1. 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
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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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
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Yokohama National University 7

8. Conventional Pairwise Markov Random Fields
Gaussian Prior
Huber Prior
Saturated Quadratic Prior x
Convex Functions near x=0 x x
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9. Supervised Learning of Pairwise Markov Random Fields
Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields:
Supervised Learning
30 Standard Images
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Yokohama National University 9

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:
Histogram from
Supervised Data
M. Yasuda, S. Kataoka and K.Tanaka, J. Phys. Soc. Jpn, Vol.81, No.4, Article No , 2012.
December, 2016
Yokohama National University 10

11. 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, , 2012.
December, 2016
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12. 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

13. 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, , 2012.
December, 2016
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14. Degradation Process in Bayesian Image Modeling
Assumption: Degraded image is generated from the original image by Additive White Gaussian Noise.
December, 2016
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15. 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

16. Noise Reductions by Generalized Sparse Priors and Loopy Belief Propagation
K. Tanaka, M. Yasuda and D. M. Titterington:
J. Phys. Soc. Jpn, 81, , 2012.
Original Image
Degraded Image
K. Tanaka, M. Yasuda and D. M. Titterington: JPSJ, 81, , 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

17. 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),
P. Arbelaez, M. Maire, C. Fowlkes and J. Malik: IEEE Trans. PAMI, 33, 898, 2011.
December, 2016
Yokohama National University

18. 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, , 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),
P. Arbelaez, M. Maire, C. Fowlkes and J. Malik:
IEEE Trans. PAMI, 33, 898, 2011.
481 x 321
December, 2016
Yokohama National University

19. Bayesian Image Segmentation
December, 2016
Yokohama National University

20. Coarse Graining K K K K K K K(1) K(1) K(1) K(1) K(1) K(1) 4 Step 12 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

21. 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

22. 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

23. 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

24. 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),
December, 2016
Yokohama National University

25. 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, , 2014.
S. Kataoka, M. Yasuda and K. Tanaka: JPSJ, 81, , 2012.
December, 2016
Yokohama National University

26. 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),
P. Arbelaez, M. Maire, C. Fowlkes and J. Malik: IEEE Trans. PAMI, 33, 898, 2011.
December, 2016
Yokohama National University

27. 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 , 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 , 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 , 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 , 共立出版, July 2016
December, 2016
Yokohama National University