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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.
List of contents
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.- 1.5 The Problem of Homology.- 1.6 Notes.- 1.7 Problems.- 2 Background Concepts and Definitions.- 2.1 Transformations on Euclidean Space.- 2.2 Differential Geometry.- 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.- 3.4 Principal Coordinate Analysis.- 3.5 An Application of Principal Coordinate Analysis.- 3.6 Hyperbolic Geometries for Shapes.- 3.7 Local Analysis of Shape Variation.- 3.8 Notes.- 3.9 Problems.- 4 Some Stochastic Geometry.- 4.1 Probability Theory on Manifolds.- 4.2 The Geometric Measure.- 4.3 Transformations of Statistics.- 4.4 Invariance and Isometries.- 4.5 Normal Statistics on Manifolds.- 4.6 Binomial and Poisson Processes.- 4.7 Poisson Processes in Euclidean Spaces.- 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.- 5.3 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.- 6.3 Shape Analysis of Post Mold Data.- 6.4 Case Studies: Aldermaston Wharf and South Lodge Camp.- 6.5 Automated Homology.- 6.6 Notes.
Summary
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.
