CoE4TN3Image Processing

Chapter 4Image Enhancement in

Frequency Domain

2

Fourier Transform • Fourier Transform (FT) of a 2-D signal:

1 1

0 0

1 1

0 0

1

2 2 1/ 2

1( , ) ( , ) exp( 2 ( / / )

( , ) ( , ) exp( 2 ( / / ))

( , ) ( , ) ( , )( , )( , ) tan [ ]( , )

( , ) [ ( , ) ( , )]( , ) ( , )

M N

x y

M N

u v

F u v f x y j xu M yv NMN

f x y F u v j xu M yv N

F u v R u v jI u vI u vu vR u v

F u v R u v I u vF u M v N F u v

π

π

φ

− −

= =

− −

= =

−

= − +

= +

= +

=

= +

+ + =

∑ ∑

∑ ∑

3

Fourier Transform • Properties of Fourier Transform (FT) of a 2-D real, signal:

1 1

0 0

*

*

1(0,0) ( , )

( , ) ( , )( , ) ( , )( , ) ( , )

M N

x yF f x y

MNF u v F u vF u v F M u N vF u v F u v

− −

= ==

= − −

= − −

= − −

∑ ∑

4

Fourier Transform • It is common to multiply input image by (-1)x+y prior to

computing the Fourier Transform.

• This shift the center of the FT to (M/2,N/2)

5

Fourier Transform

6

Frequency domain filtering 1. Multiply the input image by (-1)x+y to center the transform

Compute the DFT of the image from step 12. Multiply the result by the transfer function of the filter

(centered)3. Take the inverse transform.4. Multiply the result by (-1)x+y

7

Frequency domain filtering • G(u,v)=F(u,v)H(u,v)• Based on convolution theorem: g(x,y)=h(x,y)*f(x,y)

– f(x,y) is the input image– g(x,y) is the processed image– h(x,y): impulse response or point spread function

• Based on the form of H(u,v), the output image exhibits some features of f(x,y)

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Frequency domain filtering

f(x,y)h(x,y)

Convolution

1) Flip 2) Shift, multiply, add

f(x,y)w(x,y)

Implementing a mask

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Frequency domain filtering• Convolution with a filter and implementing a mask are very

similar• The only difference is the flipping operation• If the impulse response of the filter is symmetric about the

origin the two operations are the same.• Instead of filtering in the frequency domain, we can

approximate the impulse response of the filter by a mask, and use the mask in the spatial domain.

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11

Frequency domain filtering

• Edges and sharp transitions (e.g., noise) in the gray levels of an image contribute significantly to high-frequency content of FT.

• Low frequency in the Fourier transform of an image are responsible for the general gray-level appearance of the image over smooth areas.

• Blurring (smoothing) is achieved by attenuating range of high frequency components of FT.

• We consider 3 types of lowpass filters: ideal, Butterworth and Gaussian

12

Ideal low-pass filter• Ideal: all the frequencies inside a circle of radius D0 are

passed and all the frequencies outside this circle are completely removed.

=01

),( vuH0),( DvuD ≤

0),( DvuD >

uv

2 2 1/ 2( , ) [( / 2) ( / 2) ]D u v u M v N= − + − |H||H|

D(u,v)M/2+D0M/2-D0

13

14

15

Ideal low-pass filter• Effects of ideal low-pass filtering: blurring and ringing

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Ideal low-pass filter

• The radii of the concentric rings in h(x,y) are proportional to 1/D0

D0 1/D0 Less blurring, Less ringing (amplitude of rings drop)

D0 1/D0 More blurring, More ringing (amplitude of rings increase)

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Butterworth Lowpass Filters• Smooth transfer function, no sharp discontinuity, no clear

cutoff frequency.

n

DvuD

vuH 2

0

),(1

1),(

+

=

21

18

Butterworth Lowpass Filters

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Butterworth Lowpass Filters

20

• Smooth transfer function, smooth impulse response, no ringing

02

2

2),(

),( DvuD

evuH−

=

Gaussian Lowpass Filters

21

Gaussian Lowpass Filters

22

Applications of Lowpass Filters

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Applications of Lowpass Filters

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High-pass filtering• Image sharpening can be achieved by a high-pass filtering

process.• Hhp(u,v)=1-Hlp(u,v)

• Ideal:

• Butterworth:

• Gaussian:

n

vuDD

vuH 20

2

),(1

1|),(|

+

=

=01

),( vuH0),( DvuD >

0),( DvuD ≤

20

2 2/),(1),( DvuDevuH −−=

25

High-pass filtering

26

High-pass filtering

27

High-pass filtering

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High-pass filtering

29

High-pass filtering

30

Laplacian in Frequency Domain

)(1),(),()(),(),(

),(),(),(

)(),(

),()(]),(),([

222

22

2

221

222

2

2

2

vuvuHvuFvuvuFvuG

yxfyxfyxg

vuvuH

vuFvuyyxf

xyxf

++=

++=

∇−=

+−=

+−=∂

∂+

∂∂

ℑ

31

Laplacian in Frequency Domain

32

33

Unsharp Masking, High-boost Filtering• Unsharp masking: fhp(x,y)=f(x,y)-flp(x,y)• Hhp(u,v)=1-Hlp(u,v)

• High-boost filtering: fhb(x,y)=Af(x,y)-flp(x,y)• fhb(x,y)=(A-1)f(x,y)+fhp(x,y)• Hhb(u,v)=(A-1)+Hhp(u,v)

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Unsharp Masking, High-boost Filtering

35

Homomorphic filtering• In some images, the quality of the image has reduced because

of non-uniform illumination• Homomorphic filtering can be used to perform illumination

correction• We can view an image f(x,y) as a product of two components:

• This equation cannot be used directly in order to operate separately on the frequency components of illumination and reflectance

( ) ( ) ( )yxryxiyxf ,,, ⋅=

36

( ) ( ) ( )( )( ) ( )vuFvuHvuFvuHvuSvuZvuHvuSvuFvuFvuZ

ri

ri

,),(,),(),(,),(),(

,,,

+==

+=

( ) ( ) ( ) ( )yxryxiyxfyxz ,ln,ln,ln, +==

),(),(),(

),(),(),(

00),(),(),(

''

''

yxryxieeeyxg

yxryxiyxsyxryxiyxs ===

+=

Homomorphic filtering

37

• The key to the approach is that separation of the illumination and reflectance components is achieved. The homomorphicfilter can then operate on these components separately

• Illumination component of an image generally has slow variations, while the reflectance component vary abruptly

• By removing the low frequencies (highpass filtering) the effects of illumination can be removed

Homomorphic filtering

38

Homomorphic filtering

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41

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