The Annals of Statistics

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

Huiling Le and David G. Kendall

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The Riemannian metric structure of the shape space $\sum^k_m$ for $k$ labelled points in $\mathbb{R}^m$ was given by Kendall for the atypically simple situations in which $m = 1$ or 2 and $k \geq 2$. Here we deal with the general case $(m \geq 1, k \geq 2)$ 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 $\mathbb{R}_+$ (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.

Article information

Ann. Statist., Volume 21, Number 3 (1993), 1225-1271.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 62H99: None of the above, but in this section

Shape space Riemannian submersion warped product curvature volume element singular values decomposition Brownian motion shapes of fossils stereo plots collinearity testing Poisson-Delaunay simplexes


Le, Huiling; Kendall, David G. The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics. Ann. Statist. 21 (1993), no. 3, 1225--1271. doi:10.1214/aos/1176349259.

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