Chap4 Frequency Domain

7/30/2019 Chap4 Frequency Domain

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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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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


spectrum is shifted


the center of the

frequency square.

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Discrete

Fourier

Transform

Display


spectrum is shifted


the center of the

frequency square.

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Example

Main Image (Gray Level) DFT of Main image

(Fourier spectrum)

Logarithmic scaled

of the Fourier spectrum

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(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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(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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(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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(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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(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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(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



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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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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



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Original

image

Result of filtering

with ideal low pass

filter of radius 5

Result of filtering

with ideal low pass

filter of radius 30

Result of filtering

with ideal low pass

filter of radius 230

Result of filtering

with ideal low pass

filter of radius 80

Result of filtering

with ideal low pass

filter of radius 15


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Gaussian Lowpass Filters

The transfer function of a Gaussian lowpass

filter is defined as:


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2 2/),(

),(

DvuDevuH


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Gaussian Lowpass Filters (cont)


Original

image

Result of filtering

with Gaussian

filter with cutoff

radius 5

Result of filtering

with Gaussian filter

with cutoff radius 30

Result of filtering

with Gaussian

filter with cutoff

radius 230

Result of

filtering with

Gaussian filter

with cutoff

radius 85

Result of filtering

with Gaussian

filter with cutoff

radius 15


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Lowpass Filtering Examples

A low pass Gaussian filter is used to connect

broken text

ImagestakenfromGonzalez&Woods,Digital

ImageProcessing(2002)


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Sharpening in the Frequency Domain

Edges and fine detail in images are associated

with high frequency components

High pass filters only pass the high

frequencies, drop the low ones

High pass frequencies are precisely the reverse

of low pass filters, so:

Hhp(u, v) = 1Hlp(u, v)


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Ideal High Pass Filters

The ideal high pass filter is given as:

where D0 is the cut off distance as before

0

0

),(if1

),(if0),(

DvuD

DvuDvuH




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Ideal High Pass Filters (cont)

Results of ideal

high pass filtering

withD0 = 15

Results of ideal

high pass filtering

withD0 = 30

Results of ideal

high pass filtering

withD0 = 80




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Gaussian High Pass Filters

The Gaussian high pass filter is given as:

whereD0 is the cut off distance as before

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2 2/),(1),(

DvuDevuH




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Fast Fourier Transform

The reason that Fourier based techniques have

become so popular is the development of the

Fast Fourier Transform (FFT) algorithm

Allows the Fourier transform to be carried out in

a reasonable amount of time

Reduces the amount of time required to

perform a Fourier transform by a factor of 100

600 times!


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Frequency Domain Filtering & Spatial Domain

Filtering

Similar jobs can be done in the spatial and

frequency domains

Filtering in the spatial domain can be easier to

understand

Filtering in the frequency domain can be much

faster especially for large images


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References

Digital image processing by Rafael C.

Gonzalez and Richard E. Woods

Digital image processing by Jain

Image communication I by Bernd Girod

Lecture 3, DCS339/AMCM053 by Pengwei

Hao, University of London

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Chap4 Frequency Domain

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