StatisticalMechanical Approach to Probabilistic Image Processing Loopy Belief
Statistical-Mechanical Approach to Probabilistic Image Processing -- Loopy Belief Propagation and Advanced Mean-Field Method -- Kazuyuki Tanaka and Noriko Yoshiike Graduate School of Information Sciences Tohoku University kazu@statp. is. tohoku. ac. jp http: //www. statp. is. tohoku. ac. jp/~kazu/ Collaborators: J. Inoue (Hokkaido University), D. M. Titterington (University of Glasgow) 1
Image Processing and Magnetic Material Regular lattice consisting of a lot of nodes. Interactions among neighboring nodes Output images are determined from a priori information and given data. Similarity Ordered states are determined from interactions and external fields. Para It is difficult for conventional filters to treat fluctuation in data. Critical Ferro Fluctuation is enhanced near critical temperature. 2
Probabilistic Model and Image Restoration Noise Transmission Original Image Degraded Image 3
Degradation Process and A Priori Probability in Binary Image Restoration Degradation Process (Binary Symmetric Channel) A Priori Probability 4
A Priori Probability in Binary Image Restoration 5
Bayes Formula and A Posteriori Probability 6
Maximization of Posterior Marginal 7
Deterministic Equation of Loopy Belief Propagation 8
Message Update Rule of Loopy Belief Propagation Fixed-Point Equations Natural Iteration 9
Free Energy of A Priori Probabilitic Model in Loopy Belief Propagation (Bethe Approx. ) 0 0. 5 -0. 5 0. 4 Second Order Phase Transition 0. 3 -1. 0 0. 2 -1. 5 0. 1 -2. 0 0. 5 1. 0 10
Binary Image Restoration Original images are generated by Monte Carlo simulations in the a priori probability. Original Image Degraded Image (p=0. 2) Restored Image 11
Hyperparameter Estimation Maximization of Marginal Likelihood Marginalize 12
Binary Image Restoration Original images are generated by Monte Carlo simulations in the a priori probability. Original Image Degraded Image (p=0. 2) Restored Image 13
Binary Image Restoration Hyperparameters are determined so as to maximize the marginal likelihood. Original Image Degraded Image Loopy Belief Propagation 14
Degradation Process in Multi-Valued Image Restoration 15
A Priori Probability in Multi-Valued Image Restoration Q-Ising Model Q-state Potts Model 16
Free Energy of Q-Ising Model (Q=2) in Loopy Belief Propagation (Bethe Approx. ) 0 0. 5 -0. 5 0. 4 Second Order Phase Transition 0. 3 -1. 0 0. 2 -1. 5 0. 1 -2. 0 0 0 1. 0 2. 0 17
Free Energy of Q-Ising Model (Q=4) in Loopy Belief Propagation (Bethe Approx. ) 0 0. 5 -0. 5 0. 4 Second Order Phase Transition 0. 3 -1. 0 0. 2 -1. 5 0. 1 -2. 0 0 0 1. 0 2. 0 18
Free Energy of Q-state Potts Model (Q=2) in Loopy Belief Propagation (Bethe Approx. ) 0 0 -1. 0 -0. 2 Second Order Phase Transition -0. 4 -2. 0 -0. 6 -3. 0 -0. 8 -4. 0 -1. 0 0 1. 0 2. 0 19
Free Energy of Q-state Potts Model (Q=4) in Loopy Belief Propagation (Bethe Approx. ) 0 0 -1. 0 -0. 2 First Order Phase Transition -0. 4 -2. 0 -0. 6 -3. 0 -0. 8 -4. 0 -1. 0 0 1. 0 20
Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. Q-Ising Model Original Image Degraded Image Restored Image Q-state Potts Model 21
Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. Q-Ising Model Original Image Degraded Image Restored Image 22
Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. Q-state Potts Model Original Image Degraded Image Restored Image 23
Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. Original Image Degraded Image (3 p=0. 3) Q-state Potts Model Q-Ising Model 24
Gray-Level Image Restoration Original Image Belief Propagation Lowpass Filter MSE: 2075 MSE: 244 MSE: 217 MSE: 3469 MSE: 371 Degraded Image MSE: 523 Median Filter MSE: 135 MSE: 395 25
Summary Probabilistic Image Processing by Bayes Formula and Loopy Belief Propagation Some Numerical Experiments Future Problems Segmentation Image Compression Motion Detection Color Image EM algorithm Statistical Performance Line Fields 26
Appendix A: Graphical Probabilistic Model 27
Appendix A: Kullback-Leibler divergence 28
Appendix A: Bethe Free Energy 29
Appendix A: Basic Framework of Bethe Approximation 30
Appendix A: Propagation Rule of Bethe Approximation Update Rule is reduced to Loopy Belief Propagation 31
Appendix B: A Priori Probability in Image Restoration 32
Appendix B: Original images are generated by Monte Carlo simulations in the a priori probability. Degraded Image Mean Field (p=0. 2) Approximation Original Image Bethe Approximation 33
Appendix B: Original images are generated by Monte Carlo simulations in the a priori probability. Degraded Image Mean Field Original Image Approx. (p=0. 2) Bethe Approx. 34
Appendix B: Standard Image Original Image Degraded Image Mean-Field Approx. Bethe Approx. 35
Appendix B: Hyperparameters are determined so as to maximize the marginal likelihood. Original Image Degraded Image Mean-Field Approx. Bethe Approx. 36
Appendix C: Ising Model 37
Appendix C: Exact Results in Thermodynamic Limit 38
Appendix C: Order Parameter 1. 0 Mean Field Approx. Exact Bethe Approx. 0. 5 0 0 Kikuchi Approx. 1 2 3 4 5 39
Appendix C: Free Energy of A Priori Probabilitic Model in Loopy Belief Propagation (Bethe Approx. ) 0 0 -0. 5 -0. 1 Second Order Phase Transition -0. 2 -1. 0 -0. 3 -1. 5 -0. 4 -2. 0 -0. 5 0 0. 5 1. 0 40
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