Bayesian Image Modeling by Generalized Sparse Markov Random

Bayesian Image Modeling by Generalized Sparse Markov Random Fields and Loopy Belief Propagation Kazuyuki Tanaka GSIS, Tohoku University, Sendai, Japan http: //www. smapip. is. tohoku. ac. jp/~kazu/ Collaborators Muneki Yasuda (GSIS, Tohoku University, Japan) Sun Kataoka (GSIS, Tohoku University, Japan) D. M. Titterington (Department of Statistics, University of Glasgow, UK) 20 March, 2013 SPDSA 2013, Sendai, Japan 1

Outline 1. Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation 2. Bayesian Image Modeling by Generalized Sparse Prior 3. Noise Reductions by Generalized Sparse Prior 4. Concluding Remarks 20 March, 2013 SPDSA 2013, Sendai, Japan 2

Probabilistic Model and Belief Propagation Bayesian Networks Bayes Formulas Probabilistic Information Processing Markov Random Fields Probabilistic Models Belief Propagation =Bethe Approximation i j Message =Effective Field V: Set of all the nodes (vertices) in graph G E: Set of all the links (edges) in graph G 20 March, 2013 SPDSA 2013, Sendai, Japan 3

Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation Prior Probability of natural images is assumed to be the following pairwise Markov random fields: 20 March, 2013 SPDSA 2013, Sendai, Japan 4

Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields: Histogram from Supervised Data M. Yasuda, S. Kataoka and K. Tanaka, J. Phys. Soc. Jpn, Vol. 81, No. 4, Article No. 044801, 2012. 20 March, 2013 SPDSA 2013, Sendai, Japan 5

Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields: 20 March, 2013 SPDSA 2013, Sendai, Japan 6

Bayesian Image Modeling by Generalized Sparse Prior Assumption: Prior Probability is given as the following Gibbs distribution with the interaction a between every nearest neighbour pair of pixels: p=0: q-state Potts model p=2: Discrete Gaussian Graphical Model 20 March, 2013 SPDSA 2013, Sendai, Japan 7

Prior in Bayesian Image Modeling q=16 Loopy Belief Propagation In the region of 0<p<0. 3504…, the first order phase transition appears and the solution a * does not exist. 20 March, 2013 SPDSA 2013, Sendai, Japan 8

Bayesian Image Modeling by Generalized Sparse Prior: Conditional Maximization of Entropy Lagrange Multiplier K. Tanaka, M. Yasuda and D. M. Titterington: J. Phys. Soc. Jpn, 81, 114802, 2012. 20 March, 2013 Loopy Belief Propagation SPDSA 2013, Sendai, Japan 9

Prior Analysis by LBP and Conditional Maximization of Entropy in Bayesian Image Modeling Prior Probability Repeat until A converges LBP q=16 p=0. 2 20 March, 2013 q=16 p=0. 5 SPDSA 2013, Sendai, Japan 10

Prior Analysis by LBP and Conditional Maximization of Entropy in Generalized Sparse Prior Log-Likelihood for p when the original image f* is given q=16 Free Energy of Prior LBP K. Tanaka, M. Yasuda and D. M. Titterington: J. Phys. Soc. Jpn, 81, 114802, 2012. 20 March, 2013 SPDSA 2013, Sendai, Japan 11

Degradation Process in Bayesian Image Modeling Assumption: Degraded image is generated from the original image by Additive White Gaussian Noise. 20 March, 2013 SPDSA 2013, Sendai, Japan 12

Noise Reductions by Generalized Sparse Prior Posterior Probability Lagrange Multipliers K. Tanaka, M. Yasuda and D. M. Titterington: J. Phys. Soc. Jpn, 81, 114802, 2012. 20 March, 2013 SPDSA 2013, Sendai, Japan 13

Noise Reduction Procedures Based on LBP and Conditional Maximization of Entropy Input p Repeat until C and M converge Repeat until C converge and data g Repeat until u, B and L converge Marginals and Free Energy in LBP 20 March, 2013 SPDSA 2013, Sendai, Japan 14

Noise Reductions by Generalized Sparse Priors and Loopy Belief Propagation Original Image Degraded Image Restored Image K. Tanaka, M. Yasuda and D. M. Titterington: J. Phys. Soc. Jpn, 81, 114802, 2012. p=0. 2 20 March, 2013 p=0. 5 SPDSA 2013, Sendai, Japan p=1 15

Noise Reductions by Generalized Sparse Priors and Loopy Belief Propagation Original Image Degraded Image Restored Image K. Tanaka, M. Yasuda and D. M. Titterington: J. Phys. Soc. Jpn, 81, 114802, 2012. p=0. 2 20 March, 2013 p=0. 5 p=1 SPDSA 2013, Sendai, Japan 16

Summary Formulation of Bayesian image modeling for image processing by means of generalized sparse priors and loopy belief propagation are proposed. Our formulation is based on the conditional maximization of entropy with some constraints. In our sparse priors, although the first order phase transitions often appear, our algorithm works well also in such cases. 20 March, 2013 SPDSA 2013, Sendai, Japan 17

References 1. 2. 3. 4. 5. 6. 7. S. Kataoka, M. Yasuda, K. Tanaka and D. M. Titterington: Statistical Analysis of the Expectation-Maximization Algorithm with Loopy Belief Propagation in Bayesian Image Modeling, Philosophical Magazine: The Study of Condensed Matter, Vol. 92, Nos. 1 -3, pp. 50 -63, 2012. M. Yasuda and K. Tanaka: TAP Equation for Nonnegative Boltzmann Machine: Philosophical Magazine: The Study of Condensed Matter, Vol. 92, Nos. 1 -3, pp. 192 -209, 2012. S. Kataoka, M. Yasuda and K. Tanaka: Statistical Analysis of Gaussian Image Inpainting Problems, Journal of the Physical Society of Japan, Vol. 81, No. 2, Article No. 025001, 2012. M. Yasuda, S. Kataoka and K. Tanaka: Inverse Problem in Pairwise Markov Random Fields using Loopy Belief Propagation, Journal of the Physical Society of Japan, Vol. 81, No. 4, Article No. 044801, pp. 1 -8, 2012. M. Yasuda, Y. Kabashima and K. Tanaka: Replica Plefka Expansion of Ising systems, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2012, No. 4, Article No. P 04002, pp. 1 -16, 2012. K. Tanaka, M. Yasuda and D. M. Titterington: Bayesian image modeling by means of generalized sparse prior and loopy belief propagation, Journal of the Physical Society of Japan, Vol. 81, No. 11, Article No. 114802, 2012. M. Yasuda and K. Tanaka: Susceptibility Propagation by Using Diagonal Consistency, Physical Review E, Vol. 87, No. 1, Article No. 012134, 2013. 20 March, 2013 SPDSA 2013, Sendai, Japan 18