1. 3 September, 2009 SSP2009, Cardiff, UK 1 Probabilistic Image Processing by Extended Gauss-Markov Random Fields Kazuyuki Tanaka Kazuyuki Tanaka, Muneki YasudaNicolas Morin Muneki Yasuda, Nicolas Morin Graduate School of Information Sciences, Tohoku University, Japan and D. M. Titterington Department of Statistics, University of Glasgow, UK

2. 3 September, 2009 SSP2009, Cardiff, UK 2 Image Restoration by Bayesian Statistics Assumption 1: Original images are randomly generated by according to a prior probability. Bayes Formula Assumption 2: Degraded images are randomly generated from the original image by according to the conditional probability of degradation process. Original Image Degraded Image Transmission Noise Estimate Posterior

3. 3 September, 2009 SSP2009, Cardiff, UK 3 Bayesian Image Analysis Prior Probability Assumption 1: Prior Probability consists of a product of functions defined on the neighbouring pixels. Gibbs Sampler

4. 3 September, 2009SSP2009, Cardiff, UK4 Bayesian Image Analysis V:Set of all the pixels Assumption 2: Degraded image is generated from the original image by Additive White Gaussian Noise.

5. 3 September, 2009SSP2009, Cardiff, UK5 Bayesian Image Analysis Original Image Degraded Image Prior Probability Posterior Probability Degradation Process SmoothingData Dominant Bayesian Network Estimate

6. 3 September, 2009 SSP2009, Cardiff, UK 6 Average of Posterior Probability Gaussian Integral formula Average of the posterior probability can be calculated by using the multi-dimensional Gauss integral Formula

7. 3 September, 2009 SSP2009, Cardiff, UK 7 Degraded Image Statistical Estimation of Hyperparameters Marginalized with respect to X Original Image Marginal Likelihood Hyperparameters  are determined  so as to maximize the marginal likelihood Pr{Y=y| , ,  } with respect to , 

8. 3 September, 2009 SSP2009, Cardiff, UK 8 Statistical Estimation of Hyperparameters Gaussian Integral formula

9. 3 September, 2009SSP2009, Cardiff, UK9 Exact Expression of Marginal Likelihood in Gaussian Graphical Model Extremum Conditions for  and  Iterated Algorithm EM Algorithm

10. 3 September, 2009 SSP2009, Cardiff, UK 10 Bayesian Image Analysis by Gaussian Graphical Model Iteration Procedure of EM Algorithm in Gaussian Graphical Model EM

11. 3 September, 2009SSP2009, Cardiff, UK11 Image Restoration by Gaussian Markov Random Field (GMRF) Model and Conventional FiltersMSE Conventional GMRF  =0 309 ExtendedGMRF273 231 Simultaneous AR 225 Extended GMRF  Conventional GMRF  =0 Original Image Degraded Image RestoredImage Extended GMRF  Simultaneous AR (0.00088,33)(0.00465,36)(0.00171,38)

12. 3 September, 2009SSP2009, Cardiff, UK12 Image Restoration by Gaussian Markov Random Field (GMRF) Model and Conventional FiltersMSE Conventional GMRF  =0 313 ExtendedGMRF310 324 Simultaneous AR 383 Extended GMRF  Conventional GMRF  =0 Original Image Degraded Image RestoredImage Extended GMRF  Simultaneous AR (0.00123,39)(0.00687,41)(0.00400,43)

13. 3 September, 2009SSP2009, Cardiff, UK13 Statistical Performance by Sample Average of Numerical Experiments Posterior Probability Restored Images Degraded Images Sample Average of Mean Square Error Original Images Noise

14. 3 September, 2009SSP2009, Cardiff, UK14 Statistical Performance Estimation Additive White Gaussian Noise Posterior Probability Restored Image Original Image Degraded Image Additive White Gaussian Noise

15. 3 September, 2009SSP2009, Cardiff, UK15 Statistical Performance Estimation for Gauss Markov Random Fields = 0

16. 3 September, 2009SSP2009, Cardiff, UK16 Statistical Performance Estimation for Gauss Markov Random Fields  =40       

17. 3 September, 2009SSP2009, Cardiff, UK17 Summary We propose an extension of the Gauss-Markov random field models by introducing next-nearest neighbour interactions. Values for the hyperparameters in the proposed model are determined by using the EM algorithm in order to maximize the marginal likelihood. In addition, a measure of mean squared error, which quantifies the statistical performance of our proposed model, is derived analytically as an exact explicit expression by means of the multi-dimensional Gaussian integral formulas. Statistical performance analysis of probabilistic image processing for our extended Gauss Markov Random Fields has been shown.

18. 3 September, 2009SSP2009, Cardiff, UK18 References 1.K. Tanaka: Statistical-mechanical approach to image processing (Topical Review), Journal of Physics A: Mathematical and General, vol.35, no.37, pp.R81-R150, 2002. 2.K. Tanaka and J. Inoue: Maximum Likelihood Hyperparameter Estimation for Solvable Markov Random Field Model in Image Restoration, IEICE Transactions on Information and Systems, vol.E85-D, no.3, pp.546-557, 2002. 3.K. Tanaka, J. Inoue and D. M. Titterington: Probabilistic Image Processing by Means of Bethe Approximation for Q-Ising Model, Journal of Physics A: Mathematical and General, vol. 36, no. 43, pp.11023-11036, 2003. 4.K. Tanaka, H. Shouno, M. Okada and D. M. Titterington: Accuracy of the Bethe Approximation for Hyperparameter Estimation in Probabilistic Image Processing, Journal of Physics A: Mathematical and General, vol.37, no.36, pp.8675-8696, 2004. 5.K. Tanaka and D. M. Titterington: Statistical Trajectory of Approximate EM Algorithm for Probabilistic Image Processing, Journal of Physics A: Mathematical and Theoretical, vol.40, no.37, pp.11285-11300, 2007.