This second edition of G. Winkler's successful book on random field approaches to image analysis, related Markov Chain Monte Carlo methods, and statistical inference with emphasis on Bayesian image analysis concentrates more on general principles and models and less on details of concrete applications. Addressed to students and scientists from mathematics, statistics, physics, engineering, and computer science, it will serve as an introduction to the mathematical aspects rather than a survey. Basically no prior knowledge of mathematics or statistics is required. The second edition is in many parts completely rewritten and improved, and most figures are new. The topics of exact sampling and global optimization of likelihood functions have been added. Cover Applications of Mathematics, vol. 27 Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction (Second Edition) Copyright Dedication Preface to the Second Edition Preface to the First Edition Contents Introduction Part I. Bayesian Image Analysis: Introduction 1. The Bayesian Paradigm 1.1 Warming up for Absolute Beginners 1.2 Images and Observations 1.3 Prior and Posterior Distributions 1.4 Bayes Estimators 2. Cleaning Dirty Pictures 2.1 Boundaries and Their Information Content 2.2 Towards Piecewise Smoothing 2.3 Filters, Smoothers, and Bayes Estimators 2.4 Boundary Extraction 2.5 Dependence on Hyperparameters 3. Finite Random Fields 3.1 Markov Random Fields 3.2 Gibbs Fields and Potentials 3.3 Potentials Continued Part II. The Gibbs Sampler and Simulated Annealing 4. Markov Chains: Limit Theorems 4.1 Preliminaries 4.2 The Contraction Coefficient 4.3 Homogeneous Markov Chains 4.4 Exact Sampling 4.5 Inhomogeneous Markov Chains 4.6 A Law of Large Numbers for Inhomogeneous Chains 4.7 A Counterexample for the Law of Large Numbers 5. Gibbsian Sampling and Annealing 5.1 Sampling 5.2 Simulated Annealing 5.3 Discussion ± 6. Cooling Schedules 6.1 The ICM Algorithm 6.2 Exact MAP Estimation Versus Fast Cooling 6.3 Finite Time Annealing Part III. Variations of the Gibbs Sampler 7. Gibbsian Sampling and Annealing Revisited 7.1 A General Gibbs Sampler 7.2 Sampling and Annealing Under Constraints 8. Partially Parallel Algorithms 8.1 Synchronous Updating on Independent Sets 8.2 The Swendson-Wang Algorithm 9. Synchronous Algorithms 9.1 Invariant Distributions and Convergence 9.2 Support of the Limit Distribution 9.3 Synchronous Algorithms and Reversibility Part IV. Metropolis Algorithms and Spectral Methods 10. Metropolis Algorithms 10.1 Metropolis Sampling and Annealing 10.2 Convergence Theorems 10.3 Best Constants 10.4 About Visiting Schemes 10.5 Generalizations and Modifications 10.6 The Metropolis Algorithm in Combinatorial Optimization 11. The Spectral Gap and Convergence of Markov Chains 11.1 Eigenvalues of Markov Kernels 11.2 Geometric Convergence Rates 12. Eigenvalues, Sampling, Variance Reduction 12.1 Samplers and Their Eigenvalues 12.2 Variance Reduction 12.3 Importance Sampling 13. Continuous Time Processes 13.1 Discrete State Space 13.2 Continuous State Space Part V. Texture Analysis 14. Partitioning 14.1 How to Tell Textures Apart 14.2 Bayesian Texture Segmentation 14.3 Segmentation by a Boundary Model 14.4 Julesz's Conjecture and Two Point Processes 15. Random Fields and Texture Models 15.1 Neighbourhood Relations 15.2 Random Field Texture Models 15.3 Texture Synthesis 16. Bayesian Texture Classification 16.1 Contextual Classification 16.2 Marginal Posterior Modes Methods Part VI. Parameter Estimation 17. Maximum Likelihood Estimation 17.1 The Likelihood Function 17.2 Objective Functions 18. Consistency of Spatial ML Estimators 18.1 Observation Windows and Specifications 18.2 Pseudolikelihood Methods 18.3 Large Deviations and Full Maximum Likelihood 18.4 Partially Observed Data 19. Computation of Full ML Estimators 19.1 A Naive Algorithm 19.2 Stochastic Optimization for the Full Likelihood 19.3 Main Results 19.4 Error Decomposition 19.5 L2-Estimates Part VII. Supplement 20. A Glance at Neural Networks 20.1 Boltzmann Machines 20.2 A Learning Rule 21. Three Applications 21.1 Motion Analysis 21.2 Tomographic Image Reconstruction 21.3 Biological Shape Part VIII. Appendix A. Simulation of Random Variables A.1 Pseudorandom Numbers A.2 Discrete Random Variables A.3 Special Distributions B. Analytical Tools B.1 Concave Functions B.2 Convergence of Descent Algorithms B.3 A Discrete Gronwall Lemma B.4 A Gradient System C. Physical Imaging Systems D. The Software Package AntsInFields References Symbols Index
"This book is concerned with a probabilistic approach for image analysis, mostly from the Bayesian point of view, and the important Markov chain Monte Carlo methods commonly used....This book will be useful, especially to researchers with a strong background in probability and an interest in image analysis. The author has presented the theory with rigor...he doesn’t neglect applications, providing numerous examples of applications to illustrate the theory." — MATHEMATICAL REVIEWS
