Jean-Luc Mari is a Professor of Computer Science at Aix-Marseille University, France. His research interests relate to the extraction of information within meshes and other geometric representations. He is involved in various projects connected with planetary science, biology and computer-aided design and manufacturing.Franck Hétroy-Wheeler is a Professor of Computer Science at the University of Strasbourg, France. Working within the framework of shape analysis and understanding, his research interests focus on the acquisition, reconstruction and processing of virtual 3D shapes.Gérard Subsol is a CNRS researcher within the ICAR team at the Montpellier Laboratory of Informatics, Robotics and Microelectronics (LIRMM) in France. His work on mesh modeling has a variety of applications in anatomy, computer-aided design, environmental studies and paleoanthropology.

• Preface ixIntroduction xiChapter 1. Geometric Features based on Curvatures 11.1. Introduction 11.2. Some mathematical reminders of the differential geometry of surfaces 21.2.1. Fundamental forms and normal curvature 21.2.2. Principal curvatures and shape index 51.2.3. Principal directions and lines of curvature 61.2.4. Weingarten equations and shape operator 91.2.5. Practical computation of differential parameters 121.2.6. Euler’s theorem 131.2.7. Meusnier’s theorem 151.2.8. Local approximation of the surface 161.2.9. Focal surfaces 171.3. Computation of differential parameters on a discrete 3D mesh 191.3.1. Introduction 191.3.2. Some notations 191.3.3. Computing normal vectors 201.3.4. Locally fitting a parametric surface 221.3.5. Discrete differential geometry operators 221.3.6. Integrating 2D curvatures 281.3.7. Tensor of curvature: Taubin’s formula 281.3.8. Tensor of curvature based on the normal cycle theory 301.3.9. Integral estimators 341.3.10. Processing unstructured 3D point clouds 381.3.11. Discussion of the methods 381.4. Feature line extraction 461.4.1. Introduction 461.4.2. Lines of curvature 471.4.3. Crest/ridge lines 551.4.4. Feature lines based on homotopic thinning 791.5. Region-based approaches 841.5.1. Mesh segmentation 841.5.2. Shape description based on graphs 871.6. Conclusion 98Chapter 2. Topological Features 992.1. Mathematical background 992.1.1. A topological view on surfaces 1002.1.2. Algebraic topology 1032.2. Computation of global topological features 1062.2.1. Connected components and genus 1062.2.2. Homology groups 1072.3. Combining geometric and topological features 1112.3.1. Persistent homology 1122.3.2. Reeb graph and Morse–Smale complex 1152.3.3. Homology generators 1182.3.4. Measuring holes 1212.4. Conclusion 128Chapter 3. Applications 1313.1. Introduction 1313.2. Medicine: lines of curvature for polyp detection in virtual colonoscopy 1313.3. Paleo-anthropology: crest/ridge lines for shape analysis of human fossils 1333.4. Geology: extraction of fracture lines on virtual outcrops 1373.5. Planetary science: detection of feature lines for the extraction of impact craters on asteroids and rocky planets 1403.6. Botany: persistent homology to recover the branching structure of plants 143Conclusion 145References 149Index 169

Jean-Luc Mari is a Professor of Computer Science at Aix-Marseille University, France. His research interests relate to the extraction of information within meshes and other geometric representations. He is involved in various projects connected with planetary science, biology and computer-aided design and manufacturing.Franck Hétroy-Wheeler is a Professor of Computer Science at the University of Strasbourg, France. Working within the framework of shape analysis and understanding, his research interests focus on the acquisition, reconstruction and processing of virtual 3D shapes.Gérard Subsol is a CNRS researcher within the ICAR team at the Montpellier Laboratory of Informatics, Robotics and Microelectronics (LIRMM) in France. His work on mesh modeling has a variety of applications in anatomy, computer-aided design, environmental studies and paleoanthropology.

• Preface ixIntroduction xiChapter 1. Geometric Features based on Curvatures 11.1. Introduction 11.2. Some mathematical reminders of the differential geometry of surfaces 21.2.1. Fundamental forms and normal curvature 21.2.2. Principal curvatures and shape index 51.2.3. Principal directions and lines of curvature 61.2.4. Weingarten equations and shape operator 91.2.5. Practical computation of differential parameters 121.2.6. Euler’s theorem 131.2.7. Meusnier’s theorem 151.2.8. Local approximation of the surface 161.2.9. Focal surfaces 171.3. Computation of differential parameters on a discrete 3D mesh 191.3.1. Introduction 191.3.2. Some notations 191.3.3. Computing normal vectors 201.3.4. Locally fitting a parametric surface 221.3.5. Discrete differential geometry operators 221.3.6. Integrating 2D curvatures 281.3.7. Tensor of curvature: Taubin’s formula 281.3.8. Tensor of curvature based on the normal cycle theory 301.3.9. Integral estimators 341.3.10. Processing unstructured 3D point clouds 381.3.11. Discussion of the methods 381.4. Feature line extraction 461.4.1. Introduction 461.4.2. Lines of curvature 471.4.3. Crest/ridge lines 551.4.4. Feature lines based on homotopic thinning 791.5. Region-based approaches 841.5.1. Mesh segmentation 841.5.2. Shape description based on graphs 871.6. Conclusion 98Chapter 2. Topological Features 992.1. Mathematical background 992.1.1. A topological view on surfaces 1002.1.2. Algebraic topology 1032.2. Computation of global topological features 1062.2.1. Connected components and genus 1062.2.2. Homology groups 1072.3. Combining geometric and topological features 1112.3.1. Persistent homology 1122.3.2. Reeb graph and Morse–Smale complex 1152.3.3. Homology generators 1182.3.4. Measuring holes 1212.4. Conclusion 128Chapter 3. Applications 1313.1. Introduction 1313.2. Medicine: lines of curvature for polyp detection in virtual colonoscopy 1313.3. Paleo-anthropology: crest/ridge lines for shape analysis of human fossils 1333.4. Geology: extraction of fracture lines on virtual outcrops 1373.5. Planetary science: detection of feature lines for the extraction of impact craters on asteroids and rocky planets 1403.6. Botany: persistent homology to recover the branching structure of plants 143Conclusion 145References 149Index 169

Jean-Luc Mari is a Professor of Computer Science at Aix-Marseille University, France. His research interests relate to the extraction of information within meshes and other geometric representations. He is involved in various projects connected with planetary science, biology and computer-aided design and manufacturing.Franck Hétroy-Wheeler is a Professor of Computer Science at the University of Strasbourg, France. Working within the framework of shape analysis and understanding, his research interests focus on the acquisition, reconstruction and processing of virtual 3D shapes.Gérard Subsol is a CNRS researcher within the ICAR team at the Montpellier Laboratory of Informatics, Robotics and Microelectronics (LIRMM) in France. His work on mesh modeling has a variety of applications in anatomy, computer-aided design, environmental studies and paleoanthropology.

• Preface ixIntroduction xiChapter 1. Geometric Features based on Curvatures 11.1. Introduction 11.2. Some mathematical reminders of the differential geometry of surfaces 21.2.1. Fundamental forms and normal curvature 21.2.2. Principal curvatures and shape index 51.2.3. Principal directions and lines of curvature 61.2.4. Weingarten equations and shape operator 91.2.5. Practical computation of differential parameters 121.2.6. Euler’s theorem 131.2.7. Meusnier’s theorem 151.2.8. Local approximation of the surface 161.2.9. Focal surfaces 171.3. Computation of differential parameters on a discrete 3D mesh 191.3.1. Introduction 191.3.2. Some notations 191.3.3. Computing normal vectors 201.3.4. Locally fitting a parametric surface 221.3.5. Discrete differential geometry operators 221.3.6. Integrating 2D curvatures 281.3.7. Tensor of curvature: Taubin’s formula 281.3.8. Tensor of curvature based on the normal cycle theory 301.3.9. Integral estimators 341.3.10. Processing unstructured 3D point clouds 381.3.11. Discussion of the methods 381.4. Feature line extraction 461.4.1. Introduction 461.4.2. Lines of curvature 471.4.3. Crest/ridge lines 551.4.4. Feature lines based on homotopic thinning 791.5. Region-based approaches 841.5.1. Mesh segmentation 841.5.2. Shape description based on graphs 871.6. Conclusion 98Chapter 2. Topological Features 992.1. Mathematical background 992.1.1. A topological view on surfaces 1002.1.2. Algebraic topology 1032.2. Computation of global topological features 1062.2.1. Connected components and genus 1062.2.2. Homology groups 1072.3. Combining geometric and topological features 1112.3.1. Persistent homology 1122.3.2. Reeb graph and Morse–Smale complex 1152.3.3. Homology generators 1182.3.4. Measuring holes 1212.4. Conclusion 128Chapter 3. Applications 1313.1. Introduction 1313.2. Medicine: lines of curvature for polyp detection in virtual colonoscopy 1313.3. Paleo-anthropology: crest/ridge lines for shape analysis of human fossils 1333.4. Geology: extraction of fracture lines on virtual outcrops 1373.5. Planetary science: detection of feature lines for the extraction of impact craters on asteroids and rocky planets 1403.6. Botany: persistent homology to recover the branching structure of plants 143Conclusion 145References 149Index 169