Medial Representations: Mathematics, Algorithms and Applications

1. 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 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 Skeletons
Gabor 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 Approximation
Nina 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-reps
Stephen 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-Reps
Stephen 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 Surfaces
Kaleem 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 Small
Frederic 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