Bayesian Restoration Using a New Nonstationary Edge-Preserving Image Prior

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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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MAP formulation
for signal restoration problem

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

(Observation model)
(Prior model)
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MAP formulation
for signal restoration problem

• 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

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MAP formulation
for signal restoration problem

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

Observation model (Similarity measure)
Prior model (Reconstruction constrain,
Regularization)
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MAP formulation
for signal restoration problem

• From the potential function point of view

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 fip
(real value)
Lattice of line site SE
Labeling value fiiE (only 0 or 1)
Compound MRF
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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MRF with nonstationary image prior
(G.K. Chantas, N.P. Galatsanos and A.C. Likas,
2006)

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

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

• 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,j1)
f(i,j-1) f(i,j) f(i,j1)
f(i1,j-1) f(i1,j) f(i1,j1)
A 3x3 image patch f(i,j)
Intensity at location (i,j)
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MRF with nonstationary image prior
(G.K. Chantas, N.P. Galatsanos and A.C. Likas,
2006)

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

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

• For convenience, author introduces the following
  notation

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

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

• We could get the pdf of image f by using the fact
  that
• Then we have

Over-parameterization occurs of the proposed
model !
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MRF with nonstationary image prior
(G.K. Chantas, N.P. Galatsanos and A.C. Likas,
2006)

• To overcome the over-parameterization problem,
  the author views aik 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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MRF with nonstationary image prior
(G.K. Chantas, N.P. Galatsanos and A.C. Likas,
2006)

• More on Gamma hyper-prior aik

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

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

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

• Bayesian algorithm We are interested in true
  value of f instead of aik Marginalize aik for
  solution finding, that is
• The image is estimated by finding the mode of
  above pdf

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

• Definition for improvement signal to noise ration
  (ISNR)

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Original image
MAP non-stationary, ISNR5.63 dB, l2.2
Wiener filter, ISNR3.2dB
Bayesian non-stationary, ISNR5.22 dB, l2.2
CLS, ISNR4.65dB
Degraded image
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