Solutions for Bayesian Markov random field estimation problems
نویسندگان
چکیده
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . 1.1 Markov Random Field Models 1 1.2 An Overview of Optimal Estimators and Approximations . . . . . . . 2 1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 Goals of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.2 An Overview of Cluster Approximations . . . . . . . . . . . . 3 . . . . . . . . . . 1.3.3 An Overview of Bethe Tree Approximations 5 . . . . . . . . . . . . . . . . . . . . 1.3.4 Organization of the Thesis 6 2 . LITERATURE REVIEW FOR MARI(OVRAND0M FIELDS AND BAYESIAN ESTIMATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 . . . . . . . . . . . . 2.1 Markov Random Fields and Gibbs Distributions 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Optimal Estimators 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stochastic Algorithms 14 . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Simulated Annealing 14 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 T P M and MPM 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Suboptimal Algorithms 18 . . . . . . . . . . . . . . . 2.4.1 Iterated Conditional Modes (ICM) 18 . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Mean Field Theory 19 . . . . . . 2.5 Mean Field Model and Bethe Tree in Statistical Mechanics 23 . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Mean Field Model 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Bethe Tree 25 . . . . . . . . . . . . . . . . . . . . . . . 3 . CLUSTER APPROXIMATIONS 29 . . . . . . . . . . . . . . . . . . 3.1 Derivation of Cluster Approximations 29 3.2 Theoretical Results Concerning 4 = f (4) . . . . . . . . . . . . . . . . 33 Page 3.3 An Algorithm for the Solution of 4 = f (4) . . . . . . . . . . . . . . . 38 3.4 Concrete Examples of Image Models . . . . . . . . . . . . . . . . . . 41 3.4.1 Pixel Processes without Line Fields . . . . . . . . . . . . . . . 42 3.4.2 Pixel Processes with Line Fields . . . . . . . . . . . . . . . . . 44 4 . BETHE TREE APPROXIMATIONS . . . . . . . . . . . . . . . . . . . . . 49 4.1 The Bethe Tree Approximation . . . . . . . . . . . . . . . . . . . . . 49 4.2 Theoretical Results on Fixed-Point Problems . . . . . . . . . . . . . . 54 4.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5 . NUMERICAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1 Statistical Performance: A Comparison . . . . . . . . . . . . . . . . . 61 5.1.1 Spatial Pattern Classification Problem . . . . . . . . . . . . . 62 5.1.2 Restoration Problem . . . . . . . . . . . . . . . . . . . . . . . 66 5.2 Synthetic Numerical Examples . . . . . . . . . . . . . . . . . . . . . . 73 5.2.1 Checkerboard Image . . . . . . . . . . . . . . . . . . . . . . . 73 5.2.2 Ternary Gray Levels and Line Fields . . . . . . . . . . . . . . 76 5.2.3 Text Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 Restoration Examples: Nonlinear Observation Models and Real Images 85 5.3.1 Nonlinear Observation Processes . . . . . . . . . . . . . . . . . 85 5.3.2 A Real Image . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.4 A Spatial Classification Example: Remote Sensing . . . . . . . . . . . 90 6 . CONCLUSIONS AND DIRECTIONS FOR FUTURE STUDY . . . . . . . 97 6.1 Summary of Our Main Results . . . . . . . . . . . . . . . . . . . . . . 97 6.2 Futurestudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2.1 Segmentation and Boundary Detection . . . . . . . . . . . . . 98 6.2.2 Halftoning and Inverse Halftoning . . . . . . . . . . . . . . . . 99 6.2.3 Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . 100 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
منابع مشابه
Curve and Surface Estimation using Dynamic Step Functions
This chapter describes a nonparametric Bayesian approach to the estimation of curves and surfaces that act as parameters in statistical models. The approach is based on mixing variable dimensional piecewise constant approximations, whose ‘smoothness’ is regulated by a Markov random field prior. Random partitions of the domain are defined by Voronoi tessellations of random generating point patte...
متن کاملInverse Problems in Imaging Systems and the General Bayesian Inversion Frawework
In this paper, first a great number of inverse problems which arise in instrumentation, in computer imaging systems and in computer vision are presented. Then a common general forward modeling for them is given and the corresponding inversion problem is presented. Then, after showing the inadequacy of the classical analytical and least square methods for these ill posed inverse problems, a Baye...
متن کاملLinear-Time Algorithm in Bayesian Image Denoising based on Gaussian Markov Random Field
In this paper, we consider Bayesian image denoising based on a Gaussian Markov random field (GMRF) model, for which we propose an new algorithm. Our method can solve Bayesian image denoising problems, including hyperparameter estimation, in O(n)-time, where n is the number of pixels in a given image. From the perspective of the order of the computational time, this is a state-of-the-art algorit...
متن کاملStatistical image segmentation using Triplet Markov fields
Hidden Markov fields (HMF) are widely used in image processing. In such models, the hidden random field of interest S s s X X ∈ = ) ( is a Markov field, and the distribution of the observed random field S s s Y Y ∈ = ) ( (conditional on X ) is given by ∏ ∈ = S s s s x y p x y p ) ( ) ( . The posterior distribution ) ( y x p is then a Markov distribution, which affords different Bayesian process...
متن کاملBayesian Image Restoration and Segmentationby Constrained
A constrained optimization method, called the Lagrange-Hoppeld (LH) method, is presented for solving Markov random eld (MRF) based Bayesian image estimation problems for restoration and segmentation. The method combines the augmented Lagrangian mul-tiplier technique with the Hoppeld network to solve a constrained optimization problem into which the original Bayesian estimation problem is reform...
متن کاملEvidence Estimation for Bayesian Partially Observed MRFs
Bayesian estimation in Markov random fields is very hard due to the intractability of the partition function. The introduction of hidden units makes the situation even worse due to the presence of potentially very many modes in the posterior distribution. For the first time we propose a comprehensive procedure to address one of the Bayesian estimation problems, approximating the evidence of par...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013