈ F.

For Multiplication:

1. (Closure) x · y ∈ F, for all x, y ∈ F.

2. (Commutativity) x · y = y · x, for all x, y ∈ F.

3. (Associativity) x · (y · z) = (x · y) · z, for all x, y, z ∈ F.

4. (Multiplicative identity) There exists an element 1, in F such that x · 1 =x, for all x ∈ F.

5. (Multiplicative Inverse) There exists an element(1x ; x �= 0

), in F such that

x · 1x = 1, for all x ∈ F.

Distributive Law:

x · (y + z) = (x · y) + (x · z) for all x, y, z ∈ F.

Clearly, (R,+, ·) and (C,+, ·), form a field.

© Springer Nature Singapore Pte Ltd. 2018A. Vyas et al., Multiscale Transforms with Application to ImageProcessing, Signals and Communication Technology,https://doi.org/10.1007/978-981-10-7272-7

233

234 Appendix A: Real and Complex Number System

Ordered Field.

An ordered field is a field F with a relation< satisfying the following properties:1. (Comparison) If x, y ∈ F, then the one of the following holds: x < y, y <

x, x = y.2. (Transitivity) If x, y, z ∈ F, with x < y and y < z then x < z.3. If x, y, z ∈ F, and y < z, then x + y < x + z.4. If x, y ∈ F and x, y > 0, then x · y > 0.Clearly Q and R with the usual relation < form an ordered field.

Triangle inequality in R. If x, y ∈ R, then

|x + y| ≤ |x | + |y|.

We interpret |x − y| as the distance between the point x and y in R.

Convergence in R. Let M ∈ Z and x ∈ R. A sequence {xn}∞n=M of real numbersconverges to x ∈ R, if for all ε > 0, there exists N ∈ N such that |xn − x | < ε forall n > N .

Cauchy Sequence in R. A sequence {xn}∞n=M of real numbers is a Cauchy sequenceif, if for all ε > 0, there exists N ∈ N such that |xn − xm | < ε for all n,m > N .

Cauchy Criterion for convergence of a Sequence in R. Every Cauchy sequenceof real numbers converges.

The converse of Cauchy criterion is also true in R. Hence R with the usual addi-tion and multiplication forms a complete ordered field.

The set of complex numbersC also form a complete field but not an ordered field.We denote the elements of C in the usual way,

z = x + iy, where x, y ∈ R.

We call x the real part of z, denoted by Re(z) and y the imaginary part of z by Im(z),respectively.

Define the complex conjugate z of z by

z = x − iy.

The modulus squared of z by

|z|2 = zz = (x + iy)(x − iy) = x2 + y2

Appendix A: Real and Complex Number System 235

and the modulus or magnitude |z| of z by

|z| =√

|z|2 =√x2 + y2

Properties of Complex Numbers. Suppose z, w ∈ Z. Then

(i). z = z.(ii). Re(z) = z+z

2 , Im(z) = z−z2i .

(iii). z + w = z + w z · w = z · w.

(iv). |z| = |z|, |zw| = |z||w|.(v). |Re(z)| ≤ |z|, |Im(z)| ≤ |z|.(vi). |z + w| ≤ |z| + |w|. (Triangle inequality in C)

(vii).(zw

) = zw, w �= 0.

(viii).∣∣ zw

∣∣ = |z|

|w| .

Convergence inC. Let M ∈ Z and z ∈ C.A sequence {zn}∞n=M of complex numbersconverges to z ∈ C, if for all ε > 0, there exists N ∈ N such that |zn − z| < ε for alln > N .

Here |z − w| denotes the distance between two points z and w in the complexplane. For example, if z = x1 + iy1 and w = x2 + iy2, then

|z − w| = |(x1 − x2) + i(y1 − y2)| =√(x1 − x2)2 + (y1 − y2)2,

which is the same as the usual distance inR2 between two points (x1, y1) and (x2, y2).

Cauchy Sequence in C. A sequence {zn}∞n=M of complex numbers is a Cauchysequence if, if for all ε > 0, there exists N ∈ N such that |zn − zm | < ε for alln,m > N .

This leads to the Cauchy criterion for the convergence of a sequence of complexnumbers.

Completeness of C. A sequence of complex numbers converges if and only if it isa Cauchy sequence.

Appendix BVector Space

Vector Space Let F be a field. A vector space V over a field F is a set with operationsvector addition and scalar multiplication satisfying the following properties:

A. (i). u + v ∈ V for all u, v ∈ V .(ii). u + v = v + u for all u, v ∈ V .(iii). u + (v + w) = (u + v) + w for all u, v, w ∈ V .(iv). There exist an element in V, denoted by 0, such that u+0 = u for all u ∈ V .(v). For each u ∈ V, there exist an element in V, denoted as −u, such that

u + (−u) = 0.B. (i). α · u ∈ V, for all α ∈ F and u ∈ V .(ii). 1 · u = u for all u ∈ V and 1 is the multiplicative identity of F.(iii). α · (β · u) = (α · β) · u, for all α,β ∈ F and u ∈ V .(iv). (a) α · (u + v) = α · u + α · v for all α ∈ F and u, v ∈ V .(b) (α + β) · u = α · u + β · u for all α,β ∈ F and u ∈ V .

Linear Combination. Let V be a vector space over a field F, let n ∈ N, and letv1, v2, . . . vn ∈ V . A linear combination of vectors v1, v2, . . . vn is any vector of theform

n∑

i=1

αivi = α1v1 + α2v2 + · · · + αnvn,

where α1,α2, . . . ,αn ∈ F.

Let V be a vector space over a field F, suppose U ⊆ V . The span of U is the setof all linear combination of elements of U and is denoted by spanU.


237

238 Appendix B: Vector Space

If U is a finite set, say, U = {u1, u2, . . . , un}, then

spanU ={

n∑

i=1

αi ui : αi ∈ F for all i = 1, 2, . . . , n

}

.

Linear Dependence and Independence. Let V be a vector space over a field F andlet {v1, v2, . . . , vn} is linearly dependent if there exists α1,α2, . . . ,αn ∈ F that arenot all zero, such that,

α1v1 + α2v2 + · · · + αnvn = 0.

We say that the set {v1, v2, . . . , vn} is linearly independent if

α1v1 + α2v2 + · · · + αnvn = 0,

holds only when αi = 0 for all i = 1, 2, . . . , n.

If U is an infinite subset of V, we say U is linearly independent if every finitesubset of U is linearly independent and we say U is linearly dependent if U has afinite subset that is linearly dependent.

Basis. Let V be a vector space over a field F. A subset U of V is a basis for V if Uis a linearly independent set such that spanU = V .

Euclidean Basis for Rn or Cn .Define E = {e1, e2, . . . , en} by

e1 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

100···0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

, e2 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

010···0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

, e3 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

001···0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

, · · ·, en =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

000···1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

These vectors can be regarded as elements of Rn or Cn.

If a vector space V has a basis consisting of finitely many elements, we say thatV is finite dimensional vector space. In this case, any two basis for V have the samenumber of elements.

Dimension of Vector Space V . Suppose V is a finite dimensional vector space. Thenumber of elements in a basis for V, denoted as, dimV . If dimV = n, we say thatV is n-dimensional vector space.

Appendix CLinear Transformation, Matrices

Linear Transformation. Let U and V be two vector space over the same field F. Alinear transformation T is a function T : U → V having the following properties:

(i). T (u + v) = T (u) + T (v) for all u, v ∈ U.

(ii). T (αu) = αT (x) for all α ∈ F and u ∈ U.

Any linear transformation T : X → Y can be represented in basis for X and Yby matrix multiplication.

Matrix. For m, n ∈ N, an m × n matrix A over a field F is a rectangular array of theform

A =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

a11 a12 · · · a1na21 a22 · · · a2na31 a32 · · · a3n

· · · · · ·· · · · · ·· · · · · ·

am1 am2 · · · amn

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

where ai j ∈ F for all i = 1, 2, . . . ,m and j = 1, 2, . . . , n. We call ai j the (i, j)thentry of A.We also denote A by [ai j ].

Note that ann×1matrix is a vectorwith n-components i. e. an element ofRn ofCn .

Transpose of a Matrix. let A = [ai j ] be an m × n matrix over C. The transpose ofmatrix A, denoted by At , is the n × m matrix B = [bi j ] defined by bi j = a ji for alli, j.or the transpose At is obtained by interchanging the rows and columns of matrix A.


239

240 Appendix C: Linear Transformation, Matrices

Conjugate Transpose of a Matrix. let A = [ai j ] be an m × n matrix over C. Theconjugate transpose A∗ ofmatrix A is the n×mmatrixC = [ci j ] defined by ci j = a ji

for all i, j.or the conjugate transpose A∗ is obtained by taking the complex conjugate of all theentries of At .

Let A be an m × n matrix over C. Then

〈Az, w〉 = 〈z, A∗w〉,

for every z ∈ Cn andw ∈ C

m . Furthermore, A∗ is the only matrix with this property.

Unitary Matrix. Let A be an n × n matrix over C. Then A is called unitary matrixif A is invertible and A−1 = A∗.

For a matrix over the real numbers, the conjugate transpose is the same as thetranspose. So a real unitary matrix A is one that satisfies A−1 = A∗; such a matrixis called orthogonal.

Let A be an n × n matrix over C. Then the following statements are equivalent:(i). A is unitary.(ii). The columns of A form an orthonormal basis for Cn.

(iii). The rows of A form an orthonormal basis for Cn.

(iv). Matrix A preserves inner products, that is, 〈Az, Aw〉 = 〈z, w〉 for all z,w ∈ C

n.

(v). ||Az|| = ||z||, for all z ∈ Cn.

Appendix DInner Product Space and Orthonormal Basis

An inner product is a generalization of the dot product for vectors in Rn.

Inner Product. Let V be a vector space over C. A (complex) inner product is a map〈·, ·〉 : V × V → C with the following properties:

(i). (Additivity) 〈u + v,w〉 = 〈u, w〉 + 〈v,w〉 for all u, v, w ∈ V .(ii). (Scalar Homogeneity) 〈αu, v〉 = α〈u, v〉 for all α ∈ C and u, v ∈ V .(iii). (Conjugate Symmetry) 〈u, v〉 = 〈v, u〉 for all u, v ∈ V .(iv). (Positive Definiteness) 〈u, u〉 ≥ 0 for all u ∈ V, and 〈u, u〉 = 0 if and only

if u = 0.

A vector space V with complex inner product is called a complex inner productspace.

Conditions (i) and (iii) imply that

〈u, v + w〉 = 〈u, v〉 + 〈u, w〉 for all u, v, w ∈ V .

Conditions (ii) and (iii) imply that

〈u,αv〉 = α〈u, v〉 for all α ∈ C and for all u, v ∈ V .

An inner product always yields a norm in the following way:

Norm. Let V be a vector space overCwith a complex inner product 〈·, ·〉. For u ∈ V,define

||u|| = √〈u, u〉,

is called norm of u and denoted by || · ||.


241

242 Appendix D: Inner Product Space and Orthonormal Basis

Cauchy-Schwarz Inequality. Let V be a complex inner product space. Then forany u, v ∈ V,

|〈u, u〉| ≤ ||u||||v||.

Triangle Inequality. Let V be a complex inner product space. Then for any u, v ∈ V,

||u + v|| ≤ ||u|| + ||v||.

Orthogonality. Suppose V is a complex inner product space. For u, v ∈ V, we saythat u and v are orthogonal, if 〈u, v〉 = 0. It is denoted by u ⊥ v.

Note that if u ⊥ v, then we have

||u + v||2 = ||u||2 + 〈u, v〉 + 〈v, u〉 + ||v||2 = ||u||2 + ||v||2.

Orthogonal Set. Suppose V is a complex inner product space. Let B be a collectionof vectors in V . B is called orthogonal set if any two different elements of B areorthogonal. B is called an orthonormal set if B is an orthogonal set and ||v|| = 1 forall v ∈ B.

Orthogonal sets of nonzero vectors are linearly independent.

Suppose V is a complex inner product space. Suppose B is an orthogonal set ofvectors in V and 0 /∈ B. Then Bis a linearly independent set.

Suppose V is a complex inner product space and B = {u1, u2, . . . , un} is anorthogonal set in V with u j �= 0 for all j. If v ∈ span(B), then

v =n∑

j=1

〈v, u j 〉||u j ||2 u j .

If B is orthonormal set, then

v =n∑

j=1

〈v, u j 〉u j .

Appendix D: Inner Product Space and Orthonormal Basis 243

Orthogonal Projection. Suppose V is a complex inner product space and B ={u1, u2, . . . , un} is an orthogonal set in V with u j �= 0 for all j. Let S = span(B).For v ∈ V, define the orthogonal projection PS(v) of v on S by

PS(v) =n∑

j=1

〈v, u j 〉||u j ||2 u j .

The orthogonal projection operator PS has the following properties:(i). PS is a linear transformation.(ii). For every v ∈ V, PS(v) ∈ S.(iii). If s ∈ S, then PS(S) = S.(iv). (Orthogonality Property) For v ∈ V and s ∈ S, (v − PS(v)) ⊥ S.(v). (Best approximation property) For any v ∈ V and s ∈ S, ||v − PS(v)|| ≤

||v − s||, with equality if and only if S = PS(v).

Gram-Schmidt Procedure. Suppose V is a complex inner product space and{u1, u2, . . . , un} is a linearly independent set in V . Then there exists an orthonormalset {v1, v2, . . . , vn} with the same span.

Suppose V is a complex inner product space. An orthonormal basis for V is anorthonormal set in V that is also a basis.

Let V be a complex inner product space with finite orthonormal basis R ={u1, u2, . . . , un}.

(i). For any v ∈ V, v = ∑nj=1〈v, u j 〉u j .

(ii). (Parseval’s Relation) For any v,w ∈ V, 〈v,w〉 = ∑nj=1〈v, u j 〉〈w, u j 〉.

(iii). (Plancherel’s Formula) For any v ∈ V, ||v||2 = ∑nj=1 |〈v, u j 〉|2.

Appendix EFunctions and Convergence

E.1 Functions

Bounded (L∞) Functions. A piecewise continuous function f (x) defined on aninterval I is bounded (or L∞) on I if there is a number M > 0 such that | f (x)| ≤ Mfor all x ∈ I.

The L∞-norm of a function f (x) is defined by

|| f ||∞ = sup{| f (x)| : x ∈ I }.

If I is a closed and finite interval, then any continuous function f (x) ∈ I is alsoin L∞(I ).

Integrable (L1) Functions. A piecewise continuous function f (x) defined on aninterval I is integrable (or L1) on I if the integral

∫

I| f (x)|dx

is finite.

The L1-norm of a function f (x) is defined by

|| f ||1 =∫

I| f (x)|dx .

If f (x) is L∞(I ), then f (x) is L1(I ). Any continuous function on a finite closedinterval I is L1(I ), since such a function must be on L∞(I ). Any piecewise contin-uous function with only jump discontinuous on a finite closed interval I is on L1(I ).


245

246 Appendix E: Functions and Convergence

First Approximation Theorem. Let f (x) ∈ L1(R), and let ε > 0 be given. Thenthere exists a number R such that if

g(x) = f (x)χ[−R,R](x) ={f (x) i f x ∈ [−R, R],0 i f x /∈ [−R, R].

Then ∫ ∞

−∞| f (x) − g(x)|dx = || f − g||1 < ε.

This theorem says that any function in L1(R) can be approximated arbitrarilyclosely in the sense of the L1-norm by a function with compact support.

Square Integrable (L2) Functions. A piecewise continuous function f (x) definedon an interval I is square integrable (or L2) on I if the integral

∫

I| f (x)|2dx

is finite.

The L2-norm of a function f (x) is defined by

|| f ||2 =(∫

I| f (x)|2dx

)1/2

.

Any bounded function f (x) on a finite interval I is also in L2(I ). This includescontinuous functions on closed intervals and functions piecewise continuous onclosed intervals with only jump discontinuous. Any functions that is L∞(I ) andL1(I ), (I may be finite or infinite) is also in L2(I ).

Cauchy-Schwarz Inequality. Let f (x) and g(x) be two functions on L2(I ). Then

∣∣∣∣

∫

If (x)g(x)

∣∣∣∣ ≤(∫

I| f (x)|2dx

)1/2 (∫

I|g(x)|2dx

)1/2

.

Let I be a finite interval. Cauchy-Schwarz inequality implies that any functionf (x) ∈ L2(I ) is also in L1(I ). This is not true in case I is infinite interval. Forexample, let

f (x) ={1/x i f x ≥ 1,0 i f x < 1.

Appendix E: Functions and Convergence 247

Clearly, f (x) is in L2(R) but not in L1(R). Converse of the theorem is false for bothfinite and infinite intervals.

Minkowski’s Inequality. Let f (x) and g(x) be two functions on L2(I ). Then

(∫

I| f (x) + g(x)|2dx

)1/2

≤(∫

I| f (x)|2dx

)1/2

+(∫

I|g(x)|2dx

)1/2

.

It is also known as the triangle inequality for L2 functions; that is, it says that|| f + g||2 ≤ || f ||2 + ||g||2.

Approximation Theorem for L2(R). Let f (x) ∈ L2(R), and let ε > 0 be given.Then there exists a number R such that if

g(x) = f (x)χ[−R,R](x) ={f (x) i f x ∈ [−R, R],0 i f x /∈ [−R, R].

Then ∫ ∞

−∞| f (x) − g(x)|2dx = || f − g||22 < ε.

Differentiable Cn Functions. Given n ∈ N, we say that a function f (x) defined onan interval I is Cn on I if it is n-times continuously differentiable on I. Functionf (x) is C0 on I means that f (x) is continuous on I. Function f (x) is C∞ on I if itis Cn on I for every n ∈ N.

We say that a function f (x) defined on an interval I is Cnc on I if it is Cn on I

and compactly supported, C0c on I means it is C0 on I and compactly supported, and

C∞c on I if it is C∞ on I and compactly supported.

E.2 Convergence of Functions

Numerical Convergence. The sequence {an}n∈N converges to the number a if forevery ε > 0, there is an N > 0 such that if n ≤ N , then |an − a| < ε.

Aseries∑∞

n=1 an converges to a number S if the sequenceof partial sums {SN }N∈N,defined by, SN = ∑N

n=1 an converges to S. In this case, we write∑∞

n=1 an = S.

A series∑∞

n=1 an converges absolutely if∑∞

n=1 |an| converges.

248 Appendix E: Functions and Convergence

Pointwise Convergence. A sequence of functions { fn(x)}n∈N defined on an intervalI converges pointwise to a function f (x) if for each x0 ∈ I, the sequence { fn(x0)}n∈Nconverges to f (x0).

The series∑∞

n=1 fn(x) = f (x) pointwise on an interval I if for each x0 ∈I,

∑∞n=1 fn(x0) = f (x0).

Uniform (or L∞) Convergence. A sequence of functions { fn(x)}n∈N defined on aninterval I converges uniformly (or L∞) to the function f (x) if for each ε > 0, thereis an N > 0 such that if n ≥ N , then | fn(x) − f (x)| < ε for all x ∈ I.

The series∑∞

n=1 fn(x) = f (x) uniformly on an interval I if the sequence ofpartial sums SN (x) = ∑N

n=1 fn(x) converges uniformly to f (x) on I.

Theorem If fn(x) converges to f (x) uniformly (or L∞) on an interval I, then fn(x)converges to f (x) pointwise on I.

Theorem If fn(x) converges to f (x) uniformly (or L∞) on an interval I, and ifeach fn(x) is continuous on I, then f (x) is continuous on I.

Mean (or L1) convergence. The sequence { fn(x)}n∈N defined on an interval Iconverges in mean to the function f (x) on I if

limn→∞∫

I| fn(x) − f (x)|dx = 0 or limn→∞|| fn − f ||1 = 0.

The series∑∞

n=1 fn(x) = f (x) in mean on I if the sequence of partial sumsSN (x) = ∑∞

n=1 fn(x) converges in mean to f (x) on I.

Theorem If fn(x) converges to f (x) uniformly (or L∞) on a finite interval I, thenfn(x) converges to f (x) in mean or L1 on I.

Mean-square (or L2) convergence. The sequence { fn(x)}n∈N defined on an intervalI converges in mean-square to the function f (x) on I if

limn→∞∫

I| fn(x) − f (x)|2dx = 0 or limn→∞|| fn − f ||1 = 0.

The series∑∞

n=1 fn(x) = f (x) in mean-square on I if the sequence of partialsums SN (x) = ∑∞

n=1 fn(x) converges in mean-square to f (x) on I.

Theorem If fn(x) converges to f (x) uniformly (or L∞) on a finite interval I, thenfn(x) converges to f (x) in mean-square or L2 on I.

Appendix E: Functions and Convergence 249

Theorem If fn(x) converges to f (x) in mean-square (or L2) on a finite interval I,then fn(x) converges to f (x) in mean or L1 on I.

The following theorem gives several conditions under which interchanging thelimit and the integral is permitted.

Theorem

a. If fn(x) converges to f(x) on L1(I ), then

limn→∞∫

Ifn(x)dx =

∫

If (x)dx .

b. If fn(x) converges to f(x) on L∞(I ), where I is the finite interval, then

limn→∞∫

Ifn(x)dx =

∫

If (x)dx .

c. If fn(x) converges to f(x) on L2(I ), where I is the finite interval, then

limn→∞∫

Ifn(x)dx =

∫

If (x)dx .

If I is an infinite interval, then the conclusions of Theorem (b) and (c) are false.However, in case of infinite intervals, we have the following theorem by making anadditional assumption on the sequence { fn(x)}n∈N :

Theorem Suppose the for every R > 0, fn(x) converges to f (x) in L∞ or in L2 on[−R, R]. That is for each R > 0,

limn→∞∫ R

−R| fn(x)− f (x)|2dx = 0 or limn→∞sup{| fn(x)− f (x)| : x ∈ [−R, R]} = 0.

If f (x) is L1(I ) and if there is a function g(x) on L1(I ) such that for all x ∈ Iand for all n ∈ N, | fn(x)| ≤ g(x), then

limn→∞∫

Ifn(x)dx =

∫

If (x)dx .

Index

AAdaptive filters, 146Additive noise, 137Admissible shearlet, 114, 115Alpha-trimmed mean filter, 146Amplifier noise, 137Amplitude, 17Angular frequency, 16, 17Application of multiscale transforms, 154,

210Approximation adjoint, 60, 74Approximation coefficient, 54, 63Approximation operator, 48, 60, 71, 74Arithmetic mean filter, 143Auto white balance, 5

BBand-limited wavelet, 48, 97Basis, 70Bayer pattern, 3Biorthogonal, 70–72Biorthogonal QMF, 76Biorthogonal wavelet, 70Bivariate ridgelet, 94Butterworth LowPass filter, 206

CClassical shearlet, 115, 116Color Filter Array (CFA), 3Colour image enhancement, 209Compactly supported shearlets, 121Compactly supported wavelet, 48, 96, 97Complex Fourier series, 19

Complex-valued scaling function, 78Complex-valued wavelet, 78Complex wavelet transform, 77, 78Cone-adapted continuous shearlet trans-

form, 117Cone-adapted discrete shearlet transform,

118, 119, 121Conjugate symmetry, 24, 33Constraint least-square filter, 151Continuous Ridgelet transform, 94, 96, 98,

101Continuous shearlet, 114Continuous shearlet system, 113, 117Continuous shearlet transform, 114Continuous wavelet transform, 68, 69, 114Contourlet, 106Contourlet filter bank, 109, 111Contourlet transform, 106, 107, 111Contra-harmonic mean filter, 145Contrast enhancement, 200, 202Contrast stretching, 202, 203Convolution, 28Convolution theorem, 24, 32CurveletG1, 102, 103CurveletG1 transform, 102CurveletG2, 102–105Curvelets, 101Curvelet transform, 102, 106, 107

DDefocus blur, 141Demosaicing, 5Detail adjoint, 60, 74Detail coefficient, 54, 63


251

252 Index

Detail operator, 48, 60, 71, 74Differentiation property, 24Digital image, 3Digital image processing, 3Dilation operator, 112Direct inverse filtering, 151Directional decomposition, 108Discrete complex wavelet transform, 79Discrete contourlet transform, 106, 109–111Discrete cosine transform, 38, 39Discrete curveletG1, 102Discrete curveletG2, 102Discrete Fourier transform, 25, 26, 29–31,

38, 97Discrete quaternion wavelet transform, 88Discrete Ridgelet transform, 97, 98Discrete shearlet system, 115Discrete shearlet transform, 116Discrete sine transform, 38Discrete wavelet transform, 54, 55, 57, 62,

63, 65, 67Downsampling operator, 60Dual, 71Dual basis, 70Duality, 24Dual MRA, 71, 72Dual-tree complexwavelet transform, 79–82Dual wavelet, 72Dyadic wavelet, 47

EEfficiency, 65Euclidean basis, 25Euler identities, 19Exponential noise, 139

FFast Fourier transform, 28, 33, 34, 36, 37, 64Fast Slant Stack Ridgelet Transform, 100Fast wavelet transform, 62First-generation curvelet transform, 102Formals adjoints, 60Fourier coefficients, 18, 20Fourier integral, 21Fourier inversion formula, 27, 29Fourier series, 18–21, 53Fourier transform, 20–23, 38, 40, 41, 46, 47,

49, 51, 77, 78, 83, 97, 99, 100, 106Frame, 112Frequency domain filtering, 146

GGaussian blur, 140Gaussian distribution, 137Gaussian highpass filter, 208Gaussian lowpass filter, 206Geometric mean filter, 143Gridding in tomography, 96

HHarmonic analysis, 18Harmonic mean filter, 144Heisenberg uncertainty principle, 40Hilbert transform, 78, 80, 84, 86Histogram equalization, 203Histogram processing, 203Homomorphic filtering, 208

IIdeal highpass filter, 207Ideal lowpass filter, 205Image deblurring, 140Image degradation, 133Image denoising, 136Image enhancement, 199, 200Image negative, 201Image restoration, 133Image restoration algorithms, 142Image Signal Processing (ISP), 3Impulse train, 5Impulsive noise, 138Inverse continuous wavelet transform, 68Inverse discrete cosine transform, 38Inverse discrete Fourier transform, 29Inverse discrete wavelet transform, 62, 68Inverse dual-tree complex wavelet trans-

form, 80Inverse fast wavelet transform, 64Inverse Fourier transform, 23, 47, 96–98Inverse Radon transform, 97Inversion formula, 22Iterative blind deconvolution, 152

LLaplacian pyramid, 107Linear filters, 147Linearity, 23, 32Localization, 65Local radon transform, 109Local Ridgelet transform, 101, 102Log transformation, 201LPA-ICI algorithm, 153

Index 253

LP decomposition, 107Lucy-Richardson algorithm, 154

MMax and min filters, 145Mean filters, 142Median filter, 145Midpoint filter, 145Motion blur, 141MRA wavelet, 49Multiplicative noise, 137Multiresolution analysis, 48, 49, 55, 62Multiscale decomposition, 107Multiscale transforms, 150

NNAS-RIF, 152Neural network approach, 154Noise models, 136

OOrder-statistic filter, 145Orthonormal basis, 26, 47–49, 52, 98Orthonormal finite ridgelet transform, 100

PParseval frame, 112Parseval’s relation, 24, 27, 95Perfect reconstruction, 60Period, 16, 17, 20, 29Periodic function, 16, 18, 20, 29, 50Phase, 17Plancherel theorem, 27, 99Power-Law transformation, 202Pyramidal directional filter bank, 109

QQuadrature mirror filter, 57, 61Quantization, 5Quaternion algebra, 84, 85Quaternion Fourier transform, 84Quaternion multiresolution analysis, 88Quaternion wavelet, 83, 84, 86–88Quaternion wavelet scale function, 87Quaternion wavelet transform, 83, 88

RRadon domain, 95

Radonlets, 109Radon transform, 95, 96, 100Rayleigh noise, 138Rectangular blur, 141RectoPolar Ridgelet transform, 96Refinement equation, 49, 55, 62Restoration process, 133Reversal, 24Ridgelet, 95Ridgelet transform, 94, 96, 97, 100, 101, 103

SSampling, 5Scaling, 23Scaling coefficient, 54Scaling filter, 49, 57, 61, 72, 74Scaling function, 48, 55, 71, 72, 74, 97Scaling relation, 49Second-generation curvelet transform, 102,

103Separability, 31, 32Sharpening filters, 207Shearing matrix, 113Shearing operator, 113Shearlet, 111Shearlet generators, 120Shearlet group, 113, 115Shearlet scaling function, 120Shearlet transform, 114Shifting, 23, 33Shift operator, 60Short-time Fourier transform, 40Signal-to-noise ratio, 5Similarity, 31Sinusoidal function, 17Smoothing filters, 205Sparse representation, 101, 103, 105, 123Sparsity, 65Spatial filtering, 142, 204Spatial-frequency filtering, 147Speckle noise, 139Stationary wavelet transform, 69, 70Structure extraction, 64Superresolution, 141Superresolution restoration algorithm, 152

TThreshold filters, 147Tight frame, 112Translation operator, 112

254 Index

UUndecimated wavelet transform, 69Uniform noise, 138Upsampling operator, 60

WWavelet, 46–49, 51, 53–55, 65, 69, 94, 95Wavelet coefficient, 54

Wavelet coefficient model, 149Wavelet domain, 147Wavelet filter, 53, 57, 61, 74Wavelet series, 53, 54Wavelet transform, 53, 64, 68, 83, 96, 97,

106, 111Wiener filter, 146Windowed Fourier transform, 40, 69

multiscale transforms with application to image processing

Documents