Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T04:21:08.272Z Has data issue: false hasContentIssue false
  • English
  • Français

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, [email protected], 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, [email protected], http://www.mat.uc.pt/~fleite
Knut Hüper
Affiliation:
National ICT Australia, Canberra Research Laboratory, SEACS Program, Locked Bag 8001, Canberra ACT 2601, Australia, [email protected], 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, [email protected], http://www.rsise.anu.edu.au/~hueper

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Adler, Roy L., Dedieu, Jean-Pierre, Margulies, Joseph Y., Martens, Marco and Shub, Mike, ‘Newton's method on Riemannian manifolds and a geometric model for the human spine’, IMA J. Numer. Anal. 22 (2002) 359–390.CrossRefGoogle Scholar
2. Buss, S.R. and Fillmore, J.P., ‘Spherical averages and applications to spherical splines and interpolation’, ACM Trans. Graphics 2 (2001) 95–126.CrossRefGoogle Scholar
3. Carmo, M.P. Do, Riemannian geometry (Birkhäuser, Boston, 1992).CrossRefGoogle Scholar
4. Galperin, G.A., ‘A concept of mass center of a system of material points in the constant curvature spaces’, Comm. Math. Phys. 154 (1993) 63–84.CrossRefGoogle Scholar
5. Helgason, S., Differential geometry, Lie groups and symmetric spaces (Academic Press, Boston, 1978)Google Scholar
6. Jupp, P.E. and Kent, J.T., ‘Fitting smooth paths to spherical data’, Appl. Statist. 36 (1987) 34–46.CrossRefGoogle Scholar
7. Karcher, H., ‘Riemannian center of mass and mollifier smoothing’, Comm. Pure Appl. Math. 30 (1977) 509–541.CrossRefGoogle Scholar
8. Kobayashi, S. and Nomizu, K., Foundations of differential geometry, Interscience Tracts in Pure and Applied Mathematics 2 (Interscience Publishers, New York, 1969).Google Scholar
9. Krakowski, A.K., ‘Geometrical methods of inference’, PhD Thesis, Department of Mathematics and Statistics, University of Western Australia, Australia, 2002.Google Scholar
10. Lazard, M. and Tits, J., ‘Domaines d'injectivité de l'application exponentielle’, Topology 4 (1966) 315–322.CrossRefGoogle Scholar
11. Machado, L. and Leite, F. Silva, ‘Fitting smooth paths on Riemannian manifolds’, preprint no. 04–31, Department of Mathematics, University of Coimbra, 2004.Google Scholar
12. Milnor, J.W., Morse theory (Princeton University Press, Princeton, New Jersey, 1963).CrossRefGoogle Scholar
13. Moakher, M., ‘Means and averaging in the group of rotations’, SIAM. J. Matrix Anal. Appl. 24 (2001) 1–16.CrossRefGoogle Scholar
14. Moakher, M., ‘A differential geometric approach to the arithmetic and geometric means of operators in some symmetric spaces’, SIAM. J. Matrix Anal. Appl. 26 (2005) 735–747.CrossRefGoogle Scholar
15. Kaas, H. Munthe-, Quispel, R.G.W. and Zanna, A., ‘The polar decomposition on Lie groups with involutive automorphisms’, Found. Comp. Math. 1:3 (2001) 297–324.CrossRefGoogle Scholar