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 Σ₂^k is extended to an equivariant embedding of the size-and-shape manifold S Σ₂^k, which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory.