Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging

Authors:
Ananda Bandulasiri;Rabi N. Bhattacharya;Vic Patrangenaru
Affiliations:
Sam Houston State University, Department of Mathematics and Statistics, Huntsville, TX 77341, United States;University of Arizona, Department of Mathematics, Tucson, AZ 85721, United States;Florida State University, Department of Statistics, Tallahassee, FL 32306, United States
Venue:
Journal of Multivariate Analysis
Year:
2009

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Abstract

For all p2,kp, a size-and-reflection-shape space SR@S"p","0^k of k-ads in general position in R^p, invariant under translation, rotation and reflection, is shown to be a smooth manifold and is equivariantly embedded in a space of symmetric matrices, allowing a nonparametric statistical analysis based on extrinsic means. Equivariant embeddings are also given for the reflection-shape-manifold R@S"p","0^k, a space of orbits of scaled k-ads in general position under the group of isometries of R^p, providing a methodology for statistical analysis of three-dimensional images and a resolution of the mathematical problems inherent in the use of the Kendall shape spaces in p-dimensions, p2. The Veronese embedding of the planar Kendall shape manifold @S"2^k is extended to an equivariant embedding of the size-and-shape manifold S@S"2^k, which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory.