Means and averaging in the group of rotations

SIAM Journal on Matrix Analysis and Applications

Volumn 24, Issue 1, 2003, Pages 1-16

  Moakher, Maher ^a  

^a EPFL   (Switzerland)

Author keywords

Averaging; Geodesics; Operator means; Rotation; Special orthogonal group

Indexed keywords

EID: 0037281553     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479801383877     Document Type: Article

References (27)

1. M. BERGER AND B. GOSTIAUX, Differential Geometry: Manifolds, Curves, and Surfaces, Springer-Verlag, New York, 1988.
2. P. S. BULLEN, D. S. MITRINOVIĆ, AND P. M. VASIĆ, Means and Their Inequalities, Math. Appl. (East European Ser.) 31, D. Reidel, Dordrecht, The Netherlands, 1988.
3. M. L. CURTIS, Matrix Groups, Springer-Verlag, New York, Heidelberg, 1979.
4. T. D. DOWNS, Orientation statistics, Biometrika, 59 (1972), pp. 665-676.
5. A. EDELMAN, T. A. ARIAS, AND S. T. SMITH, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 303-353.
6. D. W. EGGERT, A. LORUSSO, AND R. B. FISHER, Estimating 3-D rigid body transformations: A comparison of four major algorithms, Machine Vision Appl., 9 (1997), pp. 272-290.
7. N. I. FISHER, Spherical medians, J. Roy. Statist. Soc. Ser. B, 47 (1985), pp. 342-348.
8. N. I. FISHER, Statistical Analysis of Circular Data, Cambridge University Press, Cambridge, UK, 1993.
9. N. I. FISHER, T. LEWIS, AND B. J. J. EMBLETON, Statistical Analysis of Spherical Data, Cambridge University Press, Cambridge, UK, 1987.
10. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, The Johns Hopkins University Press, London, 1989.
11. K. GROVE, H. KARCHER, AND E. A. RUH, Jacobi fields and Finsler metrics on compact Lie groups with an application to differentiable pinching problem, Math. Ann., 211 (1974), pp. 7-21.
12. G. H. HARDY, J. E. LITTLEWOOD, AND G. PÓLYA, Inequalities, Cambridge University Press, Cambridge, UK, 1934.
13. U. HELMKE AND J. B. MOORE, Optimization and Dynamical Systems, Springer-Verlag, London, 1994.
14. A. ISERLES, H. Z. MUNTHE-KAAS, S. P. NØRSET, AND A. ZANNA, Lie-group methods, Acta Numer., 9 (2000), pp. 215-365.
15. P. E. JUPP AND K. V. MARDIA, Maximum likelihood estimation for the matrix van Mises-Fisher and Bingham distributions, Ann. Statist., 7 (1979), pp. 599-606.
16. P. E. JUPP AND J. T. KENT, Fitting smooth paths to spherical data, Appl. Statist., 36 (1987), pp. 34-46.
17. H. KARCHER, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30 (1977), pp. 509-541.
18. C. G. KHATRI AND K. V. MARDIA, The von Mises-Fisher matrix distribution in orientation statistics, J. Roy. Statist. Soc. Ser. B, 39 (1977), pp. 95-106.
19. R. S. MANNING, J. H. MADDOCKS, AND J. D. KAHN, A continuum rod model of sequence-dependent DNA structure, J. Chem. Phys., 105 (1996), pp. 5626-5646.
20. K. V. MARDIA, Statistics of directional data, Prob. Math. Statist. 13, Academic Press, London, New York, 1972.
21. R. M. MURRAY, Z. LI, AND S. S. SASTRY, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1996.
22. M. J. PRENTICE, Fitting smooth paths to rotation data, Appl. Statist., 36 (1987), pp. 325-331.
23. J. M. SELIG, Geometrical Methods in Robotics, Springer-Verlag, New York, 1996.
24. S. T. SMITH, Optimization techniques on Riemannian manifolds, in Hamiltonian and Gradient Flows, Algorithms and Control, A. Bloch, ed., AMS, Providence, RI, 1994, pp. 113-136.
25. G. E. TRAPP, Hermitian semidefinite matrix means and related matrix inequalities-an introduction, Linear and Multilinear Algebra, 16 (1984), pp. 113-123.
26. C. UDRIŞTE, Convex Functions and Optimization Methods on Riemannian Manifolds, Math. Appl. 297, Kluwer Academic, Dordrecht, The Netherlands, 1994.
27. G. S. WATSON, Equatorial distributions on a sphere, Biometrika, 52 (1965), pp. 193-201.