Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T07:58:44.332Z Has data issue: false hasContentIssue false
  • English
  • Français

Locating Fréchet means with application to shape spaces

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Huiling Le
Huiling Le*
Affiliation:
University of Nottingham
*
∗ Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We use Jacobi field arguments and the contraction mapping theorem to locate Fréchet means of a class of probability measures on locally symmetric Riemannian manifolds with non-negative sectional curvatures. This leads, in particular, to a method for estimating Fréchet mean shapes, with respect to the distance function ρ determined by the induced Riemannian metric, of a class of probability measures on Kendall's shape spaces. We then combine this with the technique of ‘horizontally lifting’ to the pre-shape spheres to obtain an algorithm for finding Fréchet mean shapes, with respect to ρ, of a class of probability measures on Kendall's shape spaces in terms of the vertices of random shapes. This gives us, for example, an algorithm for finding Fréchet mean shapes of samples of configurations on the plane which is expressed directly in terms of the vertices.

Keywords

Kendall's shape spaceFréchet mean shapepre-shape sphereJacobi fieldLipschitz conditioncontraction mappinghorizontal lift

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 2 , June 2001 , pp. 324 - 338
Copyright
Copyright © Applied Probability Trust 2001 

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.)

References

[1] Cheeger, J. and Ebin, D. G. (1975). Comparison Theorems in Riemannian Geometry. North-Holland, Amsterdam.Google Scholar
[2] Goodall, C. R. (1991). Procrustes methods in the statistical analysis of shape (with discussion). J. R. Statist. Soc. B 53, 285–339.Google Scholar
[3] Jost, J. (1998). Riemannian Geometry and Geometric Analysis. Springer, Berlin.Google Scholar
[4] Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541.Google Scholar
[5] Kendall, D. G. (1984). Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
[6] 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
[7] Kendall, D. G., Barden, D., Carne, T. K. and Le, H. (1999). Shape and Shape Theory. John Wiley, Chichester.Google Scholar
[8] Kent, J. T. (1992). New directions in shape analysis. In The Art of Statistical Science, ed. Mardia, K. V. John Wiley, Chichester, pp. 115–127.Google Scholar
[9] Le, H. (1998). On consistency of procrustean mean shapes. Adv. Appl. Prob. 30, 53–63.Google Scholar
[10] Le, H. and Barden, D. (2001). On simplex shape spaces. To appear in J. London Math. Soc. CrossRefGoogle Scholar
[11] O'Neill, B. (1983). Semi-Riemannian Geometry. Academic Press, Orlando.Google Scholar
[12] Ziezold, H. (1977). On expected figures and a strong Law of Large Numbers for random elements in quasi-metric spaces. In Trans. 7th Prague Conf. Inf. Theory, Statist. Decision Functions, Random Proc. Vol. A. Reidel, Dordrecht, pp. 591–602.Google Scholar