A multi-dimensional scaling approach to shape analysis

We propose an alternative to Kendall's shape space for reflection shapes of configurations in with k labelled vertices, where reflection shape consists of all the geometric information that is invariant under compositions of similarity and reflection transformations. The proposed approach embed...

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Bibliographic Details
Published in:Biometrika 2008-12, Vol.95 (4), p.779-798
Main Authors: Dryden, Ian L., Kume, Alfred, Le, Huiling, Wood, Andrew T. A.
Format: Article
Language:English
Subjects:
Applications
Approximation
Biology, psychology, social sciences
Bootstrap method
Central limit theorem
Combinatorics
Combinatorics. Ordered structures
Coordinate systems
Designs and configurations
Eigenvalues
Eigenvectors
Euclidean space
Exact sciences and technology
General topics
Geometric shapes
Mathematical vectors
Mathematics
Matrices
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Probability and statistics
Procrustes mean shape
Reflection shape
Sciences and techniques of general use
Statistical analysis
Statistics
Studies
Tangent space projection
Tangents
Theorems
Vertices
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Summary:We propose an alternative to Kendall's shape space for reflection shapes of configurations in with k labelled vertices, where reflection shape consists of all the geometric information that is invariant under compositions of similarity and reflection transformations. The proposed approach embeds the space of such shapes into the space of (k − 1) × (k − 1) real symmetric positive semidefinite matrices, which is the closure of an open subset of a Euclidean space, and defines mean shape as the natural projection of Euclidean means in on to the embedded copy of the shape space. This approach has strong connections with multi-dimensional scaling, and the mean shape so defined gives good approximations to other commonly used definitions of mean shape. We also use standard perturbation arguments for eigenvalues and eigenvectors to obtain a central limit theorem which then enables the application of standard statistical techniques to shape analysis in two or more dimensions.
ISSN:0006-3444
1464-3510
DOI:10.1093/biomet/asn050