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

For all p > 2 , k > p , a size-and-reflection-shape space S R Σ 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 Σ 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, p > 2 . The Veronese embedding of the planar Kendall shape manifold Σ 2 k is extended to an equivariant embedding of the size-and-shape manifold S Σ 2 k , which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory.