The last half century has seen the development of many biological or physical t- ories 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. We bring this kno- edge and experience together into this book in order to make medial technology more widely understood and used. The book consists of an introductory chapter, two chapters on the major mat- matical results on medial representations, ?ve 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. We hope that this book will serve the science and engineering communities using medial models and will provide learning material for students entering this ?eld. We are fortunate to have recruited many of the world leaders in medial theory, algorithms, and applications to write chapters in this book. We thank them for their signi?cant effort in preparing their contributions. We have edited these chapters and have combined them with the ?ve chapters that we have written to produce an integrated whole.
Table of Contents1. Introduction
Stephen Pizer and Kaleem Siddiqi and Paul Yushkevich 1.1 Object representations 1.2 Medial representations of objects 1.2.1 The Definition of the Medial Locus 1.2.2 Structural Geometry of Medial Loci 1.2.3 Local Geometry of Medial Loci 1.2.4 Medial Atoms and M-Reps 1.3 Psychophysical and Neurophysiological Evidence for Medial Loci 1.4 Extracting Medial Loci of Objects 1.4.1 Distance Transforms, the Hessian, Thinning and Pruning 1.4.2 Skeletons via Shocks of Boundary Evolution 1.4.3 Greyscale Skeletons 1.4.4 Core Tracking 1.4.5 Skeletons from Digital Distance Transforms 1.4.6 Voronoi Skeletons 1.4.7 Skeletonization by Deformable Medial Models 1.5 Applications of medial loci in computer vision Part I Mathematics 2. Local Forms and Transitions of the Medial Axis
Peter J. Giblin and Benjamin B. Kimia 2.1 Introduction 2.2 Definitions 2.3 Contact 2.4 Local forms of the symmetry set and medial axis in 2D 2.5 Local forms of the medial axis in 3D 2.6 Local reconstruction from the symmetry set or medial axis in 2D 2.7 Local reconstruction from the symmetry set or medial axis in 3D 2.8 Symmetry sets and medial axes of families of curves 2.9 Medial axes of families of surfaces 2.10 Consistency conditions at branches 2.11 Summary 3.Geometry and Medial Structure
James Damon 3.1 Introduction 3.2 Medial Data on Skeletal Structures 3.2.1 Blum Medial Axis and General Skeletal Structures 3.2.2 Radial Flow Defined for a Skeletal Structure 3.2.3 Radial and Edge Shape Operators for 1D and 2D Medial Structures 3.2.4 Level Set Structure of a Region and Smoothness of the Boundary 3.3 Local and Relative Geometry of the Boundary 3.3.1 Intrinsic Differential Geometry of the Boundary 3.3.2 Geometric Medial Map 3.3.3 Deformations of Skeletal Structures and Boundary Smoothness and Geometry 3.4 Global Geometry of a Region and its Boundary 3.4.1 Skeletal and Medial Integrals 3.4.2 Global Integrals as Skeletal and Medial Integrals 3.4.3 Consequences for Global Geometry 3.4.4 Expansion 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 Evolution
Kaleem 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 3D
Gunilla 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 Eu