Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University

1. Physical Fluctuomatics5thand 6th Probabilistic information processing by Gaussian graphical model Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/ Physical Fluctuomatics (Tohoku University)

2. Textbooks • Kazuyuki Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese) , Chapter 7. • 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, Section 4. Physical Fluctuomatics (Tohoku University)

3. Contents • Introduction • Probabilistic Image Processing • Gaussian Graphical Model • Statistical Performance Analysis • Concluding Remarks Physical Fluctuomatics (Tohoku University)

4. Contents • Introduction • Probabilistic Image Processing • Gaussian Graphical Model • Statistical Performance Analysis • Concluding Remarks Physical Fluctuomatics (Tohoku University)

5. Markov Random Fields for Image Processing Markov Random Fields are One of Probabilistic Methods for Image processing. • S. Geman and D. Geman (1986): IEEE Transactions on PAMI • Image Processing for Markov Random Fields (MRF) (Simulated Annealing, Line Fields) • J. Zhang (1992): IEEE Transactions on Signal Processing • Image Processing in EM algorithm for Markov Random Fields (MRF) (Mean Field Methods) Physical Fluctuomatics (Tohoku University)

6. Hyperparameter Estimation Statistical Quantities Estimation of Image Markov Random Fields for Image Processing In Markov Random Fields, we have to consider not only the states with high probabilities but also ones with low probabilities. In Markov Random Fields, we have to estimate not only the image but also hyperparameters in the probabilistic model. We have to perform the calculations of statistical quantities repeatedly. We can calculate statistical quantities by adopting the Gaussian graphical model as a prior probabilistic model and by using Gaussian integral formulas. Physical Fluctuomatics (Tohoku University)

7. Purpose of My Talk • Review of formulation of probabilistic model for image processing by means of conventional statistical schemes. • Review of probabilistic image processing by using Gaussian graphical model (Gaussian Markov Random Fields) as the most basic example. K. Tanaka: Statistical-Mechanical Approach to Image Processing (Topical Review), J. Phys. A: Math. Gen., vol.35, pp.R81-R150, 2002. Section 2 and Section 4 are summarized in the present talk. Physical Fluctuomatics (Tohoku University)

8. Contents • Introduction • Probabilistic Image Processing • Gaussian Graphical Model • Statistical Performance Analysis • Concluding Remarks Physical Fluctuomatics (Tohoku University)

9. Bayes Formula and Bayesian Network Prior Probability Data-Generating Process Bayes Rule Posterior Probability A Event B is given as the observed data. Event A corresponds to the original information to estimate. Thus the Bayes formula can be applied to the estimation of the original information from the given data. B Bayesian Network Physical Fluctuomatics (Tohoku University)

10. Assumption 1: The degraded image is randomly generated from the original image by according to the degradation process. Assumption 2: The original image is randomly generated by according to the prior probability. Image Restoration by Probabilistic Model Noise Transmission Original Image Degraded Image Bayes Formula Physical Fluctuomatics (Tohoku University)

11. The original images and degraded images are represented by f = (f1,f2,…,f|V|) and g = (g1,g2,…,g|V|), respectively. Image Restoration by Probabilistic Model Original Image Degraded Image Position Vector of Pixel i i i fi: Light Intensity of Pixel i in Original Image gi: Light Intensity of Pixel i in Degraded Image Physical Fluctuomatics (Tohoku University)

12. gi gi fi fi Probabilistic Modeling of Image Restoration Assumption 1: A given degraded image is obtained from the original image by changing the state of each pixel to another state by the same probability, independently of the other pixels. or Random Fields Physical Fluctuomatics (Tohoku University)

13. Probabilistic Modeling of Image Restoration Assumption 2: The original image is generated according to a prior probability. Prior Probability consists of a product of functions defined on the neighbouring pixels. i j Random Fields Product over All the Nearest Neighbour Pairs of Pixels Physical Fluctuomatics (Tohoku University)

14. Bayesian Image Analysis Degraded Image Prior Probability Degradation Process Original Image Posterior Probability V：Set of All the pixels E：Set of all the nearest neighbour pairs of pixels Image processing is reduced to calculations of averages, variances and co-variances in the posterior probability. Physical Fluctuomatics (Tohoku University)

15. We have some choices to estimate the restored image from posterior probability. In each choice, the computational time is generally exponential order of the number of pixels. Estimation of Original Image Maximum A Posteriori (MAP) estimation (1) (2) Maximum posterior marginal (MPM) estimation (3) Thresholded Posterior Mean (TPM) estimation Physical Fluctuomatics (Tohoku University)

16. Contents • Introduction • Probabilistic Image Processing • Gaussian Graphical Model • Statistical Performance Analysis • Concluding Remarks Physical Fluctuomatics (Tohoku University)

17. Bayesian Image Analysis by Gaussian Graphical Model Prior Probability V:Set of all the pixels Patterns are generated by MCMC. E:Set of all the nearest-neighbour pairs of pixels Markov Chain Monte Carlo Method Physical Fluctuomatics (Tohoku University)

18. Bayesian Image Analysis by Gaussian Graphical Model Degradation Process is assumed to be the additive white Gaussian noise. V: Set of all the pixels Gaussian Noise n Degraded Image g Original Image f Degraded image is obtained by adding a white Gaussian noise to the original image. Histogram of Gaussian Random Numbers Physical Fluctuomatics (Tohoku University)

19. Bayesian Image Analysis Degraded Image Prior Probability Degradation Process Original Image Posterior Probability V：Set of All the pixels E：Set of all the nearest neighbour pairs of pixels Image processing is reduced to calculations of averages, variances and co-variances in the posterior probability. Physical Fluctuomatics (Tohoku University)

20. A Posteriori Probability Bayesian Image Analysis Physical Fluctuomatics (Tohoku University)

21. Hyperparameters a, s are determinedso as to maximize the marginal likelihood Pr{G=g|a,s} with respect to a, s. Statistical Estimation of Hyperparameters Original Image Degraded Image Marginalized with respect to F Marginal Likelihood Physical Fluctuomatics (Tohoku University)

22. A Posteriori Probability Bayesian Image Analysis |V|x|V| matrix Gaussian Graphical Model Physical Fluctuomatics (Tohoku University)

23. Average of Posterior Probability Gaussian Integral formula Physical Fluctuomatics (Tohoku University)

24. Posterior Probability Bayesian Image Analysis by Gaussian Graphical Model V:Set of all the pixels Average of the posterior probability can be calculated by using the multi-dimensional Gauss integral Formula E:Set of all the nearest-neghbour pairs of pixels |V|x|V| matrix Multi-Dimensional Gaussian Integral Formula Physical Fluctuomatics (Tohoku University)

25. Statistical Estimation of Hyperparameters Original Image Degraded Image Marginalized with respect to F Marginal Likelihood Physical Fluctuomatics (Tohoku University)

26. Calculations of Partition Function Gaussian Integral formula (A is a real symmetric and positive definite matrix.) Physical Fluctuomatics (Tohoku University)

27. Exact expression of Marginal Likelihood in Gaussian Graphical Model Multi-dimensional Gauss integral formula We can construct an exact EM algorithm. Physical Fluctuomatics (Tohoku University)

28. Iteration Procedure in Gaussian Graphical Model Bayesian Image Analysis by Gaussian Graphical Model Physical Fluctuomatics (Tohoku University)

29. Image Restoration by Markov Random Field Model and Conventional Filters Original Image Degraded Image V:Set of all the pixels Restored Image MRF (3x3) Lowpass (5x5) Median Physical Fluctuomatics (Tohoku University)

30. Contents • Introduction • Probabilistic Image Processing • Gaussian Graphical Model • Statistical Performance Analysis • Concluding Remarks Physical Fluctuomatics (Tohoku University)

31. Performance Analysis Sample Average of Mean Square Error Additive White Gaussian Noise Posterior Probability Signal Observed Data Estimated Results Physical Fluctuomatics (Tohoku University)

32. Statistical Performance Analysis Degraded Image Original Image Additive White Gaussian Noise Restored Image Posterior Probability Additive White Gaussian Noise Physical Fluctuomatics (Tohoku University)

33. Statistical Performance Analysis Physical Fluctuomatics (Tohoku University)

34. Statistical Performance Estimation for Gaussian Markov Random Fields = 0 Physical Fluctuomatics (Tohoku University)

35. Statistical Performance Estimation for Gaussian Markov Random Fields s=40 s=40 a a Physical Fluctuomatics (Tohoku University)

36. Contents • Introduction • Probabilistic Image Processing • Gaussian Graphical Model • Statistical Performance Analysis • Concluding Remarks Physical Fluctuomatics (Tohoku University)

37. Summary • Formulation of probabilistic model for image processing by means of conventional statistical schemes has been summarized. • Probabilistic image processing by using Gaussian graphical model has been shown as the most basic example. Physical Fluctuomatics (Tohoku University)

38. References K. Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese) . K. Tanaka: Statistical-Mechanical Approach to Image Processing (Topical Review), J. Phys. A, 35 (2002). A. S. Willsky: Multiresolution Markov Models for Signal and Image Processing, Proceedings of IEEE, 90 (2002). Physical Fluctuomatics (Tohoku University)

39. Problem 5-1: Derive the expression of the posterior probability Pr{F=f|G=g,a,s} by using Bayes formulas Pr{F=f|G=g,a,s}=Pr{G=g|F=f,s}Pr{F=f,a}/Pr{G=g|a,s}. Here Pr{G=g|F=f,s} and Pr{F=f,a} are assumed to be as follows: [Answer] Physical Fluctuomatics (Tohoku University)

40. Problem 5-2: Show the following equality. Physical Fluctuomatics (Tohoku University)

41. Problem 5-3: Show the following equality. Physical Fluctuomatics (Tohoku University)

42. Problem 5-4: Show the following equalities by using the multi-dimensional Gaussian integral formulas. Physical Fluctuomatics (Tohoku University)

43. Problem 5-5: Derive the extremum conditions for the following marginal likelihood Pr{G=g|a,s} with respect to the hyperparameters a and s. [Answer] Physical Fluctuomatics (Tohoku University)

44. Problem 5-6: Derive the extremum conditions for the following marginal likelihood Pr{G=g|a,s} with respect to the hyperparameters a and s. [Answer] Physical Fluctuomatics (Tohoku University)

45. Problem 5-7: Make a program that generate a degraded image by the additive white Gaussian noise. Generate some degraded images from a given standard images by setting s=10,20,30,40 numerically. Calculate the mean square error (MSE) between the original image and the degraded image. Gaussian Noise Degraded Image Original Image K.Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 . Sample Program: http://www.morikita.co.jp/soft/84661/ Histogram of Gaussian Random Numbers Fi -Gi~N(0,402) Physical Fluctuomatics (Tohoku University)

46. Problem 5-8: Make a program of the following procedure in probabilistic image processing by using the Gaussian graphical model and the additive white Gaussian noise. Algorithm: Repeat the following procedure until convergence K.Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 . Sample Program: http://www.morikita.co.jp/soft/84661/ Physical Fluctuomatics (Tohoku University)