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Statistical Learning Procedure in Loopy Belief Propagation for Probabilistic Image Processing

Statistical Learning Procedure in Loopy Belief Propagation for Probabilistic Image Processing. Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/. References

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Statistical Learning Procedure in Loopy Belief Propagation for Probabilistic Image Processing

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  1. Statistical Learning Procedurein Loopy Belief Propagationfor Probabilistic Image Processing Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/ References K. Tanaka, H. Shouno, M. Okada and D. M. Titterington: Accuracy of the Bethe approximation for hyperparameter estimation in probabilistic image processing, J. Phys. A, vol.37, pp.8675-8695 (2004). CIMCA2005, Vienna

  2. Bayesian Network Bayes Formula Probabilistic Information Processing Bayesian Network and Belief Propagation Probabilistic Model Belief Propagation Graphical Model J. Pearl: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference (Morgan Kaufmann, 1988). CIMCA2005, Vienna

  3. Belief Propagation • How should we treat the calculation of the summation over 256N configurations? It is very hard to calculate it exactly except some special cases. • Formulation for approximate algorithm • Accuracy of the approximate algorithm CIMCA2005, Vienna

  4. Contents • Introduction • Bayesian Image Analysis and Gaussian Graphical Model • Belief Propagation • Concluding Remarks CIMCA2005, Vienna

  5. Noise Transmission Bayesian Image Analysis Original Image Degraded Image CIMCA2005, Vienna

  6. Degradation Process Bayesian Image Analysis Additive White Gaussian Noise Transmission Original Image Degraded Image CIMCA2005, Vienna

  7. Standard Images A Priori Probability Bayesian Image Analysis Generate Similar? CIMCA2005, Vienna

  8. A Posteriori Probability Bayesian Image Analysis Gaussian Graphical Model CIMCA2005, Vienna

  9. Bayesian Image Analysis Degraded Image A Priori Probability Degraded Image Original Image Pixels A Posteriori Probability CIMCA2005, Vienna

  10. Hyperparameter Determination by Maximization of Marginal Likelihood Marginalization Degraded Image Original Image Marginal Likelihood CIMCA2005, Vienna

  11. Marginal Likelihood Maximization of Marginal Likelihood by EM (Expectation Maximization) Algorithm Q-Function Incomplete Data Equivalent CIMCA2005, Vienna

  12. EM Algorithm Iterate the following EM-steps until convergence: Marginal Likelihood Maximization of Marginal Likelihood by EM (Expectation Maximization) Algorithm Q-Function A. P. Dempster, N. M. Laird and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Roy. Stat. Soc. B, 39 (1977). CIMCA2005, Vienna

  13. Contents • Introduction • Bayesian Image Analysis and Gaussian Graphical Model • Belief Propagation • Concluding Remarks CIMCA2005, Vienna

  14. Probabilistic Model on a Graph with Loops Marginal Probability CIMCA2005, Vienna

  15. 3 1 4 2 3 8 1 2 5 4 7 5 6 Belief Propagation Message Update Rule CIMCA2005, Vienna

  16. 3 1 4 2 5 Message Passing Rule of Belief Propagation Fixed Point Equations for Massage CIMCA2005, Vienna

  17. Fixed Point Equation Fixed Point Equation and Iterative Method Iterative Method CIMCA2005, Vienna

  18. Image Restoration by Gaussian Graphical Model EM Algorithm with Belief Propagation Original Image Degraded Image MSE: 1512 MSE: 1529 CIMCA2005, Vienna

  19. Comparison of Belief Propagation with Exact Results in Gaussian Graphical Model CIMCA2005, Vienna

  20. Image Restoration by Gaussian Graphical Model Exact Original Image Degraded Image Belief Propagation MSE:315 MSE: 325 MSE: 1512 Lowpass Filter Median Filter Wiener Filter MSE: 411 MSE: 545 MSE: 447 CIMCA2005, Vienna

  21. Image Restoration by Gaussian Graphical Model Belief Propagation Original Image Degraded Image Exact MSE236 MSE: 260 MSE: 1529 Median Filter Wiener Filter Lowpass Filter MSE: 224 MSE: 372 MSE: 244 CIMCA2005, Vienna

  22. Contents • Introduction • Bayesian Image Analysis and Gaussian Graphical Model • Belief Propagation • Concluding Remarks CIMCA2005, Vienna

  23. Summary • Formulation of belief propagation • Accuracy of belief propagation in Bayesian image analysis by means of Gaussian graphical model (Comparison between the belief propagation and exact calculation) CIMCA2005, Vienna

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