Geometric and Topological Mesh Feature Extraction for 3d Shape Analysis

Preface ix

Introduction xi

Chapter 1. Geometric Features based on Curvatures 1

1.1. Introduction 1

1.2. Some mathematical reminders of the differential geometry of surfaces 2

1.2.1. Fundamental forms and normal curvature 2

1.2.2. Principal curvatures and shape index 5

1.2.3. Principal directions and lines of curvature 6

1.2.4. Weingarten equations and shape operator 9

1.2.5. Practical computation of differential parameters 12

1.2.6. Euler’s theorem 13

1.2.7. Meusnier’s theorem 15

1.2.8. Local approximation of the surface 16

1.2.9. Focal surfaces 17

1.3. Computation of differential parameters on a discrete 3D mesh 19

1.3.1. Introduction 19

1.3.2. Some notations 19

1.3.3. Computing normal vectors 20

1.3.4. Locally fitting a parametric surface 22

1.3.5. Discrete differential geometry operators 22

1.3.6. Integrating 2D curvatures 28

1.3.7. Tensor of curvature: Taubin’s formula 28

1.3.8. Tensor of curvature based on the normal cycle theory 30

1.3.9. Integral estimators 34

1.3.10. Processing unstructured 3D point clouds 38

1.3.11. Discussion of the methods 38

1.4. Feature line extraction 46

1.4.1. Introduction 46

1.4.2. Lines of curvature 47

1.4.3. Crest/ridge lines 55

1.4.4. Feature lines based on homotopic thinning 79

1.5. Region-based approaches 84

1.5.1. Mesh segmentation 84

1.5.2. Shape description based on graphs 87

1.6. Conclusion 98

Chapter 2. Topological Features 99

2.1. Mathematical background 99

2.1.1. A topological view on surfaces 100

2.1.2. Algebraic topology 103

2.2. Computation of global topological features 106

2.2.1. Connected components and genus 106

2.2.2. Homology groups 107

2.3. Combining geometric and topological features 111

2.3.1. Persistent homology 112

2.3.2. Reeb graph and Morse–Smale complex 115

2.3.3. Homology generators 118

2.3.4. Measuring holes 121

2.4. Conclusion 128

Chapter 3. Applications 131

3.1. Introduction 131

3.2. Medicine: lines of curvature for polyp detection in virtual colonoscopy 131

3.3. Paleo-anthropology: crest/ridge lines for shape analysis of human fossils 133

3.4. Geology: extraction of fracture lines on virtual outcrops 137

3.5. Planetary science: detection of feature lines for the extraction of impact craters on asteroids and rocky planets 140

3.6. Botany: persistent homology to recover the branching structure of plants 143

Conclusion 145

References 149

Index 169

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