图像分析中的模型和逆问题

1 Introduction 1

1.1 About Modeling 3

1.1.1 Bayesian Approach 3

1.1.2 Inverse Problem 8

1.1.3 Energy-Based Formulation 10

1.1.4 Models 11

1.2 Structure of the Book 14

Ⅰ Spline Models 21

2 Nonparametric Spline Models 23

2.1 Definition 23

2.2 Optimization 26

2.2.1 Bending Spline 26

2.2.2 Spline Under Tension 28

2.2.3 Robustness 31

2.3 Bayesian Interpretation 34

2.4 Choice of Regularization Parameter 36

2.5 Approximation Using a Surface 39

2.5.1 L-Spline Surface 40

2.5.2 Quadratic Energy 43

2.5.3 Finite Element Optimization 46

3 Parametric Spline Models 51

3.1 Representation on a Basis of B-Splines 51

3.1.1 Approximation Spline 53

3.1.2 Construction of B-Splines 54

3.2 Extensions 57

3.2.1 Multidimensional Case 57

3.2.2 Heteroscedasticity 62

3.3 High-Dimensional Splines 67

3.3.1 Revealing Directions 68

3.3.2 Projection Pursuit Regression 70

4 Auto-Associative Models 75

4.1 Analysis of Multidimensional Data 75

4.1.1 A Classical Approach 76

4.1.2 Toward an Alternative Approach 80

4.2 Auto-Associative Composite Models 82

4.2.1 Model and Algorithm 82

4.2.2 Properties 84

4.3 Projection Pursuit and Spline Smoothing 86

4.3.1 Proiection Index 87

4.3.2 Spline Smoothing 90

4.4 Illustration 93

Ⅱ Markov Models 97

5 Fundamental Aspects 99

5.1 Definitions 99

5.1.1 Finite Markov Fields 100

5.1.2 Gibbs Fields 101

5.2 Markov-Gibbs Equivalence 103

5.3 Examples 106

5.3.1 Bending Energy 106

5.3.2 Bernoulli Energy 107

5.3.3 Gaussian Energy 108

5.4 Consistency Problem 109

6 Bayesian Estimation 113

6.1 Principle 113

6.2 Cost Functions 118

6.2.1 Cost Function Examples 119

6.2.2 Calculation Problems 121

7 Simulation and Optimization 123

7.1 Simulation 124

7.1.1 Homogeneous Markov Chain 124

7.1.2 Metropolis Dynamic 125

7.1.3 Simulated Gibbs Distribution 127

7.2 Stochastic Optimization 130

7.3 Probabilistic Aspects 134

7.4 Deterministic Optimization 138

7.4.1 ICM Algorithm 138

7.4.2 Relaxation Algorithms 141

8 Parameter Estimation 147

8.1 Complete Data 148

8.1.1 Maximum Likelihood 149

8.1.2 Maximum Pseudolikelihood 150

8.1.3 Logistic Estimation 153

8.2 Incomplete Data 156

8.2.1 Maximum Likelihood 157

8.2.2 Gibbsian EM Algorithm 161

8.2.3 Bayesian Calibration 170

Ⅲ Modeling in Action 175

9 Model-Building 177

9.1 Multiple Spline Approximation 177

9.1.1 Choice of Data and Image Characteristics 179

9.1.2 Definition of the Hidden Field 181

9.1.3 Building an Energy 183

9.2 Markov Modeling Methodology 185

9.2.1 Details for Implementation 185

10 Degradation in Imaging 189

10.1 Denoising 190

10.1.1 Models with Explicit Discontinuities 190

10.1.2 Models with Implicit Discontinuities 198

10.2 Deblurring 201

10.2.1 A Particularly Ill-Posed Problem 202

10.2.2 Model with Implicit Discontinuities 204

10.3 Scatter 205

10.3.1 Direct Problem 206

10.3.2 Inverse Problem 211

10.4 Sensitivity Functions and Image Fusion 216

10.4.1 A Restoration Problem 217

10.4.2 Transfer Function Estimation 221

10.4.3 Estimation of Stained Transfer Function 224

11 Detection of Filamentary Entities 227

11.1 Valley Detection Principle 228

11.1.1 Definitions 228

11.1.2 Bayes-Markov Formulation 230

11.2 Building the Prior Energy 231

11.2.1 Detection Term 231

11.2.2 Regularization Term 234

11.3 Optimization 236

11.4 Extension to the Case of an Image Pair 239

12 Reconstruction and Projections 243

12.1 Projection Model 243

12.1.1 Transmission Tomography 243

12.1.2 Emission Tomography 246

12.2 Regularized Reconstruction 247

12.2.1 Regularization with Explicit Discontinuities 248

12.2.2 Three-Dimensional Reconstruction 252

12.3 Reconstruction with a Single View 256

12.3.1 Generalized Cylinder 256

12.3.2 Training the Deformations 259

12.3.3 Reconstruction in the Presence of Occlusion 261

13 Matching 269

13.1 Template and Hidden Outline 270

13.1.1 Rigid Transformations 270

13.1.2 Spline Model of a Template 272

13.2 Elastic Deformations 276

13.2.1 Continuous Random Fields 276

13.2.2 Probabilistic Aspects 282

References 289

Author Index 301

Subject Index 305