《Markov随机场模型在计算机视觉中的应用》第三版

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《马尔可夫随机场在计算机视觉中的应用》是Stan Li博士编著的第三版经典著作，属于Advances in Pattern Recognition系列的一部分，该系列由Springer出版社发行，旨在探索和提升图像分析领域的理论与实践。本书详细探讨了马尔可夫随机场（Markov Random Field, MRF）在计算机视觉中的关键作用，这是一种强大的概率模型，常用于处理各种视觉任务，如图像分割、物体识别、场景理解等。马尔可夫随机场是一种统计建模方法，其基本思想是将图像像素或区域之间的关系建构成一个概率图模型，其中每个节点代表一个变量，而边则表示变量之间的依赖性。MRF通过定义潜在函数（能量函数），对图像数据的局部一致性（局部像素间的相似性）和全局一致性（整个图像的整体结构）进行建模。在计算机视觉中，MRF的特性使得它们能够有效地捕捉到视觉数据的复杂结构，如边缘、纹理和形状等特征。在本书中，作者不仅深入介绍了MRF的基本概念和原理，还涵盖了实际应用中的一些关键算法和技术，如最大似然估计、条件随机场（Conditional Random Fields, CRF）的优化方法，以及MRF在图像分割中的图形割（Graph Cuts）算法。此外，书中还提供了丰富的实例和案例研究，展示了MRF如何解决诸如行人检测、人脸识别、纹理合成等典型问题，并讨论了MRF与其他视觉技术（如深度学习）的结合可能性。对于想要进一步了解和掌握马尔可夫随机场在计算机视觉领域的读者，这本书是不可或缺的参考资料。它不仅适合研究生和研究人员，也适用于工程师和开发人员，希望利用这些模型来增强计算机视觉系统的性能。同时，对于计算机视觉领域的课程教材，这本书也是一个理想的补充材料，可以帮助学生理解复杂的概率模型及其在实际问题中的应用。《马尔可夫随机场在图像分析中的模型》第三版是一本综合了理论深度和实践指导的宝贵资源，为读者提供了从基础到进阶的学习路径，推动了计算机视觉领域的发展和创新。

xx Contents

2.1.4 Markov-Gibbs Equivalence ............... 28

2.1.5 Normalized and Canonical Forms ............ 29

2.2 Auto-models ........................... 30

2.3 Multi-level Logistic Model .................... 32

2.4 The Smoothness Prior ...................... 34

2.5 HierarchicalGRFModel..................... 37

2.6 TheFRAMEModel ....................... 37

2.7 Multiresolution MRF Modeling ................. 40

2.8 Conditional Random Fields ................... 43

2.9 Discriminative Random Fields .................. 44

2.10 Strong MRF Model ........................ 45

2.11 K-MRF and Nakagami-MRF Models .............. 46

2.12 Graphical Models: MRF’s versus

BayesianNetworks ........................ 47

3 Low-Level MRF Models 49

3.1 Observation Models ....................... 50

3.2 Image Restoration and Reconstruction ............. 51

3.2.1 MRF Priors for Image Surfaces ............. 51

3.2.2 Piecewise Constant Restoration ............. 54

3.2.3 Piecewise Continuous Restoration ........... 56

3.2.4 Surface Reconstruction .................. 58

3.3 Edge Detection . ......................... 60

3.3.1 Edge Labeling Using Line Process ........... 61

3.3.2 Forbidden Edge Patterns ................ 63

3.4 TextureSynthesisandAnalysis................. 65

3.4.1 MRF Texture Modeling ................. 65

3.4.2 Texture Segmentation .................. 69

3.5 OpticalFlow ........................... 71

3.5.1 Variational Approach .................. 71

3.5.2 Flow Discontinuities ................... 73

3.6 StereoVision ........................... 74

3.7 Spatio-temporal Models ..................... 76

3.8 BayesianDeformableModels .................. 78

3.8.1 Formulation of EigenSnake ............... 80

3.8.2 Experiments ....................... 86

4 High-Level MRF Models 91

4.1 Matching under Relational Constraints ............ 91

4.1.1 Relational Structure Representation .......... 92

4.1.2 Work in Relational Matching .............. 96

4.2 Feature-BasedMatching..................... 98

4.2.1 Posterior Probability and Energy ............ 99

4.2.2 Matching to Multiple Objects ............. 101

4.2.3 Extensions ........................ 103

Contents xxi

4.2.4 Experiments ....................... 105

4.3 Optimal Matching to Multiple

OverlappingObjects ....................... 113

4.3.1 Formulation of MAP-MRF Estimation ......... 113

4.3.2 Computational Issues .................. 117

4.4 Pose Computation ........................ 121

4.4.1 Pose Clustering and Estimation ............. 121

4.4.2 Simultaneous Matching and Pose Estimation . . . . . 124

4.4.3 Discussion ......................... 127

4.5 Face Detection and Recognition ................. 127

5 Discontinuities in MRF’s 129

5.1 Smoothness, Regularization,

and Discontinuities ........................ 130

5.1.1 Regularization and Discontinuities ........... 131

5.1.2 Other Regularization Models .............. 135

5.2 TheDiscontinuityAdaptiveMRFModel ........... 136

5.2.1 Deﬁning the DA Model ................. 136

5.2.2 Relations with Previous Models ............. 141

5.2.3 Discrete Data and 2D Cases ............... 142

5.2.4 Solution Stability ..................... 143

5.2.5 Computational Issues .................. 144

5.3 Total Variation Models ...................... 146

5.3.1 Total Variation Norm .................. 147

5.3.2 TV Models ........................ 147

5.3.3 Multichannel TV ..................... 150

5.4 Modeling Roof Discontinuities .................. 151

5.4.1 Roof-Edge Model ..................... 152

5.4.2 MAP-MRF Solution ................... 154

5.4.3 Computational Issues .................. 155

5.5 ExperimentalResults....................... 156

5.5.1 Step-Edge-Preserving Smoothing ............ 156

5.5.2 Roof-Edge-Preserving Smoothing ............ 157

6 MRF Model with Robust Statistics 161

6.1 The DA Prior and Robust Statistics .............. 162

6.1.1 Robust M-Estimator ................... 163

6.1.2 Problems with M-Estimator ............... 165

6.1.3 Redeﬁnition of M-Estimator ............... 166

6.1.4 AM-Estimator ...................... 167

6.1.5 Convex Priors for DA and M-Estimation ........ 168

6.1.6 Half-Quadratic Minimization .............. 170

6.2 Experimental Comparison ................... 173

6.2.1 Location Estimation ................... 173

6.2.2 Rotation Angle Estimation ............... 177

xxii Contents

7 MRF Parameter Estimation 183

7.1 Supervised Estimation with Labeled Data ........... 184

7.1.1 Maximum Likelihood ................... 184

7.1.2 Pseudo-likelihood ..................... 188

7.1.3 Coding Method ..................... 188

7.1.4 Mean Field Approximations ............... 190

7.1.5 Least Squares Fit ..................... 191

7.1.6 Markov Chain Monte Carlo Methods .......... 194

7.1.7 Learning in the FRAME Model ............. 198

7.2 Unsupervised Estimation

with Unlabeled Data ....................... 199

7.2.1 Simultaneous Restoration and Estimation ....... 200

7.2.2 Simultaneous Segmentation and Estimation . . . . . 202

7.2.3 Expectation-Maximization ................ 206

7.2.4 Cross Validation ..................... 208

7.3 Estimating the Number of MRF’s ................ 210

7.3.1 Akaike Information Criterion (AIC) .......... 211

7.3.2 Reversible Jump MCMC ................. 211

7.4 Reduction of Nonzero Parameters ................ 213

8 Parameter Estimation in Optimal Object Recognition 215

8.1 Motivation . . . ......................... 215

8.2 Theory of Parameter Estimation

for Recognition . ......................... 217

8.2.1 Optimization-Based Object Recognition ........ 218

8.2.2 Criteria for Parameter Estimation ........... 219

8.2.3 Linear Classiﬁcation Function .............. 222

8.2.4 A Nonparametric Learning Algorithm ......... 225

8.2.5 Reducing Search Space .................. 227

8.3 Application in MRF Object Recognition ............ 228

8.3.1 Posterior Energy ..................... 228

8.3.2 Energy in Linear Form .................. 229

8.3.3 How the Minimal Conﬁguration Changes ....... 230

8.3.4 Parametric Estimation under Gaussian Noise . . . . . 232

8.4 Experiments ........................... 234

8.4.1 Recognition of Line Patterns .............. 234

8.4.2 Recognition of Curved Objects ............. 238

8.4.3 Convergence ....................... 240

8.5 Conclusion ............................ 241

9 Minimization – Local Methods 243

9.1 Problem Categorization ..................... 243

9.2 Classical Minimization

with Continuous Labels ..................... 246

9.3 Minimization with Discrete Labels ............... 247

Chapter 1

Introduction

Modeling problems in this book are addressed mainly from the computational

viewpoint. The primary concerns are how to deﬁne an objective function for

the optimal solution to a image analysis or computer vision problem and how

to ﬁnd the optimal solution. The solution is deﬁned in an optimization sense

because the perfect solution is diﬃcult to ﬁnd due to various uncertainties in

the process, so we usually look for an optimal one that optimizes an objective

in which constraints are encoded.

Contextual constraints are ultimately necessary in the interpretation of vi-

sual information. A scene is understood in the spatial and visual contexts; the

objects are recognized in the context of object features in a lower-level repre-

sentation; the object features are identiﬁed based on the context of primitives

at an even lower-level; and the primitives are extracted in the context of im-

age pixels at the lowest level of abstraction. The use of contextual constraints

is indispensable for a capable vision system.

Markov random ﬁeld (MRF) theory provides a convenient and consistent

way of modeling context-dependent entities such as image pixels and cor-

related features. This is achieved through characterizing mutual inﬂuences

among such entities using conditional MRF distributions. The practical use of

MRF models is largely ascribed to a theorem stating the equivalence between

MRF’s and Gibbs distributions that was established by Hammersley and Clif-

ford (1971) and further developed by Besag (1974). This is because the joint

distribution is required in most applications but deriving the joint distribu-

tion from conditional distributions turns out to be very diﬃcult for MRF’s.

The MRF-Gibbs equivalence theorem points out that the joint distribution of

an MRF is a Gibbs distribution, the latter taking a simple form. This gives us

not only a mathematically sound but also mathematically tractable means for

statistical image analysis (Grenander 1983; Geman and Geman 1984). From

the computational perspective, the local property of MRF’s leads to algo-

rithms that can be implemented in a local and massively parallel manner.

S.Z. Li, Markov Random Field Modeling in Image Analysis,1

Advances in Pattern Recognition, DOI: 10.1007/978-1-84800-279-1

 Springer-Verlag London Limited 2009

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