1 Bayesian Restoration Using a New Nonstationary Edge-Preserving Image Prior Giannis K. Chantas,...

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Bayesian Restoration Using a New Nonstationary

Edge-Preserving Image Prior

Giannis K. Chantas, Nikolaos P. Galatsanos, and Aristidis C. Likas

IEEE Transactions on Image Processing, Vol. 15, No. 10, October 2006

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Outline

Review of Markov random field (MRF) for signal restoration problem

Bayesian restoration using a new non-stationary

edge-preserving image prior

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MAP formulation for signal restoration problem

Noisy signal d Restored signal f

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The problem of the signal restoration could be modeled as the MAP estimation problem, that is,

arg max{ ( | )}

By Baye's rule:

arg max{ ( | ) ( )}

:

:

f

f

f p f d

f p d f p f

f Unknown data

d Observed data

(Prior model)

(Observation

model)

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Assume the observation is the true signal plus the independent Gaussian noise, that is

Assume the unknown data f is MRF, the prior model is:

2 2

1

( ) / 2( | )

2 2

1 1( | )

2 2

m

i i ii

f dU d f

m m

i ii m i m

p d f e e

2'

'

11 ( )( )( )

i ii i Ni

f fU f TTP f e e

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Substitute above information into the MAP estimator, we could get:

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121 1

arg max{ ( | )} arg min{ ( | ) ( )}

( )arg min{ ( ) }

2

f f

m mi i

f i ii i

f p f d U d f U f

f df f

Observation model (Similarity measure)

Prior model (Reconstruction constrain, Regularization)

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From the potential function point of view:

2

212

1 1

( )arg min{ ( ) }

2

m mi i

f i ii i

f df f f

The edge region is blurred due to the improper design of the prior model

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MRF with pixel process and line process (Geman and Geman, 1984)

Lattice of pixel site: SP Labeling value: fi

p (real value)

Lattice of line site: SE Labeling value: fii’

E (only 0 or 1)Compound MRF

2' ' '

'

( , ) ( ) (1 )P

P E P P E Ei i ii ii

i Nii S

U f f f f f f

Prior model with indicator (Line process)

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MRF with nonstationary image prior (G.K. Chantas, N.P. Galatsanos and A.C. Likas, 2006)

From image modeling point of view, the binary nature (0 or 1) of the line process (Previous prior model) is insufficient to capture the image variations

Edge pattern 1

Edge pattern 2

Edge pattern 2 is more sharper than edge pattern 1

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A linear imaging model is assumed in this paper, that is:

-1

where

: Observation (Nx1 vector)

:NxN convolution matrix

: Gassian noise with pdf (0, )

:

arg max{ ( | )} arg max{ ( | ) ( )} f f

g H f n

g

H

n N I

Goal

f p f g p g f p f

��

��

��

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For the image prior model, they assume the first order difference of the image f in four direction, 0o, 90o, 45o, 135o, respectively, are given by

f(i-1,j-1) f(i-1,j) f(i-1,j+1)

f(i,j-1) f(i,j) f(i,j+1)

f(i+1,j-1)

f(i+1,j) f(i+1,j+1)

A 3x3 image patch; f(i,j): Intensity at location (i,j)

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The previous equation can be also written in matrix vector form for the entire image, that is

1 2

1

[ ..... ] , 1, 2,3, 4

where

: Difference operator for image f

(0, ( ) ) and is a parameter of the model

(Generalization of line process )

k k k TN

k k ki i i

kk

k

k and

N a a

Q f

Q

��

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For convenience, author introduces the following notation

1 2

1 2 4

1 2 3 4

1 2 3 4

1 2 3 4

{ , ,..... }

{ , ,..... }

[ , , , ]

[ , , , ]

[( ) , ( ) , ( ) , ( ) ]

k k k kN

T

T

T T T T T

A diag a a a

A diag A A A

a a a a a

Q Q Q Q Q

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Assume the residual εik in each direction and at each

pixel location are independent. Then, the joint density for the residuals is Gaussian and is given as:

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We could get the pdf of image f by using the fact that:

Then we have:

Q f ��

Over-parameterization occurs of the proposed model !

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To overcome the over-parameterization problem, the author views ai

k as a random variable instead of parameter and introduces Gamma hyper-prior for it

Where lk and mk are parameters of the hyper-prior

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More on Gamma hyper-prior aik

2

[ ]

[ ]

k

ki

ki

if l

E a

Var a

1

[ ] (2 )

[ ] 0

k

ki k

ki

if l

E a m

Var a

Stationary prior Non-stationary prior

Pdf:

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MAP estimation – Maximize p(.) is equivalent to minimize JMAP

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Bayesian algorithm : We are interested in true value of f instead of ai

k – Marginalize aik for solution finding, that is

The image is estimated by finding the mode of above pdf

( , ; , , ) ( , , ; , , )p g f B m l p g f a B m l da

* arg max ( , ; , , )f

f p g f B m l

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Definition for improvement signal to noise ration (ISNR)

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Original image MAP non-stationary, ISNR:5.63 dB, l=2.2

Wiener filter, ISNR:3.2dB

Bayesian non-stationary, ISNR:5.22 dB, l=2.2

CLS, ISNR:4.65dBDegraded image

1 Bayesian Restoration Using a New Nonstationary Edge-Preserving Image Prior Giannis K. Chantas,...

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