Medial Representations

of Integrals in terms of Moment Integrals 3.4.5 Divergence Theorem for Fluxes with Discontinuities across the Medial Axis 3.4.6 Computing the Average Outward Flux for the Grassfire Flow 3.5 Global Structure of the Medial Axis 3.5.1 Graph Structure for Decomposition into Irreducible Medial Components 3.5.2 Graph Structure of a Single Irreducible Medial Component 3.5.3 Consequences for the Topology of the Medial Axis and Region 3.6 Summary Part II Algorithms 4. Skeletons Via Shocks of Boundary EvolutionKaleem Siddiqi and Sylvain Bouix and Jayant Shah 4.1 Overview 4.2 Optics, Mechanics and Hamilton-Jacobi Skeletons 4.2.1 Medial Loci and the Eikonal Equation 4.2.2 Hamiltonian Derivation of the Eikonal Equation 4.2.3 Divergence, Average Outward Flux and Object Angle 4.3 Homotopy Preserving Medial Loci 4.3.1 2D Simple Points 4.3.2 3D Simple Points 4.3.3 Average Outward Flux Ordered Thinning 4.3.4 The Algorithm and its Complexity 4.3.5 Labeling the Medial Set 4.3.6 Examples 4.4 An Object Angle Approach 4.4.1 Examples 4.5 Discussion and Conclusion Discrete Skeletons from Distance Transforms in 2D and 3DGunilla Borgefors and Ingela Nystrom and Gabriella Sanniti di Baja 5.1 Introduction 5.2 Definitions and Notions 5.3 Distance Transforms 5.3.1 2D Distance Transforms 5.3.2 3D Distance Transforms 5.3.3 Euclidean Distance Transforms 5.4 Centers of Maximal Disks/Balls 5.4.1 Centers of Maximal Disks 5.4.2 Centers of Maximal Balls 5.4.3 Reduced Set of Centers of Maximal Objects 5.4.4 Reverse Distance Transforms 5.4.5 Role of Centers of Maximal Objects in Skeletons 5.5 Skeletons of 2D Shapes 5.5.1 Computing the Nearly-thin 2D Skeleton 5.5.2 Post-processing, 2D case 5.6 Skeletons of 3D Shapes 5.6.1 Computing the Nearly-thin Surface Skeleton 5.6.2 Post-processing, Surface Skeleton 5.6.3 Computing the Nearly-thin Curve Skeleton 5.6.4 Post-processing, Curve Skeleton 5.7 Some Applications and Extensions 6. Voronoi SkeletonsGabor Szekely 6.1 The Voronoi Skeleton and Its Extraction in 2D 6.1.1 Basics 6.1.2 The boundary sampling problem 6.1.3 Generation of the Voronoi Diagram 6.1.4 From Voronoi Diagrams to skeletons 6.1.5 Topological organization of the 2D skeleton 6.1.6 The salience of 2D skeletal branches 6.1.7 Pruning the 2D Voronoi skeleton 6.1.8 A hierarchy of skeleton branches 6.2 The Voronoi Skeleton in 3D 6.2.1 3D Voronoi diagram generation 6.2.2 Topological organization of the 3D Voronoi skeleton 6.2.3 The salience of 3D skeletal branches 6.2.4 Pruning the 3D Voronoi skeleton 6.2.5 Interactive generation of skeletal hierarchy in 3D 6.3 Application examples 6.3.1 Skeletons of artificial 3D objects 6.3.2 Bone thickness characterization using skeletonization 6.3.3 Analysis of the cortical structure of the brain 6.4 Discussion 7. Voronoi Methods for 3D Medial Axis ApproximationNina Amenta and Sunghee Choi 7.1 Introduction 7.2 Approximating the Medial Axis 7.2.1 A Few 2D Results 7.2.2 Slivers 7.3 Sampling and approximation 7.3.1 Stable subsets of the medial axis 7.3.2 l-medial axis and uniform sampling 7.3.3 g-medial axis and scale-invariant sampling 7.4 Medial axis algorithms for input point clouds 7.4.1 Anti-crust 7.4.2 Thinning algorithms 7.4.3 Power shape 7.5 Medial axis algorithms for input surfaces 7.6 Discussion 8. Synthesis, Deformation, and Statistics of 3D Objects via M-repsStephen Pizer and Qiong Han and Sarang Joshi and P. Thomas Fletcher and Paul A. Yushkevich and Andrew Thall 8.1 Introduction 8.2 M-reps, Medial Atoms, and Figures 8.3 Object-relative coordinates 8.4 Figures, Subfigures, and Multi-Object Ensembles 8.5 Synthesis of Objects & Multi-Object Ensembles by Multiscale Figural Description 8.6 M-reps as Symmetric Spaces 8.7 The Statistical View of Objects 8.8 Discrete M-reps 8.9 Correspondence of Discrete M-reps in Families of Training Cases 8.10 Continuous M-Reps Via Splines or Other Basis Functions 8.11 Summary and Conclusion Part III Applications 9. Statistical Applications with Deformable M-RepsStephen Pizer and Martin Styner and Timothy Terriberry and Robert Broadhurst and Sarang Joshi and Edward Chaney and P. Thomas Fletcher 9.1 Introduction and Statistical Formulation 9.2 Segmentation by Posterior Optimization of Deformable M-reps: Overview 9.2.1 Segmentation Method: Posterior Optimization for Multiscale Deformation of Figurally Based Models 9.2.2 Segmentation method: user operation 9.3 Training and measuring statistical geometric typicality 9.3.1 M-rep model fitting and geometric statistics formation 9.3.2 measuring statistical geometric typicality 9.4 Training and measuring statistical geometry-to-image match 9.4.1 Transforming between figural and Euclidean coordinates 9.4.2 Geometry-to-image match via statistics on discrete regional quantile functions 9.5 Pablo Details and Results 9.5.1 The Voxel-Scale Stage of Segmentation 9.5.2 Evaluation of Segmentations 9.6 Hypothesis Testing for Localized Shape Differences between Groups 9.6.1 Tests in Euclidean space 9.6.2 Tests in Symmetric Spaces 9.7 Applications of Hypothesis Testing to Brain Structure Shape Differences in Neuro-Imaging 9.7.1 Hippocampus study in Schizophrenia 9.7.2 Lateral Ventricle Study of Healthy and Schizophrenic Twins 9.8 Discussion and Future Work 9.8.1 Are M-reps Effective? 9.8.2 Other M-rep Uses and Properties 10. 3D Model Retrieval Using Medial SurfacesKaleem Siddiqi and Juan Zhang and Diego Macrini and Sven Dickinson and Ali Shokoufandeh 10.1 Introduction 10.2 3D Model Retrieval 10.3 Medial Surfaces and DAGs 10.4 Indexing 10.5 Matching 10.6 Experimental Results 10.6.1 Matching Results 10.6.2 Indexing Results 10.7 Discussion and Conclusion 11. From The Infinitely Large to the Infinitely SmallFrederic F. Leymarie and Benjamin B. Kimia 11.1 Introduction 11.2 Formation and Description of Galaxies 11.3 Geography: Topography, Cartography, Networks 11.4 From Urbanism to Architecture and Archaeology 11.5 From Garden Layouts to the Genesis of Plants 11.6 Visual Arts: Painting, Drawing, Sculpting 11.7 Motion Analysis, Body Animation, Robotics 11.8 Machining, Metal Forging, Industrial Design, Object Registration 11.9 Medicine and Biology 11.10 Crystallography, Chemistry, Molecular Design 11.11 Perception and Cognition 11.12 Conclusion A Notation A.1 Common Notation A.2 Chapter 1 A.3 Chapter 2 A.4 Chapter 3 A.5 Chapter 4 A.6 Chapter 5 A.7 Chapter 6 A.8 Chapter 7 A.9 Chapter 9 A.10 Chapter 10 Glossary References