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EE4328, Section 005 Introduction to Digital Image
Processing
Linear Image Restoration
Zhou Wang
Dept. of Electrical EngineeringThe Univ. of Texas at Arlington
Fall 2006
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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
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Linear Blur Model
h(m,n)
blurring filter
x(m,n) y(m,n)
Gaussian blur motion blur
•Spatial domain
From Prof. Xin
Li
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Linear Blur Model
H(u,v)
blurring filter
X(u,v) Y(u,v)
Gaussian blur motion blur
•Frequency (2D-DFT) domain
From Prof. Xin
Li
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Blurring Effect
Gaussian blur
motion blur
From [Gonzalez & Woods]
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Image Restoration: Deblurring/Deconvolution
h(m,n)
blurring filter
g(m,n)x(m,n) y(m,n)
deblurring/deconvolution
filter
x(m,n)^
• Non-blind deblurring/deconvolutionGiven: observation y(m,n) and blurring function h(m,n)
Design: g(m,n), such that the distortion between x(m,n) and is minimized
• Blind deblurring/deconvolutionGiven: observation y(m,n)
Design: g(m,n), such that the distortion between x(m,n) and is minimized
x(m,n)^
x(m,n)^
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Deblurring: Inverse Filtering
h(m,n)
blurring filter
g(m,n)x(m,n) y(m,n)
inverse filter
x(m,n)^
X(u,v) H(u,v) = Y(u,v)
X(u,v) =Y(u,v
)H(u,v)
1
H(u,v)= Y(u,v)
G(u,v) =1
H(u,v)
Exact recovery!
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Deblurring: Pseudo-Inverse Filtering
h(m,n)
blurring filter
g(m,n)x(m,n) y(m,n)
deblur filter
x(m,n)^
G(u,v) =1
H(u,v)
|),(|0
|),(|),(
1),(
vuH
vuHvuHvuG
Inverse filter: What if at some (u,v), H(u,v) is 0 (or very close to 0) ?
Pseudo-inverse filter: small threshold
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Inverse and Pseudo-Inverse Filtering
),(
1),(
vuHvuG
|),(|0
|),(|),(
1),(
vuH
vuHvuHvuG
= 0.1Adapted from Prof. Xin Li
blurred image
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More Realistic Distortion Model
h(m,n) +x(m,n) y(m,n) g(m,n)
deblur filter
x(m,n)^
blurring filter
w(m,n)
),(
),(),(
),(
),(),(),(),(),(),(ˆ
vuH
vuWvuX
vuH
vuWvuHvuXvuGvuYvuX
• What happens when an inverse filter is applied?
close to zero at high frequencies
additive white Gaussian noise
Y(u,v) = X(u,v) H(u,v) + W(u,v)
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Radially Limited Inverse Filtering
Radially limited inverse filter:
Rvu
RvuvuHvuG
22
22
0
),(
1
),(
R
• Motivation– 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
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Radially Limited Inverse Filtering
Radially limitedinversefiltering:
Image size:
480x480
Original Blurred Inverse filtered
R = 40 R = 70 R = 85
From [Gonzalez & Woods]
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Wiener (Least Square) Filtering
KvuH
vuHvuG
2|),(|
),(*),(
2
2
X
WK
• Optimal in the least MSE sense, i.e.G(u, v) is the best possible linear filter that minimizes
noise power
signal power
Wiener filter:
2),(),(ˆ vuXvuXEenergyerror
•Have to estimate signal and noise power
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Weiner Filtering
Inverse filtering
Radially limited inverse
filteringR = 70
Weiner filtering
Blurred image
From [Gonzalez & Woods]
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Inverse vs. Weiner Filtering
From [Gonzalez & Woods]
distorted inverse filtering
Wiener filtering
motion
blur +noise
less noise
less noise
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Weiner Image Denoising
h(m,n) +x(m,n) y(m,n)
w(m,n)
• What if no blur, but only noise, i.e. h(m,n) is an impulse or H(u, v) = 1 ?
KvuH
vuHvuG
2|),(|
),(*),( 2
2
X
WK
Wiener filter:
22
2
22 /1
1
1
1),(
WX
X
XWKvuG
Wiener denoising
filter:
where
for H(u,v) = 1
Typically applied locally in space
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Weiner Image Denoising
adding noisenoise var = 400
local Wiener denoising
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Summary of Linear Image Restoration Filters
KvuH
vuHvuG
2|),(|
),(*),( 2
2
X
WK
Wiener filter:
22
2
),(WX
XvuG
Wiener
denoising filter:
where
G(u,v) =1
H(u,v)
|),(|0
|),(|),(
1),(
vuH
vuHvuHvuG
Inverse filter:
Pseudo-inverse filter:
Radially limited inverse filter:
Rvu
RvuvuHvuG
22
22
0
),(
1
),(
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Examples
jj
jj
jj
vuH
1.001.03.03.0
0000
1.001.03.03.0
3.03.003.03.01
),(
6252 X
• A blur filter h(m,n) has a 2D-DFT given by
• 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
4) Wiener filtering approach, with and
1252 W
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1) Inverse filter
2) Pseudo-inverse filter, with = 0.05
Examples
jInfj
InfInfInfInf
Infjj
jInfj
vuHvuG
101067.167.1
101067.167.1
67.167.167.167.11
),(
1),(
jj
jj
jj
vuH
vuHvuHvuG
1001067.167.1
0000
1001067.167.1
67.167.1067.167.11
|),(|0
|),(|),(
1),(
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Examples
00067.167.1
0000
00067.167.1
67.167.1067.167.11
|),(|0
|),(|),(
1),(
j
j
jj
vuH
vuHvuHvuG
6252 X 1252 W
3) Pseudo-inverse filter, with = 0.2
4) Wiener filter, with and
jj
jj
jj
KvuH
vuHvuG
48.0048.079.079.0
0000
48.0048.079.079.0
79.079.0079.079.083.0
|),(|
),(*),(
2
2.0625
1252
2
X
WK
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• 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
Advanced Image Restoration