Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-16T17:29:03.159Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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

[1] Goodall, C. (1991) Procrustes methods in the statistical analysis of shape. J. R. Statist. Soc. B 53, 285–339.Google Scholar
[2] Gower, J. C. (1975) Generalized procrustes analysis. Psychometrika 40, 33–51.CrossRefGoogle Scholar
[3] Karcher, H. (1977) Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30, 509–541.CrossRefGoogle Scholar
[4] Kendall, D. G. (1984) Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
[5] Kendall, W. S. (1990) Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proc. London Math. Soc. 61, 371–406.CrossRefGoogle Scholar
[6] Kristof, W. and Wingersky, B. (1971) Generalization of the orthogonal Procrustes rotation procedure for more than two matrices. Proc. 79th Annual Convention Amer. Psychol. Assoc. , 89–90.Google Scholar
[7] Le, H. (1988) Shape theory in flat and curved spaces, and shape densities with uniform generators. Ph.D. dissertation, University of Cambridge.Google Scholar
[8] Le, H. (1991) On geodesics in euclidean shape spaces. J. London Math. Soc. 44, 360–372.Google Scholar
[9] Le, H. and Kendall, D. G. (1993) The Riemannian structure of euclidean shape spaces: a novel environment for statistics. Ann. Statist. 21, 1225–1271.CrossRefGoogle Scholar
[10] Ten Berge, J. M. F. (1977) Orthogonal procrustes rotation for two or more matrices. Psychometrika 42, 267–276.CrossRefGoogle Scholar
[11] Ziezold, H. (1990) Mean figures and mean shapes in the plane. Mathematische Schriften, Kassel.Google Scholar