# Image Enhancement â€“ Frequency Domain Filter

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```							      Digital Image Processing

Image Enhancement – Frequency Domain
Filter

School of Electronics & Information Engineering
Soochow University
Digital Image Processing

Image Enhancement - 3
Outline

Frequency vs. spatial domain
Different approaches

2
Digital Image Processing

1-D Discrete Fourier Transform

f(x), x=0,1,…,M-1 . discrete function
F(u), u=0,1,…,M-1. DFT of f(x)
Forward discrete Fourier transform:
M 1                  j 2
u
1

x
F (u )                    f ( x )e              M
M         x 0

Inverse transform:
M 1                 j 2
u

 F (u )e
x
f ( x)                                M

u 0
3
Digital Image Processing

2-D DFT

2-D: x-axis then y-axis
N 1 M 1                  j 2 (
u
x
v
1
  f ( x, y)e
y)
F (u, v)                                                  M    M
MN         y 0 x 0

M 1 N 1                   j 2 (
u
x
v

  F (u , v)e
y)
f ( x, y )                                         M    M

u 0 v 0

4
Digital Image Processing

Complex Quantities to Real Quantities

Useful representation

magnitude
F (u, v)  [ R (u, v)  I (u, v)]
2             2         1/ 2

I (u , v)                   phase
 (u , v)  tan [   1
]
R (u , v)
Power spectrum

P(u, v)  F (u, v)  R 2 (u, v)  I 2 (u, v)
2

5
Digital Image Processing

DFT: example

log(F)

7
Digital Image Processing

Fourier Transform
FT – example 1

8
Digital Image Processing

Fourier Transform
FT – example 2

9
Digital Image Processing

Fourier Transform
FT – example 2

10
Digital Image Processing

Properties in the frequency domain

Fourier transform works globally
No direct relationship between a specific components in an image and
frequencies

Frequency content

Rate of change of gray levels in an image

11
Digital Image Processing
+45,-45 degree

artifacts

12
Digital Image Processing

Image Enhancement - 3
Frequency vs. spatial domain

Enhancement in frequency domain in principle is
straightforward. However, it makes more sense to filter
in the spatial domain using small filter masks.
The relationship between frequency and spatial domain
is the convolution theorem
Select H(u,v) so that the desired image g(x,y) exhibits
some highlighted features of f(x,y)
g ( x, y)  f ( x, y) * h( x, y)  G(u, v)  F (u, v) H (u, v)
g ( x, y)  F 1F (u, v) H (u, v)

13
Digital Image Processing

Image Enhancement - 3
Frequency vs. spatial domain

14
Digital Image Processing

Image Enhancement - 3
Basic steps in the frequency domain

15
Digital Image Processing

Image Enhancement - 3
Different approaches

Lowpass filters
Ideal Lowpass filters
Butterworth Lowpass filters
Gaussian Lowpass filters

Highpass filters
Homomorphic filters

16
Digital Image Processing

Image Enhancement - 3
Lowpass filters

1, if D(u, v)  D0
Ideal filters :H (u, v)  
0, if D(u, v)  D0

D(u,v) : distance from point (u,v) to the original
D0 : cutoff frequency
Ideal filter is nonphysical
Special case : notch filter
Power ratio of enhanced and original image

                
  100 P(u, v) / PT 
u v             
17
Digital Image Processing

Image Enhancement - 3
Lowpass filters – notch filter

H ( M / 2, N / 2)  0

18
Digital Image Processing

Image Enhancement - 3
Lowpass filters – ideal lowpass filter

19
Digital Image Processing

Image Enhancement - 3
Lowpass filters – ideal lowpass filter

20
Digital Image Processing

Image Enhancement - 3
Lowpass filters – ideal lowpass filter

21
Digital Image Processing

Image Enhancement - 3
Lowpass filters – ideal lowpass filter
The blurring and ringing phenomenon

22
Digital Image Processing

Image Enhancement - 3
Lowpass filters - Butterworth

Butterworth lowpass filters
1
H (u , v) 
1  D (u , v) / D0 
2n

23
Digital Image Processing

Image Enhancement - 3
Lowpass filters – Butterworth examples

24
Digital Image Processing

Image Enhancement - 3
Lowpass filters – Butterworth examples

25
Digital Image Processing

Image Enhancement - 3
Lowpass filters - Gaussian

Guassian lowpass filters
 D2 (u ,v ) / 2 2
H (u, v)  e

26
Digital Image Processing

Image Enhancement - 3
Lowpass filters - Gaussian

27
Digital Image Processing

Image Enhancement - 3
Highpass filters

H hp (u , v)  1  H lp (u , v)

0, if D(u, v)  D0
Ideal filters : H (u, v)  
1, if D(u, v)  D0
1
Butterworth highpass : H (u , v) 
1  D0 / D (u , v)
2n

 D2 (u ,v ) / 2 D0
2
Gaussian lowpass : H (u, v)  1  e

28
Digital Image Processing

Image Enhancement - 3
Highpass filters (cont’)

29
Digital Image Processing

Spatial-domain HPF

ideal           Butterworth          Gaussian

negative           30
Digital Image Processing
original
Ideal high-pass filters

0        if D(u, v)  D0
H (u, v)  
1        if D(u, v)  D0
ringing
D0=15                  D0=30                  D0=80

31
Digital Image Processing

Butterworth high-pass filters

1
H (u , v) 
1  [ D0 / D(u , v)] 2 n

n=2,   D0=15                  D0=30                  D0=80

32
Digital Image Processing

Gaussian high-pass filters

 D2 (u ,v ) / 2 D0
2
H (u, v)  1  e

D0=15                  D0=30                   D0=80

33
Digital Image Processing

Laplacian frequency-domain filters

Spatial-domain Laplacian (2nd derivative)
2 f 2 f
2 f  2  2
x   y
Fourier transform
  n f ( x) 
               ( ju) n F (u )
   x n 
  2 f ( x, y )  2 f ( x, y ) 
                                 ( ju ) 2 F (u , v)  ( jv) 2 F (u , v)
     x 2              y 2     
 (u 2  v 2 ) F (u , v)
34
Digital Image Processing

Laplacian frequency-domain filters

Input
F(u,v)
f(x,y)               F
Laplacian                                         -(u2+v2)

2 f 2 f
2 f  2  2
x   y                            -(u2+v2)F(u,v)
F

The Laplacian filter in the frequency domain is
H(u,v) = -(u2+v2)
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0             Digital Image Processing

H(u,v) = -(u2+v2) frequency

36
spatial
Digital Image Processing

original                              Laplacian

Scaled                                 original+
Laplacian                              Laplacian

37
Digital Image Processing

Image Enhancement - 3
Homomorphic filters

A simple image model: illumination–reflection model
f(x,y) : the intensity is called gray level for monochrome image
f(x,y)=i(x,y)*r(x,y)
0<i(x,y)<inf, the illumination
0<r(x,y)<1, the reflection

38
Digital Image Processing

Image Enhancement - 3
Homomorphic filters (cont’)

The illumination component
Slow spatial variations
Low frequency

The reflectance component
Vary abruptly, particularly at the junctions of dissimilar objects
High frequency

Homomorphic filters
Effect low and high frequency differently
Compress the low frequency dynamic range
Enhance the contrast in high frequency
39
Digital Image Processing

Image Enhancement - 3
Homomorphic filters (cont’)

40
Digital Image Processing

Image Enhancement - 3
Homomorphic filters (cont’)

f(x,y)=i(x,y)*r(x,y)
z(x,y)=ln f(x,y) = ln i(x,y) + ln r(x,y)
F{z(x,y)} = F{ln i(x,y)} + F{ln r(x,y)}
S(u,v) = H(u,v) I(u,v) + H(u,v) R(u,v)
s(x,y) = i’(x,y) + r’(x,y)
g(x,y) = exp[s(x,y)] = exp[i’(x,y)]exp[r’(x,y)]

41
Digital Image Processing

Image Enhancement - 3
Homomorphic filters (cont’)

42
Digital Image Processing

Image Enhancement - 3
Homomorphic filters - example

H (u, v)  ( H   L )[1  e    c ( D2 ( u ,v ) / D0
)]   L
2

43

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