Graduate School of Information Sciences, Tohoku University Physical Fluctuomatics Applied Stochastic Process 10th Probabilistic image processing by means of physical models Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/ Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Textbooks Kazuyuki Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese) , Chapter 7. Kazuyuki Tanaka: Statistical-mechanical approach to image processing (Topical Review), Journal of Physics A: Mathematical and General, vol.35, no.37, pp.R81-R150, 2002. Muneki Yasuda, Shun Kataoka and Kazuyuki Tanaka: Computer Vision and Image Processing 3 (Edited by Y. Yagi and H. Saito): Chapter 6. Stochastic Image Processing, pp.137-179, Advanced Communication Media CO., Ltd., December 2010 (in Japanese). Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Contents Probabilistic Image Processing Loopy Belief Propagation Statistical Learning Algorithm More Practical Probabilistic models for Image Processing Summary Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Image Representation in Computer Vision Digital image is defined on the set of points arranged on a square lattice. The elements of such a digital array are called pixels. We have to treat more than 100,000 pixels even in the digital cameras and the mobile phones. Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Image Representation in Computer Vision At each point, the intensity of light is represented as an integer number or a real number in the digital image data. A monochrome digital image is then expressed as a two-dimensional light intensity function and the value is proportional to the brightness of the image at the pixel. Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Noise Reduction by Conventional Filters The function of a linear filter is to take the sum of the product of the mask coefficients and the intensities of the pixels. Smoothing Filters 192 202 190 192 202 190 202 219 120 202 173 120 100 218 110 100 218 110 It is expected that probabilistic algorithms for image processing can be constructed from such aspects in the conventional signal processing. Markov Random Fields Algorithm Probabilistic Image Processing Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Fundamental Probabilistic Theory for Image Processing Marginal Probability of Event B A B C D Marginalization Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Fundamental Probabilistic Theory for Image Processing: Bayes Formula Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Fundamental Probabilistic Theory for Image Processing: Bayes Formula Prior Probability Data-Generating Process Bayes Rule Posterior Probability A 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. B Bayesian Network Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Image Restoration by Probabilistic Model 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. Noise Transmission Original Image Degraded Image Bayes Formula Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Image Restoration by Probabilistic Model The original images and degraded images are represented by f = (f1,f2,…,f|V|) and g = (g1,g2,…,g|V|), respectively. Original Image Degraded Image Position Vector of Pixel i i i fi: Light Intensity of Pixel i in Original Image gi: Light Intensity of Pixel i in Degraded Image Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Probabilistic Modeling of Image Restoration 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. i j Random Fields Product over All the Nearest Neighbour Pairs of Pixels Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Probabilistic Modeling of Image Restoration 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. fi gi fi gi or Random Fields Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

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

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. (1) Maximum A Posteriori (MAP) estimation In the image restoration, we usually have to estimate the hyperparameters alpha and p. In statistics, the maximum likelihood estimation is often employed. In the standpoint of maximum likelihood estimation, the hyperparameters are determined so as to maximize the marginal likelihood defined by marginalize the joint probability for the original image and degraded image with respect to the original image. The marginal likelihood is expressed in terms of the partition functions of the a priori probabilistic model and the a posteriori probabilistic model. We can calculate these partition functions approximately by using the Bethe approximation. (2) Maximum posterior marginal (MPM) estimation (3) Thresholded Posterior Mean (TPM) estimation Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Prior Probability of Image Processing Digital Images (fi=0,1,…,q-1) Sampling of Markov Chain Monte Carlo Method E: Set of all the nearest-neighbour pairs of pixels Paramagnetic Ferromagnetic Fluctuations increase near α=1.76…. V: Set of all the pixels Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

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

Degradation Process for Binary Image Binary Symmetric Channel V：Set of all the pixels Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

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

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

Bayesian Image Analysis Prior Probability Original Image Degradation Process Degraded Image 画素 Posterior Probability Computational Complexity Marginal Posterior Probability of Each Pixel Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Posterior Probability for Image Processing by Bayesian Analysis Binary Symmetric Channel Additive White Gaussian Noise Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Bayesian Network and Posterior Probability of Image Processing Probabilistic Model expressed in terms of Graphical Representations with Cycles V: Set of all the pixels E: Set of all the nearest neighbour pairs of pixels Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

∂i: Set of all the nearest neighbour pixels of the pixel i Markov Random Fields Markov Random Fields ∂i: Set of all the nearest neighbour pixels of the pixel i Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Contents Probabilistic Image Processing Loopy Belief Propagation Statistical Learning Algorithm More Practical Probabilistic models for Image Processing Summary Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Marginal Probability in Belief Propagation In order to compute the marginal probability Pr{F2|G=g}, we take summations over all the pixels except the pixel 2. Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Marginal Probability in Belief Propagation 2 Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Marginal Probability in Belief Propagation 2 2 In the belief propagation, the marginal probability Pr{F2|G=g} is approximately expressed in terms of the messages from the neighbouring region of the pixel 2. Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Marginal Probability in Belief Propagation 1 2 In order to compute the marginal probability Pr{F1,F2|G=g}, we take summations over all the pixels except the pixels 1 and2. Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Marginal Probability in Belief Propagation 1 2 1 2 In the belief propagation, the marginal probability Pr{F1,F2|G=g} is approximately expressed in terms of the messages from the neighbouring region of the pixels 1 and 2. Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Loopy Belief Propagation Probabilistic Model expressed in terms of Graphical Representations with Cycles V: Set of all the pixels E: Set of all the nearest neighbour pairs of pixels 2 1 3 4 5 Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Loopy Belief Propagation 2 1 3 4 5 Simultaneous fixed point equations to determine the messages Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Belief Propagation Algorithm for Image Processing Step 1: Solve the simultaneous fixed point equations for messages by using iterative method j i i Step 2: Substitute the messages to approximate expressions of marginal probabilities for each pixel and compute estimated image so as to maximize the approximate marginal probability at each pixel. Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Belief Propagation in Probabilistic Image Processing Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Contents Probabilistic Image Processing Loopy Belief Propagation Statistical Learning Algorithm More Practical Probabilistic models for Image Processing Summary Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Statistical Estimation of Hyperparameters Hyperparameters a, b are determined so as to maximize the marginal likelihood Pr{G=g|a,b} with respect to a, b. In the image restoration, we usually have to estimate the hyperparameters alpha and p. In statistics, the maximum likelihood estimation is often employed. In the standpoint of maximum likelihood estimation, the hyperparameters are determined so as to maximize the marginal likelihood defined by marginalize the joint probability for the original image and degraded image with respect to the original image. The marginal likelihood is expressed in terms of the partition functions of the a priori probabilistic model and the a posteriori probabilistic model. We can calculate these partition functions approximately by using the Bethe approximation. Original Image Degraded Image Marginalized with respect to F Marginal Likelihood Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Maximum Likelihood in Signal Processing Incomplete Data Original Image =Parameter Degraded Image =Data Marginal Likelihood Hyperparameter Extremum Condition Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Maximum Likelihood and EM algorithm Degraded Image =Data Incomplete Data Original Image =Parameter Marginal Likelihood Q -function Hyperparameter Expectation Maximization (EM) algorithm provide us one of Extremum Points of Marginal Likelihood. Extemum Condition Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Maximum Likelihood and EM algorithm Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

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

Bayesian Image Analysis by Exact Solution of Gaussian Graphical Model Iteration Procedure in Gaussian Graphical Model In the image restoration, we usually have to estimate the hyperparameters alpha and p. In statistics, the maximum likelihood estimation is often employed. In the standpoint of maximum likelihood estimation, the hyperparameters are determined so as to maximize the marginal likelihood defined by marginalize the joint probability for the original image and degraded image with respect to the original image. The marginal likelihood is expressed in terms of the partition functions of the a priori probabilistic model and the a posteriori probabilistic model. We can calculate these partition functions approximately by using the Bethe approximation. Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Image Restorations by Exact Solution and Loopy Belief Propagation of Gaussian Graphical Model 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. Loopy Belief Propagation Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Image Restorations by Gaussian Graphical Model Original Image Degraded Image Belief Propagation Exact MSE: 1512 MSE:325 MSE:315 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. Lowpass Filter Wiener Filter Median Filter MSE: 411 MSE: 545 MSE: 447 Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Image Restoration by Gaussian Graphical Model Original Image Degraded Image Belief Propagation Exact MSE: 1529 MSE: 260 MSE236 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. Lowpass Filter Wiener Filter Median Filter MSE: 224 MSE: 372 MSE: 244 Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

a(t) s(t) Image Restoration by Discrete Gaussian Graphical Model Original Image Q=4 a(t) Degraded Image s=1 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. MSE:0.98893 SNR:1.03496 (dB) Belief Propagation MSE: 0.36011 s(t) SNR: 5.42228 (dB) Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

a(t) s(t) Image Restoration by Discrete Gaussian Graphical Model Original Image Q=4 a(t) Degraded Image s=1 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. MSE: 0.98893 SNR: 1.07221 (dB) Belief Propagation MSE: 0.28796 s(t) SNR: 6.43047 (dB) Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

a(t) s(t) Image Restoration by Discrete Gaussian Graphical Model Original Image Q=8 a(t) Degraded Image s=1 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. MSE:0.98893 SNR:1.89127 (dB) Belief Propagation MSE: 0.37697 s(t) SNR: 6.07987 (dB) Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

a(t) s(t) Image Restoration by Discrete Gaussian Graphical Model Original Image Q=8 a(t) Degraded Image s=1 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. MSE:0.98893 SNR:0.93517 (dB) Belief Propagation MSE: 0.32060 s(t) SNR: 5.82715 (dB) Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Binary Image Restoration by Discrete Gaussian Graphical Model Maximization of Marginal Likelihood MSE p=0.1 0.07072 0.28981 0.07902 p=0.2 0.11864 0.38206 0.17184 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. Binary Symmetric Channel MSE p=0.1 0.04761 0.41097 0.04857 p=0.2 0.08319 0.39216 0.13290 Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Binary Image Restoration by Discrete Gaussian Graphical Model Binary Symmetric Channel Degraded Image Original Image Restored Image Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Color Image Restorations by Gaussian Graphical Model Original Image Degraded Image Restored Image Maximization of Marginal Likelihood Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Color Image Restorations by Gaussian Graphical Model Original Image Degraded Image Restored Image Maximization of Marginal Likelihood Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Contents Probabilistic Image Processing Loopy Belief Propagation Statistical Learning Algorithm More Practical Probabilistic models for Image Processing Summary Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Compound Gauss Markov Random Field Model Prior Information for Line Fields V(u) Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Compound Gauss Markov Random Field Model Original Image Degraded Image Gaussian Graphical Model (No Edge States) Compound Gauss Markov Random Field Model with Edge States Compound Gauss Markov Random Field Model with Quantized Edge States Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Compound Gauss Markov Random Field Model Original Image Degraded Image Gaussian Graphical Model (No Edge States) Compound Gauss Markov Random Field Model with Edge States Compound Gauss Markov Random Field Model with Quantized Edge States Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Contents Probabilistic Image Processing Loopy Belief Propagation Statistical Learning Algorithm More Practical Probabilistic models for Image Processing Summary Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Summary Image Processing Bayesian Network for Image Processing Design of Belief Propagation Algorithm for Image Processing Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Practice 10-1 For random fields F=(F1,F2,…,F|V|)T and G=(G1,G2,…,G|V|)T and state vectors f=(f1,f2,…,f|V|)T and g=(g1,g2,…,g|V|)T, we consider the following posterior probability distribution: Here ZPosterior is a normalization constant and E is the set of all the nearest-neighbour pairs of nodes. Prove that and where ∂i denotes the set of all the nearest neighbour pixels of the pixel i. Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Give some numerical experiments for p=0.1 and 0.2. Practice 10-2 Make a program that generate a degraded image g=(g1,g2,…,g|V|) from a given binary image f =(f1,f2,…,f|V|) in which each state variable fi takes only 0 and 1 by according to the following conditional probability distribution of the binary symmetric channel: Give some numerical experiments for p=0.1 and 0.2. Example of Numerical Experiments for p=0.2 Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Give some numerical experiments for s=20 and 40. Practice 10-3 Make a program that generate a degraded image g=(g1,g2,…,g|V|) from a given image f =(f1,f2,…,f|V|) in which each state variable fi takes integers between 0 and q-1 by according to the following conditional probability distribution of the additive white Gaussian noise: Give some numerical experiments for s=20 and 40. Histogram of Gaussian Random Numbers Original Image f Gaussian Noise n Degraded Image g Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Practice 10-4 Make a program for estimating q-valued original images f =(f1,f2,…,f|V|) from a given degraded image g =(g1,g2,…,g|V|) by using a belief propagation algorithm for the following posterior probability distribution for q valued images: Give some numerical experiments. K. Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese) , Chapter 8 . M. Yasuda, S. Kataoka and K. Tanaka: Computer Vision and Image Processing 3 (Edited by Y. Yagi and H. Saito): Chapter 6. Stochastic Image Processing, pp.137-179, Advanced Communication Media CO., Ltd., December 2010 (in Japanese). Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)

Practice 10-5 Make a program for estimating original images f =(f1,f2,…,f|V|) from a given degraded image g =(g1,g2,…,g|V|) by using a belief propagation algorithm for the following posterior probability distribution based on the Gaussian graphical model: Give some numerical experiments. Kazuyuki Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese) , Chapter 8 and Appendix G. Physical Fluctuomatics / Applied Stochansic Process (Tohoku University)