Etyngier, Patrick (2008) Statistical learning, shape manifolds and applications to image segmentation. PhD thesis, ENPC p.210.

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Abstract

Image segmentation with shape priors has received a lot of attention over the past few years. Most existing work focuses on a linearized shape space with small deformation modes around a mean shape, which is only relevant when considering similar shapes. In this thesis, we introduce a new framework that can handle more general shape priors. We model a category of shapes as a finite dimensional manifold, the shape prior manifold, which we analyze from the shape samples using dimensionality reduction techniques such as diffusion maps. An embedding function is then learned from the manifold. Unfortunately, this model does not provide an explicit projection operator onto the underlying shape manifold, and therefore, our work tackles this problem. Our solution is threefold. First, we propose different solutions to the out-of-sample problem and define three attracting forces directed towards the manifold. These forces can be used as projection operators onto the manifold:

² Projection towards the closest point

² Projection with the same embedding

² Projection at constant embedding Next, we introduce a shape prior term for the active contours/regions framework through a non-linear energy term designed to attract shapes towards the manifold. Finally, we describe a variational framework for manifold denoising. Results with real objects such as car silhouettes or anatomical structures show the potential of our method.

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Table of content

List of Figures 13

1 Introduction 19

1.1 Context - 19

1.1.1 On the image side - 19

1.1.2 On the statistical learning side - 20

1.2 Image segmentation and statistical learning - 22

1.3 Goal and organization of this dissertation - 23

2 Introduction (Version française) 31

2.1 Contexte - 31

2.1.1 L’image comme point de départ - 31

2.1.2 L’apprentissage statistique comme point de départ - 33

2.2 Segmentation d’image et apprentissage statistique - 34

2.3 But et organisation de ce manuscrit - 36

3 Shapes 41

3.1 Basic definitions for shapes - 43

3.2 Shape representations in practice - 44

3.2.1 Introduction - 44

3.2.2 Explicit form - 45

3.2.3 Implicit functions and distance functions - 48

3.3 Shape space, topology and distances - 49

3.3.1 Distances between shapes - 49

3.4 Deformation, shape manifolds & interpolation - 51

3.4.1 Introduction - 51

3.4.2 Shape gradient & Gâteaux derivatives - 51

3.4.3 Shape manifold interpolation - 55

4 Image segmentation 61

4.1 Introduction - 63

4.2 Image segmentation - 63

4.2.1 Edge detection - 63

4.2.2 Region-based / Pixel-grouping methods - 65

4.3 Segmentation with active contours - 70

4.3.1 Introduction - 70

4.3.2 Active contours: Snakes - 70

4.3.3 Within the Level Set framework - 71

4.4 Incorporating shape priors in active contours - 75

4.4.1 Introduction - 75

4.4.2 Learning linear shape priors - 75

4.4.3 Non linear shape priors - 77

5 Dimensionality reduction & manifold learning techniques 79

5.1 Maximum variance based methods - 81

5.1.1 PCA - Principal Component Analysis - 81

5.1.2 KPCA - Kernel Principal Component Analysis - 83

5.2 Distance-based methods - 88

5.2.1 MDS - Multi-Dimensional Scaling - 88

5.2.2 Isomap - 90

5.3 Laplacian-based methods - 92

5.3.1 Dimensionality reduction and Laplace-Beltrami operator on manifolds - 92

5.3.2 Discrete Laplace-Beltrami Operator - 93

5.3.3 Normalization & convergence - 95

5.3.4 LEM - Laplacian Eigenmaps - 96

5.3.5 DFM - Diffusion Maps - 97

5.4 Other manifold learning methods - 101

5.5 Estimating the dimension of the manifold - 102

6 Application of graph Laplacian to Interactive Image Retrieval 105

6.1 Introduction - 107

6.1.1 Related Work - 107

6.1.2 Motivation and Contribution - 108

6.2 Overview of the Search Process - 111

6.3 Graph Laplacian and Relevance Feedback - 113

6.3.1 s-Weighted Transductive Learner - 113

6.3.2 A Robust k-step random walk matrix - 114

6.3.3 Display Model - 116

6.4 Nyström Extension - 118

6.5 Performances - 119

6.5.1 Databases - 119

6.5.2 Benchmarking - 120

6.5.3 Comparison - 121

6.6 Conclusion - 121

7 New points & attracting forces 123

7.1 Out-of-sample problem - 125

7.1.1 Introduction - 125

7.1.2 First approach: embedding regularization - 125

7.1.3 Nyström Extensions - 126

7.2 Pre-image problem - 128

7.3 Attracting forces toward a manifold - 128

7.3.1 Introduction - 128

7.3.2 General assumptions and Delaunay triangulation - 129

7.3.3 Attracting force #1: closest projection - 129

7.3.4 Attracting force #2: same embedding - 132

7.3.5 Attracting force #3: constant embedding - 133

7.4 Conclusion - 135

8 Applications of attracting forces to manifold denoising and image segmentation with priors 139

8.1 Manifold Denoising - 141

8.2 Projections and shapes - 142

8.2.1 Attracting force #1, rectangle shape manifold & fish shape manifold - 143

8.2.2 Attracting force #2, cross shape manifold - 143

8.2.3 Attracting force #3, ventricle shape manifold - 145

8.3 Image segmentation with general non linear shape priors - 146

8.3.1 Fish shapes, attracting force #2 - 146

8.3.2 Surveillance : Cars, attracting force #3 - 146

8.3.3 Bio Medical Imaging : Ventricle Nuclei, attracting force #3 147

9 Conclusion 159

10 Conclusion (Version Française) 161

A Radon/Hough space for pose estimation 165

A.1 Introduction - 168

A.2 Feature detection, matching & tracking - 169

A.2.1 The Hough transform - 169

A.2.2 The Radon transform - 172

A.2.3 Tracking / Matching lines in the Radon space - 173

A.3 Inference from complete Hough/Radon space - 177

A.3.1 Objectives & Problem formulation - 177

A.3.2 3D-2D Line Relation through Boosting - 178

A.3.3 Line Inference & Pose estimation - 182

A.4 Conclusion & Discussion - 186

Bibliography 189

Publications of the author 213

Statistiques de consultation

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