In the name of AllahIn the name of Allah

the compassionate, the merciful

Digital Image Processing

S. Kasaei

Room CE307, SUT

E-Mail: skasaei@sharif eduE Mail: skasaei@sharif.eduHome Page: http://ce.sharif.edu

http://ipl.ce.sharif.eduhttp://sharif.edu/~skasaei

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ttp //s a edu/ s asae

Chapter 5

Image Transforms

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Image Transform

Motivation: Represents a block of image pixels as the superposition of– Represents a block of image pixels as the superposition of some typical basic patterns (transform basic functions).

General process:Forward transformation– Forward transformation

– Process on transform coefficients– Inverse transformation

+t1 t t t

Transform Basis

Transform Coefficients

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+t1 t 2 t 3 t 4

Basis Vectors & Images

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Introduction

The term image transforms usually refers to a class of unitary matrices used fora class of unitary matrices used for representing images.

Just as a 1-D signal can be represented by an orthogonal series of basis functions, an image can also be expanded in terms of a discrete set of basis arrays called basis images

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images.

Introduction

These basis images can be generated by it t iunitary matrices.

An image transform provides a set of coordinates of basis vectors to form the vector spacevector space.

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Introduction

Series coeff.

Series rep.

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1-D Basis Vectors

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2-D Unitary Transforms

Transform coeff.

Series rep.

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2-D Unitary Transforms

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2-D Unitary Transforms

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2-D Unitary Transforms

T

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2-D Unitary Transforms

Basis image

Projection of U on A

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Projection of U on A

2-D Unitary Transforms

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2-D Unitary Transforms

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2-D Unitary Transforms

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2-D Unitary Transforms

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2-D Unitary Transforms

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2-D Unitary Transforms

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2-D Unitary Transforms

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Properties of Unitary Transforms

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Properties of Unitary Transforms

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Properties of Unitary Transforms

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Properties of Unitary Transforms

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Properties of Unitary Transforms

About (5.33):}{}{ BATrABTr =

IAA T =∗

⇒ }{}{}{ uuTT

u RTrARATrAARTr == ∗∗

About (5.34):

]|)()([|)( 22 nnvEn µδ =

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]|)()([|)( nnvEn vv µδ −=

Properties of Unitary Transforms

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Properties of Unitary Transforms

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Properties of Unitary Transforms

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Properties of Unitary Transforms

A

A+B=2A=B

B

A’+B’=2A’>B’A >B

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Properties of Unitary Transforms

Cov[v(0),v(1)]Cov[v(0),v(1)]

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Discrete Fourier Transform (DFT)

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DFT (cntd)

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DFT (cntd)

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DFT (cntd)

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DFT (cntd)

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DFT (cntd)

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DFT (cntd)

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DFT (cntd)

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DFT (cntd)

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DFT (cntd)

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DFT (cntd)

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DFT (cntd)

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2-D DFT

Kasaei45A 2-D image and its Fourier spectrum.

2-D DFT

Kasaei46An image, its phase only image, and its contrast enhanced image.

2-D DFT

Kasaei47An image and its Fourier spectrum’s magnitude and phase.

2-D DFT

Kasaei48Shifted image and its Fourier spectrum’s magnitude and phase.

2-D DFT

Kasaei49Rotated image and its Fourier spectrum’s magnitude.

2-D DFT

Kasaei50 An image and its Fourier spectrum’s magnitude.

2-D DFT

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2-D DFT (cntd)

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2-D DFT (cntd)

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2-D DFT (cntd)

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2-D DFT (cntd)

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Discrete Cosine TransformDiscrete Cosine Transform(DCT)

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DCT (cntd)

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DCT (cntd)

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DCT (cntd)

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DCT (cntd)

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DCT (cntd)

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DCT (cntd)

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DCT (cntd)

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Discrete Sine TransformDiscrete Sine Transform (DST)

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DST (cntd)

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Hadamard TransformHadamard Transform(HT)

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HT (cntd)

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HT (cntd)

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HT (cntd)

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HT (cntd)

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HT (cntd)

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HT (cntd)

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HT (cntd)

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HT (cntd)

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HT (cntd)

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HT (cntd)

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HT (cntd)

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Harr TransformHarr Transform (HrT)

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HrT (cntd)

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HrT (cntd)

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HrT (cntd)

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HrT (cntd)

Haar transform takes differences of the l diff f l l fsamples or differences of local average of

the samples of the input vector.

2-D Haar transform coefficients (except for k=l=0) are the differences along rows &k=l=0), are the differences along rows & columns of the local averages of the pixels in the image

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the image.

HrT (cntd)

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Slant TransformSlant Transform (ST)

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ST (cntd)

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ST (cntd)

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ST (cntd)

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Karhunen-Loeve Transform (KLT)

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R G B

KLT (cntd)

Kasaei89PCA1 PCA2 PCA3

KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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KLT (cntd)

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The End

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Random Signals

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Random Signals

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