1. 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

2. Blur out-of-focus blur motion blur
From Prof. Xin Li
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?

3. Linear Blur Model Spatial domain x(m,n) h(m,n) y(m,n) blurring filter
From Prof. Xin Li
Gaussian blur
motion blur

4. 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

5. From [Gonzalez & Woods]
Blurring Effect
Gaussian blur
motion blur
From [Gonzalez & Woods]

6. Image Restoration: Deblurring/Deconvolution
x(m,n)
h(m,n)
y(m,n)
g(m,n)
x(m,n) ^
blurring filter
deblurring/
deconvolution
filter
Non-blind deblurring/deconvolution
Given: 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/deconvolution
Given: observation y(m,n)
x(m,n) ^
x(m,n) ^

7. Deblurring: Inverse Filtering
x(m,n)
h(m,n)
y(m,n)
g(m,n)
x(m,n) ^
blurring filter
inverse filter
X(u,v) H(u,v) = Y(u,v) 1
G(u,v) =
H(u,v)
Y(u,v) 1
X(u,v) = =
Y(u,v)
Exact recovery!
H(u,v)
H(u,v)

8. Deblurring: Pseudo-Inverse Filtering
x(m,n)
h(m,n)
y(m,n)
g(m,n)
x(m,n) ^
blurring filter
deblur filter 1
What if at some (u,v), H(u,v) is 0 (or very close to 0) ?
Inverse filter:
G(u,v) =
H(u,v)
small threshold
Pseudo-inverse filter:

9. Inverse and Pseudo-Inverse Filtering
blurred image
Adapted from Prof. Xin Li
 = 0.1

10. More Realistic Distortion Model
additive white Gaussian noise
w(m,n)
y(m,n)
x(m,n)
h(m,n) ^ +
g(m,n)
x(m,n)
blurring filter
deblur filter
Y(u,v) = X(u,v) H(u,v) + W(u,v)
What happens when an inverse filter is applied?
close to zero at high frequencies

11. Radially Limited Inverse Filtering
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 R

12. Radially Limited Inverse Filtering
Image size:
480x480
Original
Blurred
Inverse filtered
Radially limited
inverse
filtering:
R = 40
R = 70
R = 85
From [Gonzalez & Woods]

13. Wiener (Least Square) Filtering
Wiener filter:
noise power
signal power
Optimal in the least MSE sense, i.e.
G(u, v) is the best possible linear filter that minimizes
Have to estimate signal and noise power

14. Weiner Filtering Inverse filtering Blurred image
Radially limited inverse filtering
R = 70
Weiner filtering
From [Gonzalez & Woods]

15. Inverse vs. Weiner Filtering
distorted
inverse filtering
Wiener filtering
motion blur +
noise
less noise
less noise
From [Gonzalez & Woods]

16. Weiner Image Denoising Wiener denoising filter:
w(m,n)
y(m,n)
x(m,n)
h(m,n) +
What if no blur, but only noise, i.e. h(m,n) is an impulse or H(u, v) = 1 ?
Wiener filter:
where
for H(u,v) = 1
Wiener denoising filter:
Typically applied locally in space

17. Weiner Image Denoising
adding noise
noise var = 400
local Wiener denoising

18. Summary of Linear Image Restoration Filters1
Inverse filter:
G(u,v) =
H(u,v)
Pseudo-inverse filter:
Radially limited inverse filter:
Wiener filter:
where
Wiener denoising filter:

19. Examples A blur filter h(m,n) has a 2D-DFT given by
Find the deblur filter G(u,v) using
The inverse filtering approach
The pseudo-inverse filtering approach, with  = 0.05
The pseudo-inverse filtering approach, with  = 0.2
Wiener filtering approach, with and

20. Examples
Inverse filter
Pseudo-inverse filter, with  = 0.05

21. Examples
Pseudo-inverse filter, with  = 0.2
Wiener filter, with and

22. 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