マルコフ確率場の統計的機械学習の数理と データサイエンスへの展開 Statistical Machine Learning in Markov Random Field and Expansion to Data Sciences 田中和之 東北大学大学院情報科学研究科 Kazuyuki Tanaka GSIS, Tohoku University, Sendai, Japan http://www.smapip.is.tohoku.ac.jp/~kazu/ Collaborators Muneki Yasuda (Yamagata University, Japan) Masayuki Ohzeki (Tohoku University, Japan) Shun Kataoka (Tohoku University, Japan) Candy Hsu (National Tsin Hua University, Taiwan) Mike Titterington (University of Glasgow, UK) My talk is to realize an acceleration of Bayesian image segmentation by using the real space renormalization group transformation in the statistical mechanics.. Our Bayesian image segmentation modeling is based on Markov random fields and loopy belief propagation. December, 2016 Yokohama National University

Outline Introduction Probabilistic Graphical Models and Loopy Belief Propagation Statistical Machine Learning in Markov Random Fields Generalized Sparse Prior for Image Modeling Probabilistic Noise Reduction Probabilistic Image Segmentation Statistical Machine Learning by Inverse Renormalization Group Transformation Related Works Concluding Remarks December, 2016 Yokohama National University

Pairwise Markov Random Fields and Loopy Belief Propagation Bayes Formulas Maximum Likelihood KL Divergence Markov Random Fields Probabilistic Information Processing Probabilistic Models and Statistical Machine Learning Loopy Belief Propagation j i Probability distribution of Markov random field can be expressed as a product of pairwise weights over all the neighbouring pairs of pixels. In this slide, a_i is a state variable at each pixel on a square grid graph. In the loopy belief propagation, some statistical quantities can be approximately expressed in terms of messages between neighbouring pixels. Messages can be determined so as to satisfy the message passing rules which are regarded as simultaneous fixed point equations. Practical algorithms of loopy belief propagation can be realized as an iteration method to solve the simultaneous fixed point equations for messages. Message V: Set of all the nodes (vertices) in graph G E: Set of all the links (edges) in graph G December, 2016 Yokohama National University

Pairwise Markov Random Fields and Loopy Belief Propagation Message i j 1 2 3 4 5 6 7 8 9 10 11 12 j i Message V: Set of all the nodes (vertices) in graph G E: Set of all the links (edges) in graph G December, 2016 Yokohama National University

Pairwise Markov Random Fields and Bayes Inference Prior ProbabilityPairwise MRF 1 2 3 4 5 6 7 8 9 10 11 12 Data Generative Model Posterior Probability  Pairwise MRF 1 2 3 4 5 6 7 8 9 10 11 12 Maximum A Posteriori (MAP) Estimation Simulated Annealing Graph Cut Maximum Posterior Marginal (MPM) Estimation Loopy Belief Propagation Markov Chain Monte Carlo V: Set of all the nodes (vertices) E: Set of all the links (edges) December, 2016 Yokohama National University

Hyperparameter Estimation of Pairwise Markov Random Fields Prior ProbabilityPairwise MRF Data Generative Model 1 2 3 4 5 6 7 8 9 10 11 12 Joint Probability Probability of Data d  Likelihood of a and b Marginal Likelihood Maximization of Marginal Likelihood Expectation Maximization (EM) Algorithm V: Set of all the nodes (vertices) E: Set of all the links (edges) December, 2016 Yokohama National University

Pairwise Markov Random Fields for Bayesian Image Modeling In Bayesian image modeling, natural images are often assumed to be generated by according to the following pairwise Markov random fields: Hamiltonian of Classical Spin System December, 2016 Yokohama National University 7

Conventional Pairwise Markov Random Fields Gaussian Prior Huber Prior Saturated Quadratic Prior x Convex Functions near x=0 x x December, 2016 Yokohama National University 8

Supervised Learning of Pairwise Markov Random Fields Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields: Supervised Learning 30 Standard Images December, 2016 Yokohama National University 9

Yokohama National University 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. December, 2016 Yokohama National University 10

Yokohama National University Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields: q=256 K. Tanaka, M. Yasuda and D. M. Titterington: J. Phys. Soc. Jpn, 81, 114802, 2012. December, 2016 Yokohama National University 11

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: Potts model p=2: Discrete Gaussian Graphical Model December, 2016 Yokohama National University 12

Yokohama National University Prior Analysis by LBP and Conditional Maximization of Entropy in Generalized Sparse Prior Log-Likelihood for p and a when the original image f* is given K. Tanaka, M. Yasuda and D. M. Titterington: JPSJ, 81, 114802, 2012. December, 2016 Yokohama National University 13

Degradation Process in Bayesian Image Modeling Assumption: Degraded image is generated from the original image by Additive White Gaussian Noise. December, 2016 Yokohama National University 14

Hyperparameter Estimation of Pairwise Markov Random Fields Prior ProbabilityPairwise MRF Data Generative Model 1 2 3 4 5 6 7 8 9 10 11 12 Joint Probability 1 2 3 4 5 6 7 8 9 10 11 12 Probability of Data d  Likelihood of a and b Marginal Likelihood Maximization of Marginal Likelihood Expectation Maximization (EM) Algorithm V: Set of all the nodes (vertices) E: Set of all the links (edges) December, 2016 Yokohama National University

Noise Reductions by Generalized Sparse Priors and Loopy Belief Propagation K. Tanaka, M. Yasuda and D. M. Titterington: J. Phys. Soc. Jpn, 81, 114802, 2012. Original Image Degraded Image K. Tanaka, M. Yasuda and D. M. Titterington: JPSJ, 81, 114802, 2012. Restored Image Finally, we show only the results for the gray-level image restoration. For each numerical experiments, the loopy belief propagation ca give us better results than the ones by conventional filters. Continuous Gaussian Graphical Model p=0.5 p=2.0 December, 2016 Yokohama National University p=2.0

Bayesian Modeling for Image Segmentation Image Segmentation using MRF Segment image data into some regions using belief propagation and EM algorithm Data Parameter ai = 0,1,…,q-1 Posterior Probability Dstribution 12q+1 Hyperparameters Potts Prior Data Generative Model Likelihood of Hyperparameter Now we consider Bayesian modelling of image segmentation problems for color images. Segmentation problems can be regarded as one of clustering of pixels from one observed color image. In this slide, d is a color image and is data point in our problem. a is a state variable of labeling at each pixel and takes all the possible integer from 0 to q-1. The number of all the possible states for labeling is denoted by q. The posterior probability distribution in our Bayesian modeling of image segmentation problems is expressed as the following Markov random field model. The first factor corresponds to data generative model and is a product of three-dimensional Gaussian distributions over all the pixels.. The second factor is a prior probability distributions which is the Potts model with spatially uniform ferromagnetic interactions between all the neighbouring pixels on the square grid graph. The posterior probability distribution also can be regarded as the ferromagnetic Potts model with random external fields, in which d corresponds to a kind of external fields.and a is a state variable in the classical spin system. This model include 12q+1 hyperparameters. They are determined by the maximum likelihood framework. In the maximum likelihood framework, we regard the probability of data d when the hyperparameters are given is regarded as a likelihood function of the 12q+1 hyperparameters when the data d is given. And the hyperparameters are determined so as to maximize the likelihood function. After determing the hyperparameters, the label state a at each pixel is determined so as to maximize the one-body marginal of the posterior probability distribution at the pixel,. Maximization of Posterior Marginal (MPM) Berkeley Segmentation Data Set 500 (BSDS500), http://www.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/ P. Arbelaez, M. Maire, C. Fowlkes and J. Malik: IEEE Trans. PAMI, 33, 898, 2011. December, 2016 Yokohama National University

Bayesian Image Segmentation Intel® Core™ i7-4600U CPU with a memory of 8 GB Potts Prior Hyperparameter q=8 Hyperparameter Estimation in Maximum Likelihood Framewrok and Maximization of Posterior Marginal (MPM) Estimation with Loopy Belief Propagation for Observed Color Image d 321 x 481 K. Tanaka, S. Kataoka, M. Yasuda, Y. Waizumi and C.-T. Hsu: JPSJ,.83, 124002, 2014. This is one of our numerical experiments. The number of labeling is 8. We spend about 30 minutes. Berkeley Segmentation Data Set 500 (BSDS500), http://www.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/ P. Arbelaez, M. Maire, C. Fowlkes and J. Malik: IEEE Trans. PAMI, 33, 898, 2011. 481 x 321 December, 2016 Yokohama National University

Bayesian Image Segmentation December, 2016 Yokohama National University

Coarse Graining K K K K K K K(1) K(1) K(1) K(1) K(1) K(1) 4 Step 1 2 3 4 5 6 7 Step 1 K(1) K(1) K(1) In order to realize a sub-linear computational time modeling of our image segmentation algorithm, we apply a coarse graining techniques for our Potts prior.. For simplicity, we first consider a Potts prior on a one-dimensional chain graph.. We take summations over all the even-numbered nodes. After this procedure, we generate another Potts prior. New interaction is expressed in terms of previous interaction.. 1 3 5 7 Step 2 K(1) K(1) K(1) 1 2 3 4 December, 2016 Yokohama National University

Coarse Graining y y x x x If K(2) is given, the original value of K can be estimated by iterating By repeating the same procedure, we can realize one of coarse graining procedure. This procedure is referred to as a real space renormalization group transformation in the statistical physics. If we can estimate the interaction parameter of coarse grained Potts prior, we can estimate the interaction parameter of original Potts prior by considering the inverse transformation. We regard it as an inverse real space renormalization group transformation for Potts prior on a one-dimensional chain graph. December, 2016 Yokohama National University

Coarse Graining q = 8 In the square grid graph, we can consider the similar coarse graining procedure approximately. This schemes can be regarded as an inverse real space renormalization group transformation in Bayesian modeling of image segmentation problems. December, 2016 Yokohama National University

Bayesian Image Segmentation Hyperparameter Estimation in Maximization of Marginal Likelihood with Belief Propagation for Original Image Potts Prior Hyperparameter Hyperparameter Estimation in Maximization of Marginal Likelihood with Belief Propagation after Coarse Graining Procedures Segmentation by Belief Propagation for Original Image 20 x 30 20 x 30 Labeled Image Coarse Graining Procedures (r =8) 321 x 481 December, 2016 Yokohama National University

Bayesian Image Segmentation Hyperparameter Estimation in Maximum Likelihood Framework with Belief Propagation for Original Image q=8 Potts Prior Hyperparameter Hyperparameter Estimation in Maximum Likelihood Framework with Belief Propagation after Coarse Graining Procedures 481 x 321 Segmentation by Belief Propagation for Original Image Intel® Core™ i7-4600U CPU with a memory of 8 GB MPM with LBP 30 x 20 30 x 20 Labeled Image This is one of numerical experiments. First we generate small size image from our original image. By applying our EM algorithm with belief propagation to the small size image, we estimate the hyperparameter for coarse grained Potts prior. By applying the inverse real space renormalization transformation, we can estimate the hyperparameter of original Potts prior. Our method spends less than 2 minutes although our original EM algorithm takes 30 minutes. Coarse Graining Procedures (r =8) Berkeley Segmentation Data Set 500 (BSDS500), http://www.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/ December, 2016 Yokohama National University

Reconstruction of Traffic Density プレゼンテーション資料 2013/8/13 Our related Works Image Inpainting Reconstruction of Traffic Density True data reconstruction from 80% missing data S. Kataoka, M. Yasuda, C. Furtlehner and K. Tanaka: Inverse Problems, 30, 025003, 2014. S. Kataoka, M. Yasuda and K. Tanaka: JPSJ, 81, 025001, 2012. December, 2016 Yokohama National University

Summary Bayesian Image Modeling by Loopy Belief Propagation and EM algorithm. By introducing Inverse Real Space Renormalization Group Transformations, the computational time can be reduced. Labeled Image by our proposed algorithm Ground Truth Observed Image In the first part of my talk, we show the Bayesian image segmentation modeling by the loopy belief propagation and EM algorithm in the statistical machine learning theory. In the second part, we show that a sub-linear computational time modelling can be realized by introducing inverse real space renormalization group transformations in our problem. It is expected that prior can be learned from data base set of ground truths. Berkeley Segmentation Data Set 500 (BSDS500), http://www.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/ P. Arbelaez, M. Maire, C. Fowlkes and J. Malik: IEEE Trans. PAMI, 33, 898, 2011. December, 2016 Yokohama National University

References 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, vo.11, article no.114802, November 2012. K. Tanaka, S. Kataoka, M. Yasuda, Y. Waizumi and C.-T. Hsu: Bayesian image segmentations by Potts prior and loopy belief propagation, Journal of the Physical Society of Japan, vol.83, no.12, article no.124002, December 2014. K. Tanaka, S. Kataoka, M. Yasuda and M. Ohzeki: Inverse renormalization group transformation in Bayesian image segmentations, Journal of the Physical Society of Japan, vol.84, no.4, article no.045001, April 2015. 田中和之 (分担執筆): 確率的グラフィカルモデル(鈴木譲, 植野真臣編), 第8章 マルコフ確率場と確率的画像処理, pp.195-228, 共立出版, July 2016 December, 2016 Yokohama National University