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Riemannian Means as Solutions of Variational Problems

Published online by Cambridge University Press:  01 February 2010

Luís Machado ,
F. Silva Leite  and
Knut Hüper
Luís Machado
Affiliation:
Department of Mathematics, University of Trás-os-Montes e Alto Douro, 5000-911 Vila Real, Portugal, lmiguel@utad.pt, http://home.utad.pt/~lmiguel
F. Silva Leite
Affiliation:
Department of Mathematics and Institute of Systems and Robotics, University of Coimbra, 3001-454 Coimbra, Portugal, fleite@mat.uc.pt, http://www.mat.uc.pt/~fleite
Knut Hüper
Affiliation:
National ICT Australia, Canberra Research Laboratory, SEACS Program, Locked Bag 8001, Canberra ACT 2601, Australia, knut.hueper@nicta.com.au, http://www.rsise.anu.edu.au/~hueper Department of Information Engineering, Research School of Information Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia, knut.hueper@nicta.com.au, http://www.rsise.anu.edu.au/~hueper

Abstract

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We formulate a variational problem on a Riemannian manifold M whose solutions are piecewise smooth geodesies that best fit a given data set of time labelled points in M. By a limiting process, these solutions converge to a single point in M. which we prove to be the Riemannian mean of the given points for some particular Riemannian manifolds such as Euclidean spaces, connected and compact Lie groups, and spheres.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 9 , 2006 , pp. 86 - 103
Copyright
Copyright © London Mathematical Society 2006

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