Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Goal of Restoration : Improve an image in some predefined
sense. i.e.
ˆ
f (x, y)  f (x, y)
g(x,y)=h(x,y)*f(x,y)+η(x,y)
G(u,v)=H(u,v)F(u,v)+N(u,v)
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
a ) p( z ) 
1
2
2
e  ( z  ) / 2 
GaussianNoise
2
2
2
 (z  a )e ( z a ) / b for z  a
b) p( z )   b
RayleighNoise
 0
for z  a
 a b z b 1  z

for z  0
c) p(z)   (b  1)! e
0
for z  0

ae az
d ) p( z )  
 0
 1

e) p ( z )   b  a
 0
 Pa

f ) p(z)   Pb
0

© 2002 R. C. Gonzalez & R. E. Woods
for z  0
for z  0
Gamma Noise
Exponential Noise
for a  z  b
Uniform Noise
otherwise
for z  a
for z  b
otherwise
Salt  P epper Noise
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
This test pattern is well-suited for illustrating the noise models,
because it is composed of simple, constant areas that span the
grey scale from black to white in only three increments. This
facilitates visual analysis of the characteristics of the various noise
components added to the image.
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Periodic Noise Reduction by Frequency Domain Filtering
LPF & HPF  Image Enhancement
1) Bandreject Filters
1-1) Ideal :

 1

Hu, v    0

 1

if Du, v   D 0 
if D 0 
W
2
W
W
 Du, v   D 0 
2
2
W
if Du, v   D 0 
2
D(u,v): distance from the origin of the centered frequency rectangle
W:
bandwidth
D0:
radial center
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Periodic Noise Reduction by Frequency Domain Filtering
1-2) Butterworth of order n :
1-3) Gaussian :
© 2002 R. C. Gonzalez & R. E. Woods
Hu, v   1  e
Hu, v  
1
 Du, v  W 
1 

2
2
 D u, v   D 0 
1  D 2 u , v  D0 2 
 

2  Du , v  W 
2
2n
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Periodic Noise Reduction by Frequency Domain Filtering
2) Bandreject Filters
Hbp(u,v)=1-Hbr(u,v)
It helps isolate the noise pattern.
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Periodic Noise Reduction by Frequency Domain Filtering
3) Notch Filters
3-1) Ideal :
0 if D1 u, v   D 0 or D 2 u, v   D 0
Hu, v   
otherwise
1
1
22
2

M
N
 

D1 u, v    u   u 0    v   v 0  
2
2

 
 
1
22
2

M
N
 

D1 u, v    u   u 0    v   v 0  
2
2

 
 
The center of the frequency rectangle has been shifted to the point (M/2,N/2)
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Periodic Noise Reduction by Frequency Domain Filtering
Notch Filters
3-2) Butterworth of order n :
3-3) Gaussian :
© 2002 R. C. Gonzalez & R. E. Woods
Hu, v   1  e
Hu, v  
1


D0 2
1 

 D1 u, v  D 2 u, v  
1  D1 u , v  D2 u , v  
 

2 
D0 2

n
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Periodic Noise Reduction by Frequency Domain Filtering
4) Optimum Notch Filters
f x, y   gx, y   w x , y  x, y 
x, y   F 1Hu, v  G u, v 
w x , y  
gx , y x, y   g x, y x, y 
2 x , y    2 x, y 
To obtain w(x,y) the goal is to minimize the variance in the neighborhood of
x,y in the image.
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Linear, Position-Invariant Degradations
gx, y  hx, y f x, y  x, y
Gu, v  Hu, vFu, v  Nu, v
Estimation the Degradation Function
1) Estimation by Image Observation
H s u, v  
G s u, v 
Fˆ u, v 
s
Assuming that the effect of noise is negligible.
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Linear, Position-Invariant Degradations
gx, y  hx, y f x, y  x, y
Gu, v  Hu, vFu, v  Nu, v
Estimation the Degradation Function
2) Estimation by Experimentation
Hu, v 
Gu, v
A
A : impulse Fourier transform which is constant.
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Linear, Position-Invariant Degradations
Estimation the Degradation Function
3) Estimation by modeling
Turbulence model :
Hu, v   e
Mathematical model :
© 2002 R. C. Gonzalez & R. E. Woods

K u  v
2
2

5
6
T
Hu, v 
Sinua  vb e  jua vb
ua  vb
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
How to get F^(u,v) from degraded image G(u,v) :
1) Inverse Filtering
ˆFu, v  Gu, v  Fu, v  Nu, v
Hu, v
Hu, v
We should know N(u,v) to use this method.
We should use this method near origin because H(u,v) is
near zero in other areas.
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
How to get F^(u,v) from degraded image G(u,v) :
2) Wiener (Minimum mean square Error) Filtering


e 2  E  f  fˆ

2   0


2


Hu , v 
1
ˆ
Fu , v   

 G u , v 


S
u
,
v
 Hu , v  Hu , v  2  

Sf u , v  



H(u,v) : degradation function
Sn(u,v)=|N(u,v)|2
power spectrum of the noise
Sf(u,v)=|F(u,v)|2
power spectrum of the undergraded image
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
If Sf(u,v) is not known :


e 2  E  f  fˆ

2   0
2




H
u
,
v
1
ˆ u , v   
 G u , v 
F

 Hu , v  Hu , v  2  K 


© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
How to get F^(u,v) from degraded image G(u,v) :
3) Constrained Least Squares Filtering

*

H
u, v 
ˆ

Fu , v  
 H u , v  2   Pu , v  2

1 0 
0
px , y    1 4
 1
 0
 1 0 
© 2002 R. C. Gonzalez & R. E. Woods

 G u , v 


www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
www.imageprocessingbook.com
Chapter 5
Image Restoration
Geometric Transformations : (rubber-sheet transformations)
1) Spatial Transformations
2) Gray-level Transformations
© 2002 R. C. Gonzalez & R. E. Woods
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com
Digital Image Processing, 2nd ed.
Chapter 5
Image Restoration
© 2002 R. C. Gonzalez & R. E. Woods
www.imageprocessingbook.com