画像処理における確率伝搬法と em アルゴリズムの統計的性能評価
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画像処理における確率伝搬法と EM アルゴリズムの統計的性能評価. 東北大学大学院情報科学研究科 田中和之 http://www.smapip.is.tohoku.ac.jp/~kazu/. 共同研究者 : D. M. Titterington (University of Glasgow) 皆川まりか ( 東北大 ). Reference 田中和之 : ガウシアングラフィカルモデルにもとづく確率的情報処理における一般化された信念伝搬法 , 電子情報通信学会論文誌 (D-II), Vol.J88-D-II, No.12, pp.2368-2379, 2005. - PowerPoint PPT PresentationTRANSCRIPT
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EMhttp://www.smapip.is.tohoku.ac.jp/~kazu/Reference: , (D-II), Vol.J88-D-II, No.12, pp.2368-2379, 2005 : D. M. Titterington (University of Glasgow) ()
Kyoto University
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ContentsIntroductionGaussian Graphical Model and EM AlgorithmLoopy Belief PropagationGeneralized Belief PropagationConcluding Remarks
Kyoto University
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MRF, Belief Propagation and Statistical PerformanceGeman and Geman (1986): IEEE Transactions on PAMIImage Processing for Markov Random Fields (MRF) (Simulated Annealing, Line Fields)Nishimori and Wong (1999): Physical Review EStatistical Performance Estimation for MRF (Infinite Range Model and Replica Theory)Tanaka and Morita (1995): Physics Letters ACluster Variation Method for MRF in Image ProcessingCluster Variation Method (CVM)= Generalized Belief Propagation (GBP)Is it possible to estimate the performance of loopy belief propagation statistically?
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ContentsIntroductionGaussian Graphical Model and EM AlgorithmLoopy Belief PropagationGeneralized Belief PropagationConcluding Remarks
Kyoto University
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Bayesian Image RestorationOriginal ImageDegraded ImagetransmissionNoise
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Bayes Formula and Probabilistic Image ProcessingOriginal ImageDegraded ImagePrior ProbabilityPosteriorProbabilityDegradation ProcessPixel
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Prior Probability in Probabilistic Image ProcessingSamples are generated by MCMC.Markov Chain Monte Carlo Method
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Degradation ProcessAdditive White Gaussian NoiseHistogram of Gaussian Random Numbers
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Degradation Process and PriorDegradation ProcessPrior Probability Density FunctionPosterior Probability Density FunctionMulti-Dimensional Gaussian Integral Formula
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Statistical Performance by Sample AveragePrior ProbabilityDegradation ProcessPosterior Probability
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Statistical Performance AnalysisPrior ProbabilityDegradation ProcessPosterior Probability
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Statistical Performance AnalysisMulti-Dimentional Gaussian Integral FormulaNishimori (2000)
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Probabilistic Image ProcessingMarginalizedMarginal LikelihoodPosterior Probability Density Function
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Marginal Likelihood in Probabilistic Image Processing
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Maximization of Marginal Likelihood by EM (Expectation Maximization) AlgorithmMarginal LikelihoodIterate the following EM-steps until convergence:EM AlgorithmQ-FunctionA. 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).
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Maximization of Marginal Likelihood by EM (Expectation Maximization) AlgorithmMarginal LikelihoodIncomplete DataEquivalentQ-FunctionPixel
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Maximization of Marginal Likelihood by EM (Expectation Maximization) Algorithm
Kyoto University
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Statistical Behaviour of EM (Expectation Maximization) AlgorithmNumerical Experiments for Standard ImageStatistical Behaviour of EM Algorithm
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wk
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emmml-gauss-exact07
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emmml-gauss-exact07
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ContentsIntroductionGaussian Graphical Model and EM AlgorithmLoopy Belief PropagationGeneralized Belief PropagationConcluding Remarks
Kyoto University
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Belief Propagation and Markov Random FieldGraphical Model with CyclesFixed Point EquationMarginal Probability
Kyoto University
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Gaussian Graphical Model and Loopy Belief PropagationY. Weiss and W. T. Freeman, Correctness of belief propagation in Gaussian graphical models of arbitrary topology, Neural Computation, 13, 2173 (2001). 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, Math. & Gen., 37, 8675 (2004).Dynamics of Algorithm in LBP?Loopy Belief Propagation for Gaussian Graphical ModelStatistical Analysis
Kyoto University
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Kullback-Leibler Divergence of Gaussian Graphical ModelEntropy Term
Kyoto University
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Loopy Belief PropagationTrial FunctionTractable Form
Kyoto University
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Loopy Belief PropagationTrial FunctionMarginal Distribution of GGM is also GGM
Kyoto University
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Loopy Belief PropagationBethe Free Energy in GGM
Kyoto University
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Loopy Belief PropagationVii and Vij do not depend on pixel i and link ij
Kyoto University
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Iteration ProcedureFixed Point EquationIteration
Kyoto University
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Maximization of Marginal Likelihood by EM (Expectation Maximization) AlgorithmLoopy Belief PropagationExact
Kyoto University
Graph6
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emmml-gauss-exact07
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Maximization of Marginal Likelihood by EM (Expectation Maximization) Algorithm
Kyoto University
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Maximization of Marginal Likelihood by EM (Expectation Maximization) AlgorithmStatistical Behaviour of EM AlgorithmNumerical Experiments for Standard Image
Kyoto University
Graph5
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