Mean size-and-shapes and mean shapes: a geometric point of view
- 1 March 1995
- journal article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 27 (1) , 44-55
- https://doi.org/10.2307/1428094
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 ofklabelled points in ℝmsince 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
This publication has 8 references indexed in Scilit:
- The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for StatisticsThe Annals of Statistics, 1993
- On Geodesics in Euclidean Shape SpacesJournal of the London Mathematical Society, 1991
- Procrustes Methods in the Statistical Analysis of ShapeJournal of the Royal Statistical Society Series B: Statistical Methodology, 1991
- Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine ExistenceProceedings of the London Mathematical Society, 1990
- Shape Manifolds, Procrustean Metrics, and Complex Projective SpacesBulletin of the London Mathematical Society, 1984
- Riemannian center of mass and mollifier smoothingCommunications on Pure and Applied Mathematics, 1977
- Orthogonal Procrustes Rotation for Two or More MatricesPsychometrika, 1977
- Generalized Procrustes AnalysisPsychometrika, 1975