EE4328, Section 005
Introduction to Digital Image Processing
Linear Image Restoration
Zhou Wang
Dept. of Electrical Engineering
The Univ. of Texas at Arlington
Fall 2006
Blur
out-of-focus blur
motion blur
Question 1: How do you know they are blurred?
I’ve not shown you the originals!
Question 2: How do I deblur an image?
From Prof.
Xin Li
Linear Blur Model
• Spatial domain
x(m,n)
h(m,n)
y(m,n)
blurring filter
From Prof.
Xin Li
Gaussian blur
motion blur
Linear Blur Model
• Frequency (2D-DFT) domain
X(u,v)
H(u,v)
Y(u,v)
blurring filter
From Prof.
Xin Li
Gaussian blur
motion blur
Blurring Effect
Gaussian blur
motion blur
From [Gonzalez & Woods]
Image Restoration: Deblurring/Deconvolution
x(m,n)
h(m,n)
blurring filter
y(m,n)
g(m,n)
^
x(m,n)
deblurring/
deconvolution
filter
• Non-blind deblurring/deconvolution
Given: observation y(m,n) and blurring function h(m,n)
x(m,n)
Design: g(m,n), such that the distortion between x(m,n) and ^
is minimized
• Blind deblurring/deconvolution
Given: observation y(m,n)
^
Design: g(m,n), such that the distortion between x(m,n) and x(m,n)
is minimized
Deblurring: Inverse Filtering
x(m,n)
h(m,n)
y(m,n)
blurring filter
X(u,v) H(u,v) = Y(u,v)
X(u,v) =
Y(u,v)
1
=
Y(u,v)
H(u,v)
H(u,v)
g(m,n)
^
x(m,n)
inverse filter
1
G(u,v) =
H(u,v)
Exact recovery!
Deblurring: Pseudo-Inverse Filtering
x(m,n)
h(m,n)
y(m,n)
blurring filter
1
Inverse filter: G(u,v) =
H(u,v)
g(m,n)
^
x(m,n)
deblur filter
What if at some (u,v), H(u,v)
is 0 (or very close to 0) ?
 1
| H (u, v) | 

Pseudo-inverse filter: G(u, v)   H (u, v)

| H (u, v) | 
 0
small
threshold
Inverse and Pseudo-Inverse Filtering
G (u, v) 
1
H (u, v)
blurred image
 1
| H (u, v) | 

G(u, v)   H (u, v)

| H (u, v) | 
 0
Adapted from Prof. Xin Li
 = 0.1
More Realistic Distortion Model
w(m,n)
x(m,n)
h(m,n)
blurring filter
+
additive white Gaussian noise
y(m,n)
g(m,n)
^
x(m,n)
deblur filter
Y(u,v) = X(u,v) H(u,v) + W(u,v)
• What happens when an inverse filter is applied?
X (u , v) H (u , v)  W (u , v)
ˆ
X (u , v)  Y (u , v)G (u , v) 
H (u , v)
W (u , v)
close to zero at high frequencies
 X (u , v) 
H (u , v)
Radially Limited Inverse Filtering

Radially limited

G
(
u
,
v
)

 H (u, v)
inverse filter:


1
u 2  v2  R
0
u 2  v2  R
• Motivation
R
– Energy of image signals is
concentrated at low frequencies
– Energy of noise uniformly is
distributed over all frequencies
– Inverse filtering of image signal
dominated regions only
Radially Limited Inverse Filtering
Image
size:
480x480
Original
Blurred
Inverse filtered
R = 40
R = 70
R = 85
Radially
limited
inverse
filtering:
From [Gonzalez & Woods]
Wiener (Least Square) Filtering
Wiener filter: G (u, v) 
H * (u, v)
| H (u, v) |2  K
 W2
K 2
X
noise power
signal power
• Optimal in the least MSE sense, i.e.
G(u, v) is the best possible linear filter that minimizes
2

ˆ
error energy  E  X (u, v)  X (u, v) 



• Have to estimate signal and noise power
Weiner Filtering
Blurred
image
Inverse
filtering
Radially limited
inverse filtering
R = 70
Weiner
filtering
From [Gonzalez & Woods]
Inverse vs. Weiner Filtering
distorted
inverse filtering Wiener filtering
motion blur
+
noise
less noise
less noise
From [Gonzalez & Woods]
Weiner Image Denoising
w(m,n)
x(m,n)
h(m,n)
+
y(m,n)
• What if no blur, but only noise, i.e. h(m,n) is an impulse
or H(u, v) = 1 ?
Wiener filter:
H * (u, v)
G (u, v) 
| H (u, v) |2  K
where
 W2
K 2
X
for H(u,v) = 1
Wiener
denoising filter:
1
1
 X2
G (u, v) 

 2
2
2
1  K 1   W /  X  X   W2
Typically applied locally in space
Weiner Image Denoising
adding noise
noise var = 400
local Wiener
denoising
Summary of Linear Image Restoration Filters
Inverse filter:
Pseudo-inverse
filter:
Radially limited
inverse filter:
Wiener filter:
Wiener
denoising filter:
1
G(u,v) =
H(u,v)
 1
| H (u, v) | 

G(u, v)   H (u, v)

| H (u, v) | 
 0
 1

G (u, v)   H (u, v)
 0

u 2  v2  R
u 2  v2  R
 W2
H * (u, v)
where K  2
G (u, v) 
2
X
| H (u, v) |  K
 X2
G (u, v)  2
 X   W2
Examples
• A blur filter h(m,n) has a 2D-DFT given by
1
 0.3  0.3 j

 0.3  0.3 j
0.1 j

H (u, v) 

0
0

0.1
 0.3  0.3 j
0  0.3  0.3 j 

0
0.1


0
0

0
 0.1 j 
• Find the deblur filter G(u,v) using
1) The inverse filtering approach
2) The pseudo-inverse filtering approach, with  = 0.05
3) The pseudo-inverse filtering approach, with  = 0.2
2

4) Wiener filtering approach, with   625 and W  125
2
X
Examples
1) Inverse filter
1
 1.67  1.67 j

 1.67  1.67 j
 10 j
1

G (u, v) 

Inf
Inf
H (u, v) 

10
 1.67  1.67 j
Inf
Inf
Inf
Inf
 1.67  1.67 j 

10


Inf

10 j

2) Pseudo-inverse filter, with  = 0.05
1
 1.67  1.67 j


 1
 10 j
| H (u, v) |   1.67  1.67 j

G (u, v)   H (u, v)


0
0
 0
| H (u, v) |  
10
 1.67  1.67 j
0  1.67  1.67 j 

0
10


0
0

0
10 j

Examples
3) Pseudo-inverse filter, with  = 0.2
1
 1.67  1.67 j


 1
0
| H (u, v) |   1.67  1.67 j

G (u, v)   H (u, v)


0
0
 0
| H (u, v) |  
0
 1.67  1.67 j
0  1.67  1.67 j 

0
0


0
0

0
0

4) Wiener filter, with  X2  625 and  W2  125
 W2 125
K 2 
 0.2
 X 625
0.83
 0.79  0.79 j

 0.79  0.79 j
 0.48 j
H * (u, v)

G (u, v) 

0
0
| H (u, v) |2  K 

0.48
 0.79  0.79 j
0  0.79  0.79 j 

0
0.48


0
0

0
0.48 j

Advanced Image Restoration
• Adaptive Processing
– Spatial adaptive
– Frequency adaptive
• Nonlinear Processing
– Thresholding, coring …
– Iterative restoration
• Advanced Transformation / Modeling
– Advanced image transforms, e.g., wavelet …
– Statistical image modeling
• Blind Deblurring / Deconvolution