The Statistical Theory of Shape

1 Introduction.- 1.1 Background of Shape Theory.- 1.2 Principles of Allometry.- 1.3 Defining and Comparing Shapes.- 1.4 A Few More Examples.- 1.4.1 A Simple Example in One Dimension.- 1.4.2 Dinosaur Trackways From Mt. Tom, Massachusetts.- 1.4.3 Bronze Age Post Mold Configurations in England.- 1.5 The Problem of Homology.- 1.6 Notes.- 1.7 Problems.- 2 Background Concepts and Definitions.- 2.1 Transformations on Euclidean Space.- 2.1.1 Properties of Sets.- 2.1.2 Affine Transformations.- 2.1.3 Orthogonal Transformations.- 2.1.4 Unitary Transformations.- 2.1.5 Singular Value Decompositions.- 2.1.6 Inner Products.- 2.1.7 Groups of Transformations.- 2.1.8 Euclidean Motions and Isometries.- 2.1.9 Similarity Transformations and the Shape of Sets.- 2.2 Differential Geometry.- 2.2.1 Homeomorphisms and Diffeomorphisms.- 2.2.2 Topological Spaces.- 2.2.3 Introduction to Manifolds.- 2.2.4 Topological and Differential Manifolds.- 2.2.5 An Introduction to Tangent Vectors.- 2.2.6 Tangent Vectors and Tangent Spaces.- 2.2.7 Metric Tensors and Riemannian Manifolds.- 2.2.8 Geodesic Paths and Geodesic Distance.- 2.2.9 Affine Connections.- 2.2.10 Example.- 2.2.11 New Manifolds From Old: Product Manifolds.- 2.2.12 New Manifolds From Old: Submanifolds.- 2.2.13 Derivatives of Functions between Manifolds.- 2.2.14 Example: The Sphere.- 2.2.15 Example: Real Projective Spaces.- 2.2.16 Example: Complex Projective Spaces.- 2.2.17 Example: Hyperbolic Half Spaces.- 2.3 Notes.- 2.4 Problems.- 3 Shape Spaces.- 3.1 The Sphere of Triangle Shapes.- 3.2 Complex Projective Spaces of Shapes.- 3.3 Landmarks in Three and Higher Dimensions.- 3.3.1 Introduction.- 3.3.2 Riemannian Submersions.- 3.4 Principal Coordinate Analysis.- 3.5 An Application of Principal Coordinate Analysis.- 3.6 Hyperbolic Geometries for Shapes.- 3.6.1 Singular Values and the Poincaré Plane.- 3.6.2 A Generalization into Higher Dimensions.- 3.6.3 Geodesic Distance in UT(2).- 3.6.4 The Geometry of Tetrahedral Shapes.- 3.7 Local Analysis of Shape Variation.- 3.7.1 Thin-Plate Splines.- 3.7.2 Local Anisotropy of Nonlinear Transformations.- 3.7.3 Another Measure of Local Shape Variation.- 3.8 Notes.- 3.9 Problems.- 4 Some Stochastic Geometry.- 4.1 Probability Theory on Manifolds.- 4.1.1 Sample Spaces and Sigma-Fields..- 4.1.2 Probabilities.- 4.1.3 Statistics on Manifolds.- 4.1.4 Induced Distributions on Manifolds.- 4.1.5 Random Vectors and Distribution Functions.- 4.1.6 Stochastic Independence.- 4.1.7 Mathematical Expectation.- 4.2 The Geometric Measure.- 4.2.1 Example: Surface Area on Spheres.- 4.2.2 Example: Volume in Hyperbolic Half Spaces.- 4.3 Transformations of Statistics.- 4.3.1 Jacobians of Diffeomorphisms.- 4.3.2 Change of Variables Formulas.- 4.4 Invariance and Isometries.- 4.4.1 Example: Isometries of Spheres.- 4.4.2 Example: Isometries of Real Projective Spaces.- 4.4.3 Example: Isometries of Complex Projective Spaces.- 4.5 Normal Statistics on Manifolds.- 4.5.1 Multivariate Normal Distributions.- 4.5.2 Helmert Transformations.- 4.5.3 Projected Normal Statistics on Spheres.- 4.6 Binomial and Poisson Processes.- 4.6.1 Uniform Distributions on Open Sets.- 4.6.2 Binomial Processes.- 4.6.3 Example: Binomial Processes of Lines.- 4.6.4 Poisson Processes.- 4.7 Poisson Processes in Euclidean Spaces.- 4.7.1 Nearest Neighbors in a Poisson Process.- 4.7.2 The Nonsphericity Property of the PP.- 4.7.3 The Delaunay Tessellation.- 4.7.4 Pre-Size-and-Shape Distribution of Delaunay Simplexes.- 4.8 Notes.- 4.9 Problems.- 5 Distributions of Random Shapes.- 5.1 Landmarks from the Spherical Normal: IID Case.- 5.2 Shape Densities under Affine Transformations.- 5.2.1 Introduction.- 5.2.2 Shape Density for the Elliptical Normal Distribution.- 5.2.3 Broadbent Factors and Collinear Shapes.- 5.3 Tools for the Ley Hunter.- 5.4 Independent Uniformly Distributed Landmarks.- 5.5 Landmarks from the Spherical Normal: Non-IID Case.- 5.6 The Poisson-Delaunay Shape Distribution.- 5.7 Notes.- 5.8 Problems.- 6 Some Examples of