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This book presents the state of the art, together with the most recentresearch results, in the use of Quaternion Fourier Transforms (QFT) forthe processing of color images and complex valued signals. It is basedon the work of the authors in this area since the 1990s and presents themathematical concepts, computational issues and applications onimages and signals. The book, together with the MATLAB toolboxdeveloped by two of the authors (QTFM, http://qtfm.sourceforge.net/),allows the reader to make use of the presented concepts andexperiment with them in practice through the examples provided in thebook.

Following the Introduction, Chapter 1 introduces the quaternion algebraH and presents some properties which will be of use in the subsequentchapters. Chapter 2 gives an overview of the geometric transformationswhich can be represented using quaternions. Chapter 3 provides thedefinition and properties of QFT. The signals and images considered arethose with vector-valued samples/pixels.

The fourth and final chapter is dedicated to the illustration of the use ofQFT to process color images and complex improper signals. Theconcepts presented in this chapter are illustrated on simulated and realimages and signals.

Todd A. Ell is an Engineering Fellow at UTC Aerospace Systems,Burnsville, MN, USA. He is also a Visiting Fellow at the University ofEssex, Colchester, UK. His interests include the study and application ofhypercomplex algebras to dynamic systems analysis.

Nicolas Le Bihan is currently a Chargé de Recherche at the CNRSworking in the Department of Images & Signals (DIS) at the GIPSA-Labin Grenoble, France. His research interests include polarized signalprocessing, statistical signal processing on groups and manifolds,geometric (Berry) phases, random processes on non-commutativealgebraic structures and applications in physics and geophysics.

Stephen J. Sangwine is a Senior Lecturer with the School of ComputerScience and Electronic Engineering, University of Essex, Colchester, UK.His interests include linear vector filtering and transforms of vectorsignals and images, color image processing, and digital hardwaredesign.

Quaternion FourierTransforms for Signaland Image Processing

FOCUS

Todd A. Ell, Nicolas Le Bihan and Stephen J. Sangwine

DIGITAL SIGNAL AND IMAGE PROCESSING SERIES

FOCUS SERIES in DIGITAL SIGNAL AND IMAGE PROCESSING

www.iste.co.uk Z(7ib8e8-CBEHIB(

W478-Sangwine.qxp_Layout 1 10/04/2014 16:25 Page 1

Quaternion Fourier Transforms for

Signal and Image Processing

FOCUS SERIES

Series Editor Francis Castanié

Quaternion FourierTransforms for Signaland Image Processing

Todd A. EllNicolas Le Bihan

Stephen J. Sangwine

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, aspermitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,or in the case of reprographic reproduction in accordance with the terms and licenses issued by theCLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at theundermentioned address:

ISTE Ltd John Wiley & Sons, Inc.27-37 St George’s Road 111 River StreetLondon SW19 4EU Hoboken, NJ 07030UK USA

www.iste.co.uk www.wiley.com

© ISTE Ltd 2014The rights of Todd A. Ell, Nicolas Le Bihan and Stephen J. Sangwine to be identified as the author of thiswork have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2014934161

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISSN 2051-2481 (Print)ISSN 2051-249X (Online)ISBN 978-1-84821-478-1

Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY

Contents

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

CHAPTER 1. QUATERNION ALGEBRA . . . . . . . . . . . . . . . . . . . . . 1

1.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3. Exponential and logarithm of a quaternion . . . . . . . . . . . . . . . . 7

1.3.1. Exponential of a pure quaternion . . . . . . . . . . . . . . . . . . . . 71.3.2. Exponential of a full quaternion . . . . . . . . . . . . . . . . . . . . 91.3.3. Logarithm of a quaternion . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4. Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.1. Polar forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.2. The Cj-pair notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.3. R and C matrix representations . . . . . . . . . . . . . . . . . . . . . 17

1.5. Powers of a quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.6. Subfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

CHAPTER 2. GEOMETRIC APPLICATIONS . . . . . . . . . . . . . . . . . . . 21

2.1. Euclidean geometry (3D and 4D) . . . . . . . . . . . . . . . . . . . . . . 212.1.1. 3D reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.2. 3D rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.3. 3D shears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1.4. 3D dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

vi Quaternion Fourier Transforms for Signal and Image Processing

2.1.5. 4D reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1.6. 4D rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2. Spherical geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3. Projective space (3D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1. Systems of linear quaternion functions . . . . . . . . . . . . . . . . 312.3.2. Projective transformations . . . . . . . . . . . . . . . . . . . . . . . . 33

CHAPTER 3. QUATERNION FOURIER TRANSFORMS . . . . . . . . . . . . 35

3.1. 1D quaternion Fourier transforms . . . . . . . . . . . . . . . . . . . . . 383.1.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.2. Basic transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1.3. Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1.4. Inter-relationships between definitions . . . . . . . . . . . . . . . . . 453.1.5. Convolution and correlation theorems . . . . . . . . . . . . . . . . . 47

3.2. 2D quaternion Fourier transforms . . . . . . . . . . . . . . . . . . . . . 483.2.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.2. Basic transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.3. Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.4. Inter-relationships between definitions . . . . . . . . . . . . . . . . . 55

3.3. Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3.1. Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3.2. Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.3. Verification of transforms . . . . . . . . . . . . . . . . . . . . . . . . 62

CHAPTER 4. SIGNAL AND IMAGE PROCESSING . . . . . . . . . . . . . . . 67

4.1. Generalized convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1.1. Classical grayscale image convolution filters . . . . . . . . . . . . . 674.1.2. Color images as quaternion arrays . . . . . . . . . . . . . . . . . . . 704.1.3. Quaternion convolution . . . . . . . . . . . . . . . . . . . . . . . . . 704.1.4. Quaternion image spectrum . . . . . . . . . . . . . . . . . . . . . . . 73

4.2. Generalized correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.1. Classical correlation and phase correlation . . . . . . . . . . . . . . 814.2.2. Quaternion correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2.3. Quaternion phase correlation . . . . . . . . . . . . . . . . . . . . . . 88

4.3. Instantaneous phase and amplitude of complex signals . . . . . . . . . 914.3.1. Important properties of 1D QFT of a complex signal z(t) . . . . . . 914.3.2. Hilbert transform and right-sided quaternion spectrum . . . . . . . 964.3.3. The quaternion signal associated with a complex signal . . . . . . . 984.3.4. Instantaneous amplitude and phase . . . . . . . . . . . . . . . . . . . 1014.3.5. The instantaneous frequency of a complex signal . . . . . . . . . . . 102

Contents vii

4.3.6. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.3.7. The quaternion Wigner-Ville distribution of a complex signal . . . 1094.3.8. Time marginal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.3.9. The mean frequency formula . . . . . . . . . . . . . . . . . . . . . . 113

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Nomenclature

Symbol Meaning Section EquationR Set of real numbers section 1.1C Set of complex numbers section 1.1H Set of quaternions section 1.1V(H) Set of pure quaternions section 1.1Cμ Subfield of H section 1.6I Complex root of −1 (See note below.)(.) Real part section 1.1(.) Imaginary part section 1.1

i, j,k Quaternion basis elements section 1.1 [1.2]μ Pure quaternion section 1.1 [1.2]

i i-imaginary part section 1.1 [1.4]j j-imaginary part section 1.1 [1.4]k k-imaginary part section 1.1 [1.4]

MR(.) Real 4× 4 matrix representation section 1.4 [1.69]MC(.) Complex 2× 2 matrix representation section 1.4 [1.70][[.]] Alternate real 4× 4 matrix representation section 2.3 [2.20][.] Real 4× 1 vector representation section 2.3 [2.20]S(.) Scalar part section 1.1 [1.3]V(.) Vector part section 1.1 [1.3](q1, q2) Cj-pair notation of q section 1.4AB Arc of great circle between A and B section 2.2q Quaternion conjugate section 1.2 [1.18]z Complex conjugate section 1.4 [1.54]q μ Involution section 1.2 [1.23]q−1 Inverse section 1.2 [1.26]p, q Inner product section 1.2 [1.10]μ× η Cross product section 1.2 [1.9]μ⊥η Orthogonality section 1.2

x Quaternion Fourier Transforms for Signal and Image Processing

q p Bicomplex product section 1.4 [1.65]{1, i, j,k} Canonical basis in H section 1.1q Norm section 1.2 [1.14]

|q| Modulus section 1.2 [1.16]q̃ Unit quaternion |q̃| = 1 section 1.2 [1.30]f ∗ g Convolution section 3 [3.1]f g Correlation section 3 [3.1]F {.} Fourier transform section 3 [3.1]Pη [.] Reflection operator section 2.1 [2.1]Rq [.] Rotation operator section 2.1 [2.3]Sα,β,μ [.] Shear operator section 2.1 [2.9]Dμ,α [.] Dilation operator section 2.1 [2.10]L1(G;K) Space of absolutely integrable K-valued

functions taking arguments in G section 3.1L2(G;K) Space of square integrable K-valued

functions taking arguments in G section 4.3sgn(.) Signum function section 4.3 [4.31]p.v.(.) Principal value of an integral section 4.3 [4.32]

NOTE.– The complex root of −1 which is usually denoted i, or, in engineering textsj, is denoted throughout this book by a capital letter I , in order to avoid anyconfusion with the first of the three quaternion roots of −1, all three of which aredenoted throughout in bold font like this: i, j, k.

Preface

This book aims to present the state of the art, together with the most recentresearch results in the use of quaternion Fourier transforms (QFTs) for the processingof color images and complex-valued signals. It is based on the work of the authors inthis area since the 1990s and presents the mathematical concepts, computationalissues and some applications to signals and images. The book, together with theMATLAB® toolbox developed by the authors, [SAN 13b] allows the readers tomake use of the presented concepts and experiment with them in practice through theexamples provided.

Todd A. ELLMinneapolis, MN, USA

Nicolas le BIHANMelbourne, Australia

Stephen J. SANGWINEColchester, UK

April 2014

Introduction

This book covers a topic that combines two branches of mathematical theory toprovide practical tools for the analysis and processing of signals (or images) withthree- or four-dimensional samples (or pixels). The two branches of mathematics arenot recent developments, but their combination has occurred only within the last25–30 years, and mostly since just before the millennium.

I.1. Fourier analysis

Fourier analysis was, in 1822, with Joseph Fourier’s development of techniques,the first to analyze mathematical functions into sinusoidal components. In signal andimage processing, Fourier’s ideas underpin the two fundamental representations of asignal: one in the time (or image) domain where the signal (or image) is representedby samples (or pixels) with amplitudes and the other in the frequency domain wherethe signal (or image) is represented by sinusoidal frequency components, each withan amplitude and a phase. Mathematically, these concepts are not limited to time andfrequency: one can use Fourier analysis on a function of any variable, resulting in arepresentation in terms of sinusoidal functions of that variable. However, this book isconcerned with signal and image processing, and we will therefore use the terms timeand frequency rather than more general concepts. It should be understood throughoutthat when we talk of images, the concept of time is replaced by the two spatialcoordinates that define pixel position within an image.

Today, Fourier analysis is classically taught to mathematicians, scientists andengineers in several related ways, each applicable to a specific subset ofmathematical functions or signals:

1) Fourier series analysis [SNE 61] in which continuous periodic functions oftime, with infinite duration, are represented as sums of cosine and sine functions,each with infinite duration;