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

For all p>2,k>pp>2,k>p, a size-and-reflection-shape space SRΣp,0k of kk-ads in general position in RpRp, 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 RΣp,0k, a space of orbits of scaled kk-ads in general position under the group of isometries of RpRp, 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 pp-dimensions, p>2p>2. The Veronese embedding of the planar Kendall shape manifold Σ2k is extended to an equivariant embedding of the size-and-shape manifold SΣ2k, which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory.