The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics

• Published in: The Annals of statistics 1993-09, Vol.21 (3), p.1225-1271
• Main Authors: Le, Huiling, Kendall, David G.
• Format: Article
• Language: English
• Subjects: 60D05; 62H99; Brownian motion; Collinearity; collinearity testing; Curvature; Exact sciences and technology; Geometric probability. Stochastic geometry. Random sets; Geometric shapes; Geometry and Statistics; Mathematics; Poisson-Delaunay simplexes; Probability and statistics; Probability theory and stochastic processes; Riemannian submersion; Scalars; Sciences and techniques of general use; Shape space; shapes of fossils; singular values decomposition; stereo plots; Tensors; Tessellations; Tetrahedrons; Vector fields; Vertices; volume element; warped product
• ISSN: 0090-5364; 2168-8966
• DOI: 10.1214/aos/1176349259

The Riemannian metric structure of the shape space ∑k mfor k labelled points in Rmwas given by Kendall for the atypically simple situations in which m = 1 or 2 and k ≥ 2. Here we deal with the general case (m ≥ 1, k ≥ 2) by using the properties of Riemannian submersions and warped products as studied by O'Neill. The approach is via the associated size-and-shape space that is the warped product of the shape space and the half-line R+(carrying size), the warping function being equal to the square of the size. When combined with parallel studies by Le of the corresponding global geodesic geometry, the results obtained here determine the environment in which shape-statistical calculations have to be acted out. Finally three different applications are discussed that illustrate the theory and its use in practice.