1 Physical Fluctuomatics 5th and 6th Probabilistic information processing by Gaussian graphical model Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University Physical Fluctuomatics (Tohoku University)

2 Textbooks Kazuyuki Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese), Chapter 7. K. Tanaka: Statistical-mechanical approach to image processing (Topical Review), Journal of Physics A: Mathematical and General, vol.35, no.37, pp.R81-R150, 2002, Section 4.

Physical Fluctuomatics (Tohoku University)3 Contents 1.Introduction 2.Probabilistic Image Processing 3.Gaussian Graphical Model 4.Statistical Performance Analysis 5.Concluding Remarks

Physical Fluctuomatics (Tohoku University)4 Contents 1.Introduction 2.Probabilistic Image Processing 3.Gaussian Graphical Model 4.Statistical Performance Analysis 5.Concluding Remarks

Physical Fluctuomatics (Tohoku University) 5 Markov Random Fields for Image Processing S. Geman and D. Geman (1986): IEEE Transactions on PAMI Image Processing for Markov Random Fields (MRF) (Simulated Annealing, Line Fields) J. Zhang (1992): IEEE Transactions on Signal Processing Image Processing in EM algorithm for Markov Random Fields (MRF) (Mean Field Methods) Markov Random Fields are One of Probabilistic Methods for Image processing.

Physical Fluctuomatics (Tohoku University) 6 Markov Random Fields for Image Processing In Markov Random Fields, we have to consider not only the states with high probabilities but also ones with low probabilities. In Markov Random Fields, we have to estimate not only the image but also hyperparameters in the probabilistic model.  We have to perform the calculations of statistical quantities repeatedly. Hyperparameter Estimation Statistical Quantities Estimation of Image We can calculate statistical quantities by adopting the Gaussian graphical model as a prior probabilistic model and by using Gaussian integral formulas.

Physical Fluctuomatics (Tohoku University) 7 Purpose of My Talk Review of formulation of probabilistic model for image processing by means of conventional statistical schemes. Review of probabilistic image processing by using Gaussian graphical model (Gaussian Markov Random Fields) as the most basic example. K. Tanaka: Statistical-Mechanical Approach to Image Processing (Topical Review), J. Phys. A: Math. Gen., vol.35, pp.R81-R150, Section 2 and Section 4 are summarized in the present talk.

Physical Fluctuomatics (Tohoku University)8 Contents 1.Introduction 2.Probabilistic Image Processing 3.Gaussian Graphical Model 4.Statistical Performance Analysis 5.Concluding Remarks

Physical Fluctuomatics (Tohoku University) 9 Bayes Formula and Bayesian Network Posterior Probability Bayes Rule Prior Probability Event B is given as the observed data. Event A corresponds to the original information to estimate. Thus the Bayes formula can be applied to the estimation of the original information from the given data. A B Bayesian Network Data-Generating Process

Physical Fluctuomatics (Tohoku University) 10 Image Restoration by Probabilistic Model Original Image Degraded Image Transmission Noise Assumption 1: The degraded image is randomly generated from the original image by according to the degradation process. Assumption 2: The original image is randomly generated by according to the prior probability. Bayes Formula

Physical Fluctuomatics (Tohoku University) 11 Image Restoration by Probabilistic Model Degraded Image i f i : Light Intensity of Pixel i in Original Image Position Vector of Pixel i g i : Light Intensity of Pixel i in Degraded Image i Original Image The original images and degraded images are represented by f = (f 1,f 2,…,f |V| ) and g = (g 1,g 2,…,g |V| ), respectively.

Physical Fluctuomatics (Tohoku University) 12 Probabilistic Modeling of Image Restoration Random Fields fifi gigi fifi gigi or Assumption 1: A given degraded image is obtained from the original image by changing the state of each pixel to another state by the same probability, independently of the other pixels.

Physical Fluctuomatics (Tohoku University) 13 Probabilistic Modeling of Image Restoration Random Fields Assumption 2: The original image is generated according to a prior probability. Prior Probability consists of a product of functions defined on the neighbouring pixels. ij Product over All the Nearest Neighbour Pairs of Pixels

Physical Fluctuomatics (Tohoku University) 14 Bayesian Image Analysis Original Image Degraded Image Prior Probability Posterior Probability Degradation Process Image processing is reduced to calculations of averages, variances and co-variances in the posterior probability. E ： Set of all the nearest neighbour pairs of pixels V ： Set of All the pixels

Physical Fluctuomatics (Tohoku University) 15 Estimation of Original Image We have some choices to estimate the restored image from posterior probability. In each choice, the computational time is generally exponential order of the number of pixels. Thresholded Posterior Mean (TPM) estimation Maximum posterior marginal (MPM) estimation Maximum A Posteriori (MAP) estimation (1) (2) (3)

Physical Fluctuomatics (Tohoku University)16 Contents 1.Introduction 2.Probabilistic Image Processing 3.Gaussian Graphical Model 4.Statistical Performance Analysis 5.Concluding Remarks

Physical Fluctuomatics (Tohoku University) 17 Bayesian Image Analysis by Gaussian Graphical Model Patterns are generated by MCMC. Markov Chain Monte Carlo Method Prior Probability E:Set of all the nearest-neighbour pairs of pixels V:Set of all the pixels

Physical Fluctuomatics (Tohoku University) 18 Bayesian Image Analysis by Gaussian Graphical Model Histogram of Gaussian Random Numbers Degraded image is obtained by adding a white Gaussian noise to the original image. Degradation Process is assumed to be the additive white Gaussian noise. V: Set of all the pixels Original Image f Gaussian Noise n Degraded Image g

Physical Fluctuomatics (Tohoku University) 19 Bayesian Image Analysis Original Image Degraded Image Prior Probability Posterior Probability Degradation Process Image processing is reduced to calculations of averages, variances and co-variances in the posterior probability. E ： Set of all the nearest neighbour pairs of pixels V ： Set of All the pixels

Physical Fluctuomatics (Tohoku University) 20 Bayesian Image Analysis A Posteriori Probability

Physical Fluctuomatics (Tohoku University) 21 Statistical Estimation of Hyperparameters Marginalized with respect to F Original Image Marginal Likelihood Degraded Image Hyperparameters  are determined  so as to maximize the marginal likelihood Pr{G=g| ,  } with respect to , 

Physical Fluctuomatics (Tohoku University) 22 Bayesian Image Analysis A Posteriori Probability Gaussian Graphical Model |V|x|V| matrix

Physical Fluctuomatics (Tohoku University) 23 Average of Posterior Probability Gaussian Integral formula

Physical Fluctuomatics (Tohoku University) 24 Bayesian Image Analysis by Gaussian Graphical Model Multi-Dimensional Gaussian Integral Formula Posterior Probability Average of the posterior probability can be calculated by using the multi- dimensional Gauss integral Formula |V|x|V| matrix E:Set of all the nearest-neghbour pairs of pixels V:Set of all the pixels

Physical Fluctuomatics (Tohoku University) 25 Statistical Estimation of Hyperparameters Marginalized with respect to F Original Image Marginal Likelihood Degraded Image

Physical Fluctuomatics (Tohoku University) 26 Calculations of Partition Function (A is a real symmetric and positive definite matrix.) Gaussian Integral formula

Physical Fluctuomatics (Tohoku University) 27 Exact expression of Marginal Likelihood in Gaussian Graphical Model Multi-dimensional Gauss integral formula We can construct an exact EM algorithm.

Physical Fluctuomatics (Tohoku University) 28 Bayesian Image Analysis by Gaussian Graphical Model Iteration Procedure in Gaussian Graphical Model

Physical Fluctuomatics (Tohoku University) 29 Image Restoration by Markov Random Field Model and Conventional Filters MSE Statistical Method315 Lowpass Filter (3x3)388 (5x5)413 Median Filter (3x3)486 (5x5)445 (3x3) Lowpass (5x5) Median MRF Original Image Degraded Image RestoredImage V:Set of all the pixels

Physical Fluctuomatics (Tohoku University)30 Contents 1.Introduction 2.Probabilistic Image Processing 3.Gaussian Graphical Model 4.Statistical Performance Analysis 5.Concluding Remarks

Physical Fluctuomatics (Tohoku University) 31 Performance Analysis Posterior Probability Estimated Results Observed Data Sample Average of Mean Square Error Signal Additive White Gaussian Noise

Physical Fluctuomatics (Tohoku University) 32 Statistical Performance Analysis Additive White Gaussian Noise Posterior Probability Restored Image Original Image Degraded Image Additive White Gaussian Noise

Physical Fluctuomatics (Tohoku University) 33 Statistical Performance Analysis

Physical Fluctuomatics (Tohoku University) 34 Statistical Performance Estimation for Gaussian Markov Random Fields = 0

Physical Fluctuomatics (Tohoku University) 35 Statistical Performance Estimation for Gaussian Markov Random Fields   =40 

Physical Fluctuomatics (Tohoku University)36 Contents 1.Introduction 2.Probabilistic Image Processing 3.Gaussian Graphical Model 4.Statistical Performance Analysis 5.Concluding Remarks

Physical Fluctuomatics (Tohoku University) 37 Summary Formulation of probabilistic model for image processing by means of conventional statistical schemes has been summarized. Probabilistic image processing by using Gaussian graphical model has been shown as the most basic example.

References K. Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese). K. Tanaka: Statistical-Mechanical Approach to Image Processing (Topical Review), J. Phys. A, 35 (2002). A. S. Willsky: Multiresolution Markov Models for Signal and Image Processing, Proceedings of IEEE, 90 (2002). Physical Fluctuomatics (Tohoku University) 38

Physical Fluctuomatics (Tohoku University) 39 Problem 5-1: Derive the expression of the posterior probability Pr{F=f|G=g, ,  } by using Bayes formulas Pr{F=f|G=g, ,  } =Pr{G=g|F=f,  }Pr{F=f,  }/Pr{G=g| ,  }. Here Pr{G=g|F=f,  } and Pr{F=f,  } are assumed to be as follows: [Answer]

Physical Fluctuomatics (Tohoku University) 40 Problem 5-2: Show the following equality.

Physical Fluctuomatics (Tohoku University) 41 Problem 5-3: Show the following equality.

Physical Fluctuomatics (Tohoku University) 42 Problem 5-4: Show the following equalities by using the multi-dimensional Gaussian integral formulas.

Physical Fluctuomatics (Tohoku University) 43 Problem 5-5: Derive the extremum conditions for the following marginal likelihood Pr{G=g  } with respect to the hyperparameters  and . [Answer]

Physical Fluctuomatics (Tohoku University) 44 Problem 5-6: Derive the extremum conditions for the following marginal likelihood Pr{G=g  } with respect to the hyperparameters  and . [Answer]

Physical Fluctuomatics (Tohoku University) 45 Problem 5-7: Make a program that generate a degraded image by the additive white Gaussian noise. Generate some degraded images from a given standard images by setting  =10,20,30,40 numerically. Calculate the mean square error (MSE) between the original image and the degraded image. Histogram of Gaussian Random Numbers F i  G i ~N(0,40 2 ) Original Image Gaussian Noise Degraded Image Sample Program: K.Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006.

Physical Fluctuomatics (Tohoku University) 46 Problem 5-8: Make a program of the following procedure in probabilistic image processing by using the Gaussian graphical model and the additive white Gaussian noise. Algorithm: Repeat the following procedure until convergence Sample Program: K.Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006.