Table of Contents
Conference Papers in Mathematics
Volume 2013 (2013), Article ID 425608, 7 pages
http://dx.doi.org/10.1155/2013/425608
Conference Paper

Approximating Sets of Symmetric and Positive-Definite Matrices by Geodesics

1Institute of Systems and Robotics, University of Coimbra, Pólo II, Pinhal de Marrocos, 3030-290 Coimbra, Portugal
2Department of Mathematics, University of Trás-os-Montes and Alto Douro, Apartado 1013, 5001-801 Vila Real, Portugal
3Department of Mathematics, University of Coimbra, Largo D. Dinis, 3001-454 Coimbra, Portugal

Received 14 June 2013; Accepted 10 July 2013

Academic Editors: G. S. F. Frederico, N. Martins, D. F. M. Torres, and A. J. Zaslavski

This Conference Paper is based on a presentation given by L. Machado at “The Cape Verde International Days on Mathematics 2013” held from 22 April 2013 to 25 April 2013 in Praia, Cape Verde.

Copyright © 2013 L. Machado and F. Silva Leite. 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.

Linked References

  1. F. Porikli, O. Tuzel, and P. Meer, “Covariance tracking using model update based on Lie algebra,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR '06), pp. 728–735, June 2006. View at Publisher · View at Google Scholar · View at Scopus
  2. O. Tuzel, F. Porikli, and P. Meer, “Region covariance: a fast descriptor for detection and classification,” in Proceedings of the 9th European Conference on Computer Vision, 2006.
  3. P. Filiard, X. Pennec, V. Arsigny, and N. Ayache, “Clinical DT-MRI estimation, smoothing, and fiber tracking with log-euclidean metrics,” IEEE Transactions on Medical Imaging, vol. 26, no. 11, pp. 1472–1482, 2007. View at Publisher · View at Google Scholar · View at Scopus
  4. M. Moakher and P. Batchelor, “Symmetric positive-definite matrices: from geometry to applications and visualization,” in Visualization and Image Processing of Tensor Fields, J. Weickert and H. Hagen, Eds., Chapter 17, Springer, Berlin, Germany, 2005. View at Google Scholar
  5. M. Moakher, “On the averaging of symmetric positive-definite tensors,” Journal of Elasticity, vol. 82, no. 3, pp. 273–296, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. L. Machado and F. Silva Leite, “Fitting smooth paths on riemannian manifolds,” International Journal of Applied Mathematics & Statistics, no. J06, pp. 25–53, 2006. View at Google Scholar
  7. L. Machado and T. Monteiro, “Geodesic regression on spheres: a numerical optimization approach,” in Proceedings of the 13th International Conference on Mathematical Methods in Science and Engineering (CMMSE '13), Almería, Spain.
  8. M. P. do Carmo, Riemannian Geometry, Mathematics: Theory and Applications, Birkäuser, 1992.
  9. J. Jost, Riemannian Geometry and Geometric Analysis, Universitext; Springer, 6 edition, 2011.
  10. J. W. Milnor, Morse Theory, Princeton University Press, Princeton, NJ, USA, 1963.
  11. H. Karcher, “Riemannian center of mass and mollifier smoothing,” Communications on Pure and Applied Mathematics, vol. 30, pp. 509–541, 1977. View at Google Scholar
  12. E. Batzies, L. Machado, and F. Silva Leite, The Geometric Mean and the Geodesic Fitting Problem on the Grassmann Manifold, Department of Mathematics; University of Coimbra, 2013.
  13. L. Machado and F. Silva Leite, “Interpolation and polynomial fitting on the SPD manifold,” in Proceedings of the 52nd IEEE Conference on Decision and Control (CDC '13), Florence, Italy, December 2013.
  14. F. Hiai and D. Petz, “Riemannian metrics on positive definite matrices related to means,” Linear Algebra and Its Applications, vol. 430, no. 11-12, pp. 3105–3130, 2009. View at Publisher · View at Google Scholar · View at Scopus
  15. M. Moakher, “A differential geometric approach to the geometric mean of symmetric positive-definite matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 3, pp. 735–747, 2005. View at Publisher · View at Google Scholar · View at Scopus
  16. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, NY, USA, 1991.
  17. D. H. Sattinger and O. L. Weaver, Lie Groups and Algebras With Applications to Physics, Geometry, and Mechanics, Springer, 1980.
  18. V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Geometric means in a novel vector space structure on symmetric positive-definite matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 29, no. 1, pp. 328–347, 2007. View at Publisher · View at Google Scholar · View at Scopus
  19. P. E. Jupp and J. T. Kent, “Fitting smooth paths to spherical data,” Applied Statistics, vol. 36, no. 1, pp. 34–46, 1987. View at Google Scholar
  20. P. Crouch and F. S. Leite, “The dynamic interpolation problem: on Riemannian manifolds, Lie groups, and symmetric spaces,” Journal of Dynamical and Control Systems, vol. 1, no. 2, pp. 177–202, 1995. View at Publisher · View at Google Scholar · View at Scopus
  21. L. MacHado, F. Silva Leite, and K. Krakowski, “Higher-order smoothing splines versus least squares problems on Riemannian manifolds,” Journal of Dynamical and Control Systems, vol. 16, no. 1, pp. 121–148, 2010. View at Publisher · View at Google Scholar · View at Scopus
  22. L. Noakes, G. Heinzinger, and B. Paden, “Cubic splines on curved spaces,” IMA Journal of Mathematical Control and Information, vol. 6, no. 4, pp. 465–473, 1989. View at Publisher · View at Google Scholar · View at Scopus
  23. R. Giambò, F. Giannoni, and P. Piccione, “An analytical theory for Riemannian cubic polynomials,” IMA Journal of Mathematical Control and Information, vol. 19, no. 4, pp. 445–460, 2002. View at Publisher · View at Google Scholar · View at Scopus