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September, 1993 The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics
Huiling Le, David G. Kendall
Ann. Statist. 21(3): 1225-1271 (September, 1993). DOI: 10.1214/aos/1176349259


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.


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Huiling Le. David G. Kendall. "The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics." Ann. Statist. 21 (3) 1225 - 1271, September, 1993.


Published: September, 1993
First available in Project Euclid: 12 April 2007

zbMATH: 0831.62003
MathSciNet: MR1241266
Digital Object Identifier: 10.1214/aos/1176349259

Primary: 60D05
Secondary: 62H99

Rights: Copyright © 1993 Institute of Mathematical Statistics


Vol.21 • No. 3 • September, 1993
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