In general terms, the shape of an object, data set, or image can be de­ fined as the total of all information that is invariant under translations, rotations, and isotropic rescalings. Thus two objects can be said to have the same shape if they are similar in the sense of Euclidean geometry. For example, all equilateral triangles have the same shape, and so do all cubes. In applications, bodies rarely have exactly the same shape within measure­ ment error. In such cases the variation in shape can often be the subject of statistical analysis. The last decade has seen a considerable growth in interest in the statis­ tical theory of shape. This has been the result of a synthesis of a number of different areas and a recognition that there is considerable common ground among these areas in their study of shape variation. Despite this synthesis of disciplines, there are several different schools of statistical shape analysis. One of these, the Kendall school of shape analysis, uses a variety of mathe­ matical tools from differential geometry and probability, and is the subject of this book. The book does not assume a particularly strong background by the reader in these subjects, and so a brief introduction is provided to each of these topics. Anyone who is unfamiliar with this material is advised to consult a more complete reference. As the literature on these subjects is vast, the introductory sections can be used as a brief guide to the literature.

Inhalt

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 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.