Medial Representations: Mathematics, Algorithms and Applications by Kaleem SiddiqiMedial Representations: Mathematics, Algorithms and Applications by Kaleem Siddiqi

Medial Representations: Mathematics, Algorithms and Applications

EditorKaleem Siddiqi, Stephen Pizer

Paperback | October 22, 2010

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The last half century has seen the development of many biological or physical theories that have explicitly or implicitly involved medial descriptions of objects and other spatial entities in our world. Simultaneously, mathematicians have studied the properties of these skeletal descriptions of shape, and, stimulated by the many areas where medial models are useful, computer scientists and engineers have developed numerous algorithms for computing and using these models.The book consists of an introductory chapter, two chapters on the major mathematical results on medial representations, five chapters on algorithms for extracting medial models from boundary or binary image descriptions of objects, and three chapters on applications in image analysis and other areas of study and design. This book will serve the science and engineering communities using medial models and will provide learning material for students entering this field.
Title:Medial Representations: Mathematics, Algorithms and ApplicationsFormat:PaperbackDimensions:458 pages, 9.25 × 6.1 × 0.03 inPublished:October 22, 2010Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048179467

ISBN - 13:9789048179466

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Table of Contents

1. IntroductionStephen Pizer and Kaleem Siddiqi and Paul Yushkevich1.1 Object representations1.2 Medial representations of objects1.2.1 The Definition of the Medial Locus1.2.2 Structural Geometry of Medial Loci 1.2.3 Local Geometry of Medial Loci1.2.4 Medial Atoms and M-Reps1.3 Psychophysical and Neurophysiological Evidence for Medial Loci1.4 Extracting Medial Loci of Objects1.4.1 Distance Transforms, the Hessian, Thinning and Pruning1.4.2 Skeletons via Shocks of Boundary Evolution1.4.3 Greyscale Skeletons1.4.4 Core Tracking1.4.5 Skeletons from Digital Distance Transforms1.4.6 Voronoi Skeletons1.4.7 Skeletonization by Deformable Medial Models1.5 Applications of medial loci in computer visionPart I Mathematics2. Local Forms and Transitions of the Medial AxisPeter J. Giblin and Benjamin B. Kimia2.1 Introduction2.2 Definitions2.3 Contact2.4 Local forms of the symmetry set and medial axis in 2D2.5 Local forms of the medial axis in 3D2.6 Local reconstruction from the symmetry set or medial axis in 2D2.7 Local reconstruction from the symmetry set or medial axis in 3D2.8 Symmetry sets and medial axes of families of curves2.9 Medial axes of families of surfaces2.10 Consistency conditions at branches2.11 Summary3.Geometry and Medial StructureJames Damon3.1 Introduction3.2 Medial Data on Skeletal Structures3.2.1 Blum Medial Axis and General Skeletal Structures3.2.2 Radial Flow Defined for a Skeletal Structure3.2.3 Radial and Edge Shape Operators for 1D and 2D Medial Structures3.2.4 Level Set Structure of a Region and Smoothness of the Boundary3.3 Local and Relative Geometry of the Boundary3.3.1 Intrinsic Differential Geometry of the Boundary3.3.2 Geometric Medial Map3.3.3 Deformations of Skeletal Structures and Boundary Smoothness and Geometry3.4 Global Geometry of a Region and its Boundary3.4.1 Skeletal and Medial Integrals3.4.2 Global Integrals as Skeletal and Medial Integrals3.4.3 Consequences for Global Geometry3.4.4 Expansion of Integrals in terms of Moment Integrals3.4.5 Divergence Theorem for Fluxes with Discontinuities across the Medial Axis3.4.6 Computing the Average Outward Flux for the Grassfire Flow3.5 Global Structure of the Medial Axis3.5.1 Graph Structure for Decomposition into Irreducible Medial Components3.5.2 Graph Structure of a Single Irreducible Medial Component3.5.3 Consequences for the Topology of the Medial Axis and Region3.6 SummaryPart II Algorithms4. Skeletons Via Shocks of Boundary EvolutionKaleem Siddiqi and Sylvain Bouix and Jayant Shah4.1 Overview4.2 Optics, Mechanics and Hamilton-Jacobi Skeletons4.2.1 Medial Loci and the Eikonal Equation4.2.2 Hamiltonian Derivation of the Eikonal Equation 4.2.3 Divergence, Average Outward Flux and Object Angle4.3 Homotopy Preserving Medial Loci4.3.1 2D Simple Points4.3.2 3D Simple Points4.3.3 Average Outward Flux Ordered Thinning4.3.4 The Algorithm and its Complexity4.3.5 Labeling the Medial Set4.3.6 Examples4.4 An Object Angle Approach4.4.1 Examples4.5 Discussion and ConclusionDiscrete Skeletons from Distance Transforms in 2D and 3DGunilla Borgefors and Ingela Nystrom and Gabriella Sanniti di Baja5.1 Introduction5.2 Definitions and Notions5.3 Distance Transforms5.3.1 2D Distance Transforms5.3.2 3D Distance Transforms5.3.3 Euclidean Distance Transforms5.4 Centers of Maximal Disks/Balls5.4.1 Centers of Maximal Disks5.4.2 Centers of Maximal Balls5.4.3 Reduced Set of Centers of Maximal Objects5.4.4 Reverse Distance Transforms5.4.5 Role of Centers of Maximal Objects in Skeletons5.5 Skeletons of 2D Shapes5.5.1 Computing the Nearly-thin 2D Skeleton5.5.2 Post-processing, 2D case5.6 Skeletons of 3D Shapes5.6.1 Computing the Nearly-thin Surface Skeleton5.6.2 Post-processing, Surface Skeleton5.6.3 Computing the Nearly-thin Curve Skeleton5.6.4 Post-processing, Curve Skeleton5.7 Some Applications and Extensions6. Voronoi SkeletonsGabor Szekely6.1 The Voronoi Skeleton and Its Extraction in 2D6.1.1 Basics6.1.2 The boundary sampling problem6.1.3 Generation of the Voronoi Diagram6.1.4 From Voronoi Diagrams to skeletons6.1.5 Topological organization of the 2D skeleton6.1.6 The salience of 2D skeletal branches6.1.7 Pruning the 2D Voronoi skeleton6.1.8 A hierarchy of skeleton branches6.2 The Voronoi Skeleton in 3D6.2.1 3D Voronoi diagram generation6.2.2 Topological organization of the 3D Voronoi skeleton6.2.3 The salience of 3D skeletal branches6.2.4 Pruning the 3D Voronoi skeleton6.2.5 Interactive generation of skeletal hierarchy in 3D6.3 Application examples6.3.1 Skeletons of artificial 3D objects6.3.2 Bone thickness characterization using skeletonization6.3.3 Analysis of the cortical structure of the brain6.4 Discussion7. Voronoi Methods for 3D Medial Axis ApproximationNina Amenta and Sunghee Choi7.1 Introduction7.2 Approximating the Medial Axis7.2.1 A Few 2D Results7.2.2 Slivers7.3 Sampling and approximation7.3.1 Stable subsets of the medial axis7.3.2 l-medial axis and uniform sampling7.3.3 g-medial axis and scale-invariant sampling7.4 Medial axis algorithms for input point clouds7.4.1 Anti-crust7.4.2 Thinning algorithms7.4.3 Power shape7.5 Medial axis algorithms for input surfaces7.6 Discussion8. 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 Thall8.1 Introduction8.2 M-reps, Medial Atoms, and Figures8.3 Object-relative coordinates8.4 Figures, Subfigures, and Multi-Object Ensembles8.5 Synthesis of Objects & Multi-Object Ensembles by Multiscale Figural Description8.6 M-reps as Symmetric Spaces8.7 The Statistical View of Objects8.8 Discrete M-reps8.9 Correspondence of Discrete M-reps in Families of Training Cases8.10 Continuous M-Reps Via Splines or Other Basis Functions8.11 Summary and ConclusionPart III Applications9. 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 Fletcher9.1 Introduction and Statistical Formulation9.2 Segmentation by Posterior Optimization of Deformable M-reps:Overview9.2.1 Segmentation Method: Posterior Optimization for Multiscale Deformation of Figurally Based Models9.2.2 Segmentation method: user operation9.3 Training and measuring statistical geometric typicality9.3.1 M-rep model fitting and geometric statistics formation9.3.2 measuring statistical geometric typicality9.4 Training and measuring statistical geometry-to-image match9.4.1 Transforming between figural and Euclidean coordinates9.4.2 Geometry-to-image match via statistics on discrete regional quantile functions9.5 Pablo Details and Results9.5.1 The Voxel-Scale Stage of Segmentation9.5.2 Evaluation of Segmentations9.6 Hypothesis Testing for Localized Shape Differences between Groups9.6.1 Tests in Euclidean space9.6.2 Tests in Symmetric Spaces9.7 Applications of Hypothesis Testing to Brain Structure Shape Differences in Neuro-Imaging9.7.1 Hippocampus study in Schizophrenia9.7.2 Lateral Ventricle Study of Healthy and Schizophrenic Twins9.8 Discussion and Future Work9.8.1 Are M-reps Effective?9.8.2 Other M-rep Uses and Properties10. 3D Model Retrieval Using Medial SurfacesKaleem Siddiqi and Juan Zhang and Diego Macrini and Sven Dickinson and Ali Shokoufandeh10.1 Introduction10.2 3D Model Retrieval10.3 Medial Surfaces and DAGs10.4 Indexing10.5 Matching10.6 Experimental Results10.6.1 Matching Results10.6.2 Indexing Results10.7 Discussion and Conclusion11. From The Infinitely Large to the Infinitely SmallFrederic F. Leymarie and Benjamin B. Kimia11.1 Introduction11.2 Formation and Description of Galaxies11.3 Geography: Topography, Cartography, Networks11.4 From Urbanism to Architecture and Archaeology11.5 From Garden Layouts to the Genesis of Plants11.6 Visual Arts: Painting, Drawing, Sculpting11.7 Motion Analysis, Body Animation, Robotics11.8 Machining, Metal Forging, Industrial Design, Object Registration11.9 Medicine and Biology11.10 Crystallography, Chemistry, Molecular Design11.11 Perception and Cognition11.12 ConclusionA NotationA.1 Common NotationA.2 Chapter 1A.3 Chapter 2A.4 Chapter 3A.5 Chapter 4A.6 Chapter 5A.7 Chapter 6A.8 Chapter 7A.9 Chapter 9A.10 Chapter 10GlossaryReferences