Variational and Level Set Methods in Image Segmentation (eBook)

Contents 4
1 INTRODUCTION 8
References 17
2 INTRODUCTORY BACKGROUND 21
2.1 Euler-Lagrange equations 21
2.1.1 Definite integrals 21
2.1.2 Variable domain of integration 23
2.2 Descent methods for unconstrained optimization 26
2.2.1 Real functions 26
2.2.2 Integral functionals 26
2.3 Level sets 28
2.4 Optical flow 31
2.4.1 The gradient equation 31
2.4.2 The Horn and Schunck formulation 32
2.4.3 The Aubert, Kornprobst, and Deriche formulation 34
2.4.4 Optical flow of rigid body motion 34
References 37
3 BASIC METHODS 38
3.1 The Mumford and Shah model 38
3.1.1 Bayesian interpretation 39
3.1.2 Graduated non convexity implementation 40
3.2 The minimum description length method of Leclerc 41
3.2.1 MDL and MAP 41
3.2.2 The piecewise constant image model 42
3.2.3 Numerical implementation 44
3.3 The region competition algorithm 45
3.3.1 Optimization 46
3.4 A level set formulation of the piecewise constant Mumford-Shah model 50
3.4.1 Curve evolution minimization of the Chan-Vese functional 51
3.4.2 Level set representation of curve evolution 53
3.4.3 Algorithm summary 54
3.4.4 Numerical implementation details of the level set evolution equation 55
3.5 Edge-based approaches 56
3.5.1 The Kass-Witkin-Terzopoulos Snakes model 56
3.5.2 The Geodesic active contour 57
3.5.3 Examples 59
References 62
4 MULTIREGION SEGMENTATION 64
4.1 Introduction 64
4.2 Multiregion segmentation using a partition constraint functional term 66
4.3 Multiphase level set image segmentation 67
4.4 Level set multiregion competition 71
4.4.1 Representation of a partition into a fixed but arbitrary number of regions 71
4.4.2 Curve evolution equations 72
4.4.3 Level set implementation 74
4.5 Multiregion level set segmentation as regularized clustering 75
4.5.1 Curve evolution equations 76
4.5.2 Level set implementation 78
4.6 Embedding a partition constraint directly in the minimization equations 79
4.6.1 Two-region segmentation: first order analysis 79
4.6.2 Extension to multiregion segmentation 81
4.6.3 Example 83
References 85
5 IMAGE MODELS 87
5.1 Introduction 87
5.2 Segmentation by maximizing the image likelihood 88
5.2.1 The Gaussian model 89
5.2.2 The Gamma image model 93
5.2.3 Generalization to distributions of the exponential family 95
5.2.4 The Weibull image Model 97
5.2.5 The Complex Wishart Model 99
5.2.6 MDL interpretation of the smoothness term coefficient 102
5.2.7 Generalization to multiregion segmentation 103
5.2.8 Examples 105
5.3 Maximization of the mutual information between the segmentation and the image 108
5.3.1 Curve evolution equation 110
5.3.2 Statistical interpretation 112
5.3.3 Algorithm summary 112
5.4 Segmentation by maximizing the discrepancy between the regions image distributions 113
5.4.1 Statistical interpretation 114
5.4.2 The kernel width 114
5.4.3 Algorithm summary 115
5.4.4 Example 115
5.5 Image segmentation using a region reference distribution 115
5.5.1 Statistical interpretation 117
5.5.2 Summary of the algorithms 118
5.5.3 Example 118
5.6 Segmentation with an overlap prior 118
5.6.1 Statistical interpretation 121
5.6.2 Example 121
References 124
6 REGION MERGING PRIORS 127
6.1 Introduction 127
6.2 Definition of a region merging prior 129
6.3 A minimum description length prior 130
6.4 An entropic region merging prior 130
6.4.1 Entropic interpretation 131
6.4.2 Segmentation functional 131
6.4.3 Minimization equations 132
6.4.4 A region merging interpretation of the level set evolution equations 134
6.4.5 The weight of the entropic prior 134
6.5 Example 136
6.5.1 Segmentation with the entropic region merging prior 136
6.5.2 Segmentation with the MDL region merging prior 137
6.5.3 Computation time 137
References 141
7 MOTION BASED IMAGE SEGMENTATION 142
7.1 Introduction 142
7.2 Piecewise constant MDL estimation and segmentation of optical flow 144
7.2.1 Numerical implementation 146
7.2.2 Example 148
7.3 Joint segmentation and linear parametric estimation of optical flow 148
7.3.1 Formulation 150
7.3.2 Functional minimization 154
7.3.3 Level set implementation 158
7.3.4 Multiregion segmentation 158
7.3.5 Examples 158
References 161
8 IMAGE SEGMENTATION ACCORDING TO THE MOVEMENT OF REAL OBJECTS 164
8.1 Introduction 164
8.2 The functionals 167
8.3 Minimization of E1 169
8.3.1 Minimization with respect to the screws of motion 169
8.3.2 Minimization with respect to depth 170
8.3.3 Minimization with respect to the active curve 170
8.3.4 Algorithm 171
8.3.5 Uncertainty of scale in 3D interpretation 171
8.3.6 Multiregion segmentation 172
8.3.7 Example 172
8.4 Minimization of E2 172
8.4.1 Minimization with respect to the essential parameter vectors 172
8.4.2 Minimization with respect to optical flow 174
8.4.3 Minimization with respect to 174
8.4.4 Recovery of regularized relative depth 174
8.4.5 Algorithm 175
8.4.6 Example 176
8.5 Minimization of E3 177
8.5.1 Example 178
References 181
9 APPENDIX 184
9.1 The Horn and Schunck optical flow estimation algorithm 184
9.1.1 Iterative resolution by the Jacobi and Gauss-Seidel iterations 186
9.1.2 Evaluation of derivatives 187
9.2 The Aubert, Deriche, and Kornprobst algorithm 187
9.3 Construction of stereoscopic images of a computed 3D interpretation 189
References 191
Index 192