1 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

2 Outline Review of Markov random field (MRF) for signal restoration problem Bayesian restoration using a new non-stationary edge-preserving image prior

3 MAP formulation for signal restoration problem Noisy signal dRestored signal f

4 MAP formulation for signal restoration problem The problem of the signal restoration could be modeled as the MAP estimation problem, that is, (Prior model) (Observation model)

5 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:

6 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)

7 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

8 MRF with pixel process and line process (Geman and Geman, 1984) Lattice of pixel site: S P Labeling value: f i p (real value) Lattice of line site: S E Labeling value: f ii’ E (only 0 or 1) Compound MRF Prior model with indicator (Line process)

9 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

10 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:

11 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, 0 o, 90 o, 45 o, 135 o, 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)

12 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

13 MRF with nonstationary image prior (G.K. Chantas, N.P. Galatsanos and A.C. Likas, 2006) For convenience, author introduces the following notation

14 MRF with nonstationary image prior (G.K. Chantas, N.P. Galatsanos and A.C. Likas, 2006) Assume the residual ε i k in each direction and at each pixel location are independent. Then, the joint density for the residuals is Gaussian and is given as:

15 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 !

16 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 a i k as a random variable instead of parameter and introduces Gamma hyper-prior for it Where l k and m k are parameters of the hyper-prior

17 MRF with nonstationary image prior (G.K. Chantas, N.P. Galatsanos and A.C. Likas, 2006) More on Gamma hyper-prior a i k Stationary prior Non-stationary prior Pdf:

18 MRF with nonstationary image prior (G.K. Chantas, N.P. Galatsanos and A.C. Likas, 2006) MAP estimation – Maximize p(.) is equivalent to minimize J MAP

19 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 a i k – Marginalize a i k for solution finding, that is The image is estimated by finding the mode of above pdf

20 MRF with nonstationary image prior (G.K. Chantas, N.P. Galatsanos and A.C. Likas, 2006) Definition for improvement signal to noise ration (ISNR)

21 Original imageMAP 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

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