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Chapter 4:

Frequency Domain Processing

Image Transformations

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Introduction

Although we discuss other transforms in some

detail in this chapter, we emphasize the Fourier

transform because of its wide range of

applications in image processing problems.

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Fourier Transform (1-D)

uXuXsin

AXuF

euXsinu

Adxux2jexpxfuF uXj

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Fourier Transform (2-D)

vYvYsin

uX

uXsinAXYv,uFdydxvyux2jexpy,xf)v,u(F

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Discrete Fourier Transform

In the two-variable case the discrete Fourier transform pair is

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Discrete Fourier Transform

When images are sampled in a squared array, i.e. M=N,

we can write

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Discrete

Fourier

Transform

Examples

At all of theseexamples, the Fourier

spectrum is shifted

from top left corner to

the center of the

frequency square.

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Discrete

Fourier

Transform

Examples

At all of theseexamples, the Fourier

spectrum is shifted

from top left corner to

the center of the

frequency square.

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Discrete

Fourier

Transform

Display

At all of theseexamples, the Fourier

spectrum is shifted

from top left corner to

the center of the

frequency square.

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Discrete Fourier Transform

Example

Main Image (Gray Level) DFT of Main image

(Fourier spectrum)

Logarithmic scaled

of the Fourier spectrum

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Discrete Fourier Transform

(Properties)

Separability

1N

0v

1N

0u

1N

0y

1N

0x

N/vy2jexpv,uFN/ux2jexpN

1y,xf

N/vy2jexpy,xfN/ux2jexpN

1v,uF

The discrete Fourier transform pair can be expressed in the seperable forms:

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Discrete Fourier Transform

(Properties)

Translation

N/vyux2jexpv,uFyy,xxfand

vv,uuFN/yvxu2jexpy,xf

0000

0000

The translation properties of the

Fourier transform pair are :

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Discrete Fourier Transform

(Properties)

Periodicity

The discrete Fourier transform and its

inverse areperiodic with period N; that is,

F(u,v)=F(u+N,v)=F(u,v+N)=F(u+N,v+N)

If f(x,y) is real, the Fourier transform also

exhibits conjugate symmetry:

F(u,v)=F*(-u,-v)

Or, more interestingly:

|F(u,v)|=|F(-u,-v)|

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Discrete Fourier Transform

(Properties)

Rotation

sinvcosu

sinrycosrx

If we introduce the polar coordinates

Then we can write:

00 ,F,rf

In other words, rotating F(u,v)

rotates f(x,y) by the same angle.

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Discrete Fourier Transform

(Properties)

Convolution

v,uG*v,uFy,xgy,xfand

v,uGv,uF)y,x(g*y,xf

The convolution theorem in

two dimensions is expressed

by the relations :

Note :

ddy,xg,fy,xg*y,xf

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Discrete Fourier Transform

(Properties)

Correlation

dxgfxgxf

*

The correlation of two continuous

functions f(x) and g(x) is defined

by the relation

So we can write:

v,uGv,uFy,xgy,xf

and

v,uGv,uFy,xgy,xf

*

*

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Discrete

Fourier

Transform

Sampling

(Properties)

1-D

f(x) : a given function

F(u): Fourier Transform of f(x)

which is band-limited

s(x) : sampling function

S(u): Fourier Transform of s(x)

G(u): window for recovery of the

main function F(u) and f(x).

Recovered f(x) from sampled data

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Discrete

Fourier

Transform

Sampling

0000 y,xfdydxyy,xxy,xf

(Properties)

2-D

The sampling process for 2-Dfunctions can be formulated

mathematically by making use

of the 2-D impulse function

(x,y), which is defined as

A 2-D sampling function is

consisted of a train of impulses

separated x units in the x

direction and y units in the y

direction as shown in the figure.

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Discrete

Fourier

Transform

Sampling

0

1

v,uG

(Properties)

2-D

If f(x,y) is band limited (that is, its

Fourier transform vanishes outsidesome finite region R) the result of

covolving S(u,v) and F(u,v) might

look like the case in the case shown

in the figure. The function shown is

periodic in two dimensions.

(u,v) inside one of the rectangles

enclosing R

elsewhere

The inverse Fourier transform of

G(u,v)[S(u,v)*F(u,v)] yields f(x,y).

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The Fast Fourier Transform (FFT) Algorithm21

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Other Seperable Image Transforms

AFAT

BTBF

AB

BAFABBTB

1

1N

0x

1N

0y

v,u,y,xgy,xfv,uT

Where F is the NN image matrix,

A is an NN symmetric transformation matrix

T is the resulting N

N transform.

If the kernel g(x,y,u,v) is seperable and symmetric,

also may be expressed in matrix form:

And for inverse transform we have:

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Discrete Cosine Transform23

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Discrete Cosine Transform (DCT)

Each block consists

of 44 elements,

corresponding to x

and y varying from 0to 3. The highest

value is shown in

white. Other values

are shown in grays,

with darker meaning

smaller.

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Discrete Cosine Transform

Example

Main Image (Gray Level) DCT of Main image

(Cosine spectrum)

Logarithmic scaled

of the Cosine spectrum

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DCT and Fourier Transform26

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DCT Example27

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Blockwise DCT Example28

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Comparison Of Various Transforms29

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Comparison Of Various Transforms30

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The DFT and Image Processing

To filter an image in the frequency domain:1. Compute F(u,v) the DFT of the image

2. Multiply F(u,v) by a filter functionH(u,v)

3. Compute the inverse DFT of the result

ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002)

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Some Basic Frequency Domain Filters

ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002) Low Pass Filter

High Pass Filter

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Smoothing Frequency Domain Filters

Smoothing is achieved in the frequency domain by

dropping out the high frequency components

The basic model for filtering is:

G(u,v) = H(u,v)F(u,v)where F(u,v) is the Fourier transform of the imagebeing filtered andH(u,v) is the filter transformfunction

Low pass filters only pass the low frequencies,drop the high ones

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Ideal Low Pass Filter

Simply cut off all high frequency components thatare a specified distance D0 from the origin of the

transform

changing the distance changes the behaviour of

the filter

ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002)

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Ideal Low Pass Filter (cont)

The transfer function for the ideal low pass filter

can be given as:

whereD(u,v) is given as:

0

0

),(if0

),(if1

),( DvuD

DvuD

vuH

2/122 ])2/()2/[(),( NvMuvuD

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Ideal Low Pass Filter (cont)

Above we show an image, its Fourier spectrum

and a series of ideal low pass filters of radius 5,

15, 30, 80 and 230 superimposed on top of it

ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002)

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Ideal Low Pass Filter (cont)

ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002)

Original