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 Poincare 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 Shape Analysis. - 6.1 Introduction. - 6.2 Mt. Tom Dinosaur Trackways. - 6.2.1 Orientation Analysis. - 6.2.2 Scale Analysis. - 6.2.3 Shape Analysis. - 6.2.4 Fitting the Mardia-Dryden Density. - 6.3 Shape Analysis of Post Mold Data. - 6.3.1 A Few General Remarks. - 6.3.2 The Number of Patterns in a Poisson Process. - 6.3.3 An Annular Coverage Criterion for Post Molds. - 6.4 Case Studies: Aldermaston Wharf and South Lodge Camp. - 6.4.1 Scale Analysis. - 6.4.2 Shape Analysis. - 6.4.3 Conclusions. - 6.5 Automated Homology. - 6.5.1 Introduction. - 6.5.2 Automated Block Homology. - 6.5.3 An Application to Three Brooches. - 6.6 Notes. - 6.6.1 Anthropology, Archeology, and Paleontology. - 6.6.2 Biology and Medical Sciences. - 6.6.3 Earth and Space Sciences. - 6.6.4 Geometric Probability and Stochastic Geometry. - 6.6.5 Industrial Statistics. - 6.6.6 Mathematical Statistics and Multivariate Analysis. - 6.6. 7 Pattern Recognition, Computer Vision, and Image Processing. - 6.6. 8 Stereology and Microscopy. - 6.6. 9 Topics on Groups and Invariance.