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The Scientific World Journal
Volume 2013 (2013), Article ID 292787, 9 pages
http://dx.doi.org/10.1155/2013/292787
Research Article

Riemannian Means on Special Euclidean Group and Unipotent Matrices Group

1School of Mathematics, Beijing Institute of Technology, Beijing 100081, China
2Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK

Received 1 August 2013; Accepted 16 September 2013

Academic Editors: R. Abu-Saris and P. Bracken

Copyright © 2013 Xiaomin Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results.