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Mean size-and-shapes and mean shapes: a geometric point of view

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Hulling Le
Hulling Le*
Affiliation:
University of Nottingham
*
* Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
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Abstract

Unlike the means of distributions on a euclidean space, it is not entirely clear how one should define the means of distributions on the size-and-shape or shape spaces of k labelled points in ℝm since these spaces are all curved. In this paper, we discuss, from a shape-theoretic point of view, some questions which arise in practice while using procrustean methods to define mean size-and-shapes or shapes. We obtain sufficient conditions for such means to be unique and for the corresponding generalized procrustean algorithms to converge to them. These conditions involve the curvature of the size-and-shape or shape spaces and are much less restrictive than asking for the data to be concentrated.

Keywords

PROCRUSTEAN METHODKARCHER MEAN

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 44 - 55
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

The original version of this paper was presented at the International Workshop on Stochastic Geometry, Stereology and Image Analysis held at the Universidad Internacional Menendez Pelayo, Valencia, Spain on 21–24 September 1993.

References

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