ee4328, section 005 introduction to digital image processing linear image restoration zhou wang...

EE4328, Section 005 Introduction to Digital Image

Processing

Linear Image Restoration

Zhou Wang

Dept. of Electrical EngineeringThe 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

h(m,n)

blurring filter

x(m,n) y(m,n)

Gaussian blur motion blur

•Spatial domain

From Prof. Xin

Li

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

Blurring Effect

Gaussian blur

motion blur

From [Gonzalez & Woods]

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)^

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!

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

Inverse and Pseudo-Inverse Filtering

),(

1),(

vuHvuG

|),(|0

|),(|),(

1),(

vuH

vuHvuHvuG

= 0.1Adapted from Prof. Xin Li

blurred image

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)

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

Radially Limited Inverse Filtering

Radially limitedinversefiltering:

Image size:

480x480

Original Blurred Inverse filtered

R = 40 R = 70 R = 85


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

Weiner Filtering

Inverse filtering

Radially limited inverse

filteringR = 70

Weiner filtering

Blurred image


Inverse vs. Weiner Filtering


distorted inverse filtering

Wiener filtering

motion

blur +noise

less noise

less noise

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

Weiner Image Denoising

adding noisenoise var = 400

local Wiener denoising

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

),(

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

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),(

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

• 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

ee4328, section 005 introduction to digital image processing linear image restoration zhou wang...

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