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Statistical performance analysis by loopy belief propagation in probabilistic image processing

Statistical performance analysis by loopy belief propagation in probabilistic image processing. Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University, Japan http://www.smapip.is.tohoku.ac.jp/~kazu/. Collaborators D. M. Titterington (University of Glasgow, UK)

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Statistical performance analysis by loopy belief propagation in probabilistic image processing

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  1. Statistical performance analysis by loopy belief propagation in probabilistic image processing Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University, Japan http://www.smapip.is.tohoku.ac.jp/~kazu/ Collaborators D. M. Titterington (University of Glasgow, UK) M. Yasuda (Tohoku University, Japan) S. Kataoka (Tohoku University, Japan) LRI Seminar 2010 (Univ. Paris-Sud)

  2. Introduction Bayesian network is originally one of the methods for probabilistic inferences in artificial intelligence. Some probabilistic models for information processing are also regarded as Bayesian networks. Bayesian networks are expressed in terms of products of functions with a couple of random variables and can be associated with graphical representations. Such probabilistic models for Bayesian networks are referred to as Graphical Model. LRI Seminar 2010 (Univ. Paris-Sud)

  3. Probabilistic Image Processing by Bayesian Network Probabilistic image processing systems are formulated on square grid graphs. Averages, variances and covariances of the Bayesian network are approximately computed by using the belief propagation on the square grid graph. LRI Seminar 2010 (Univ. Paris-Sud)

  4. MRF, Belief Propagation and Statistical Performance • Geman and Geman (1986): IEEE Transactions on PAMI • Image Processing by Markov Random Fields (MRF) • Tanaka and Morita (1995): Physics Letters A • Cluster Variation Method for MRF in Image Processing • CVM= Generalized Belief Propagation (GBP) • Nishimori and Wong (1999): Physical Review E • Statistical Performance Estimation for MRF • (Infinite Range Ising Model and Replica Theory) Is it possible to estimate the performance of belief propagation statistically? IW-SMI2010 (Kyoto)

  5. Outline • Introduction • Bayesian Image Analysis by Gauss Markov Random Fields • Statistical Performance Analysis for Gauss Markov Random Fields • Statistical Performance Analysis in Binary Markov Random Fields by Loopy Belief Propagation • Concluding Remarks LRI Seminar 2010 (Univ. Paris-Sud)

  6. Outline • Introduction • Bayesian Image Analysis by Gauss Markov Random Fields • Statistical Performance Analysis for Gauss Markov Random Fields • Statistical Performance Analysis in Binary Markov Random Fields by Loopy Belief Propagation • Concluding Remarks LRI Seminar 2010 (Univ. Paris-Sud)

  7. Image Restoration by Bayesian Statistics Original Image LRI Seminar 2010 (Univ. Paris-Sud)

  8. Image Restoration by Bayesian Statistics Noise Transmission Original Image Degraded Image LRI Seminar 2010 (Univ. Paris-Sud)

  9. Image Restoration by Bayesian Statistics Noise Transmission Original Image Degraded Image Estimate LRI Seminar 2010 (Univ. Paris-Sud)

  10. Image Restoration by Bayesian Statistics Noise Transmission Posterior Original Image Degraded Image Estimate Bayes Formula 10 LRI Seminar 2010 (Univ. Paris-Sud)

  11. Image Restoration by Bayesian Statistics Noise Transmission Posterior Original Image Degraded Image Estimate Assumption 1: Original images are randomly generated by according to a prior probability. Bayes Formula 11 LRI Seminar 2010 (Univ. Paris-Sud)

  12. Image Restoration by Bayesian Statistics Noise Assumption 2: Degraded images are randomly generated from the original image by according to a conditional probability of degradation process. Transmission Posterior Original Image Degraded Image Estimate Bayes Formula 12 LRI Seminar 2010 (Univ. Paris-Sud) 7July, 2010

  13. Assumption: Prior Probability consists of a product of functions defined on the neighbouring pixels. Bayesian Image Analysis Prior Probability LRI Seminar 2010 (Univ. Paris-Sud)

  14. Bayesian Image Analysis Assumption: Prior Probability consists of a product of functions defined on the neighbouring pixels. Prior Probability 14 LRI Seminar 2010 (Univ. Paris-Sud)

  15. Bayesian Image Analysis Assumption: Prior Probability consists of a product of functions defined on the neighbouring pixels. Prior Probability Patterns by MCMC. 15 LRI Seminar 2010 (Univ. Paris-Sud)

  16. Bayesian Image Analysis Assumption: Degraded image is generated from the original image by Additive White Gaussian Noise. V:Set of all the pixels LRI Seminar 2010 (Univ. Paris-Sud)

  17. Bayesian Image Analysis Assumption: Degraded image is generated from the original image by Additive White Gaussian Noise. LRI Seminar 2010 (Univ. Paris-Sud) 17

  18. Bayesian Image Analysis Assumption: Degraded image is generated from the original image by Additive White Gaussian Noise. LRI Seminar 2010 (Univ. Paris-Sud) 18

  19. Bayesian Image Analysis Degraded Image Original Image Degradation Process Prior Probability Posterior Probability Estimate LRI Seminar 2010 (Univ. Paris-Sud)

  20. Bayesian Image Analysis Degraded Image Original Image Degradation Process Prior Probability Posterior Probability Estimate Data Dominant Smoothing LRI Seminar 2010 (Univ. Paris-Sud) 20

  21. Bayesian Image Analysis Degraded Image Original Image Degradation Process Prior Probability Posterior Probability Estimate Bayesian Network Data Dominant Smoothing 11 March, 2010 IW-SMI2010 (Kyoto) LRI Seminar 2010 (Univ. Paris-Sud) 21

  22. Bayesian Image Analysis Degraded Image Original Image Degradation Process Prior Probability Posterior Probability Estimate Bayesian Network Data Dominant Smoothing LRI Seminar 2010 (Univ. Paris-Sud) 22

  23. Image Restorations by Gaussian Markov Random Fields and Conventional Filters Original Image Degraded Image Restored Image V: Set of all the pixels Gauss Markov Random Field (3x3) Lowpass (5x5) Median LRI Seminar 2010 (Univ. Paris-Sud)

  24. Outline • Introduction • Bayesian Image Analysis by Gauss Markov Random Fields • Statistical Performance Analysis for Gauss Markov Random Fields • Statistical Performance Analysis in Binary Markov Random Fields by Loopy Belief Propagation • Concluding Remarks LRI Seminar 2010 (Univ. Paris-Sud)

  25. Statistical Performance by Sample Average of Numerical Experiments Original Images LRI Seminar 2010 (Univ. Paris-Sud)

  26. Statistical Performance by Sample Average of Numerical Experiments Noise Pr{G|F=f,s} Original Images Observed Data LRI Seminar 2010 (Univ. Paris-Sud) 26

  27. Statistical Performance by Sample Average of Numerical Experiments Noise Pr{G|F=f,s} Posterior Probability Pr{F|G=g,a,s} Original Images Observed Data Estimated Results 11 March, 2010 IW-SMI2010 (Kyoto) LRI Seminar 2010 (Univ. Paris-Sud) 27

  28. Statistical Performance by Sample Average of Numerical Experiments Sample Average of Mean Square Error Noise Pr{G|F=f,s} Posterior Probability Pr{F|G=g,a,s} Original Images Observed Data Estimated Results LRI Seminar 2010 (Univ. Paris-Sud) 28

  29. Statistical Performance Estimation Original Image Degraded Image Additive White Gaussian Noise Restored Image Posterior Probability Additive White Gaussian Noise LRI Seminar 2010 (Univ. Paris-Sud)

  30. Statistical Performance Estimation for Gauss Markov Random Fields s=40 s=40 a a LRI Seminar 2010 (Univ. Paris-Sud)

  31. Outline • Introduction • Bayesian Image Analysis by Gauss Markov Random Fields • Statistical Performance Analysis for Gauss Markov Random Fields • Statistical Performance Analysis in Binary Markov Random Fields by Loopy Belief Propagation • Concluding Remarks LRI Seminar 2010 (Univ. Paris-Sud)

  32. Marginal Probability in Belief Propagation In order to compute the marginal probability Pr{F2|G=g}, we take summations over all the pixels except the pixel 2. LRI Seminar 2010 (Univ. Paris-Sud)

  33. 2 Marginal Probability in Belief Propagation LRI Seminar 2010 (Univ. Paris-Sud)

  34. 2 2 Marginal Probability in Belief Propagation In the belief propagation, the marginal probability Pr{F2|G=g} is approximately expressed in terms of the messages from the neighbouring region of the pixel 2. LRI Seminar 2010 (Univ. Paris-Sud)

  35. 1 2 Marginal Probability in Belief Propagation In order to compute the marginal probability Pr{F1,F2|G=g}, we take summations over all the pixels except the pixels 1 and2. LRI Seminar 2010 (Univ. Paris-Sud)

  36. 1 2 1 2 Marginal Probability in Belief Propagation In the belief propagation, the marginal probability Pr{F1,F2|G=g} is approximately expressed in terms of the messages from the neighbouring region of the pixels 1 and 2. LRI Seminar 2010 (Univ. Paris-Sud)

  37. Belief Propagation in Probabilistic Image Processing LRI Seminar 2010 (Univ. Paris-Sud)

  38. Belief Propagation in Probabilistic Image Processing LRI Seminar 2010 (Univ. Paris-Sud)

  39. Image Restorations by Gaussian Markov Random Fields and Conventional Filters Original Image Degraded Image Restored Image V: Set of all the pixels Belief Propagation Exact LRI Seminar 2010 (Univ. Paris-Sud)

  40. Gray-Level Image Restoration(Spike Noise) Original Image Belief Propagation Degraded Image Lowpass Filter Median Filter MSE: 244 MSE: 217 MSE:135 MSE: 2075 MSE: 3469 MSE: 371 MSE: 523 MSE: 395 LRI Seminar 2010 (Univ. Paris-Sud)

  41. Binary Image Restoration byLoopy Belief Propagation a*=0.465 LRI Seminar 2010 (Univ. Paris-Sud)

  42. Statistical Performance by Sample Average of Numerical Experiments Sample Average of Mean Square Error Noise Pr{G|F=f,s} Posterior Probability Pr{F|G=g,a,s} Original Images Observed Data Estimated Results LRI Seminar 2010 (Univ. Paris-Sud) 42

  43. = > = Statistical Performance Estimation for Binary Markov Random Fields Light intensities of the original image can be regarded as spin states of ferromagnetic system. = > = Free Energy of Ising Model with Random External Fields It can be reduced to the calculation of the average of free energy with respect to locally non-uniform external fields g1, g2,…,g|V|. LRI Seminar 2010 (Univ. Paris-Sud)

  44. Statistical Performance Estimation for Binary Markov Random Fields LRI Seminar 2010 (Univ. Paris-Sud)

  45. Statistical Performance Estimation for Markov Random Fields Multi-dimensional Gauss Integral Formulas Spin Glass Theory in Statistical Mechanics Loopy Belief Propagation 0.8 0.6 0.4 s=40 0.2 s=1 a a LRI Seminar 2010 (Univ. Paris-Sud)

  46. Statistical Performance Estimation for Markov Random Fields LRI Seminar 2010 (Univ. Paris-Sud)

  47. Outline • Introduction • Bayesian Image Analysis by Gauss Markov Random Fields • Statistical Performance Analysis for Gauss Markov Random Fields • Statistical Performance Analysis in Binary Markov Random Fields by Loopy Belief Propagation • Concluding Remarks LRI Seminar 2010 (Univ. Paris-Sud)

  48. Summary • Formulation of probabilistic model for image processing by means of conventional statistical schemes has been summarized. • Statistical performance analysis of probabilistic image processing by using Gauss Markov Random Fields has been shown. • One of extensions of statistical performance estimation to probabilistic image processing with discretestates has been demonstrated. LRI Seminar 2010 (Univ. Paris-Sud)

  49. Image Impainting by Gauss MRF and LBP Our framework can be extended to erase a scribbling. Gauss MRF and LBP LRI Seminar 2010 (Univ. Paris-Sud)

  50. References • 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. • M. Yasuda and K. Tanaka: The Mathematical Structure of the Approximate Linear Response Relation, Journal of Physics A: Mathematical and Theoretical, vol.40, no.33, pp.9993-10007, 2007. • K. Tanaka and K. Tsuda: A Quantum-Statistical-Mechanical Extension of Gaussian Mixture Model, Journal of Physics: Conference Series, vol.95, article no.012023, pp.1-9, January 2008 • K. Tanaka: Mathematical Structures of Loopy Belief Propagation and Cluster Variation Method, Journal of Physics: Conference Series, vol.143, article no.012023, pp.1-18, 2009 • M. Yasuda and K. Tanaka: Approximate Learning Algorithm in Boltzmann Machines, Neural Computation, vol.21, no.11, pp.3130-3178, 2009. • S. Kataoka, M. Yasuda and K. Tanaka: Statistical Performance Analysis in Probabilistic Image Processing, Journal of the Physical Society of Japan, vol.79, no.2, article no.025001, 2010. LRI Seminar 2010 (Univ. Paris-Sud)

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