Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 6316 LNCS, Issue PART 6, 2010, Pages 43-56

  Sommer, Stefan ^a   Lauze, François ^a   Hauberg, Søren ^a   Nielsen, Mads ^a,b  

^a UNIVERSITY OF COPENHAGEN   (Denmark)

^b NORDIC BIOSCIENCE A S   (Denmark)

Author keywords

Geodesic PCA; linearization; manifold valued statistics; manifolds; Principal Geodesic Analysis (PGA); Riemannian metrics

Indexed keywords

AUTOMOBILE ENGINE MANIFOLDS; COMPUTER VISION; GEODESY; LINEARIZATION; STEREO IMAGE PROCESSING;

COMPUTER VISION APPLICATIONS; GEODESIC PCA; INTRINSIC DISTANCES; LINEAR APPROXIMATIONS; MANIFOLD-VALUED DATUM; PRINCIPAL GEODESIC ANALYSIS; RIEMANNIAN METRICS; STATISTICS ON MANIFOLDS;

PRINCIPAL COMPONENT ANALYSIS;

EID: 78149288043     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-642-15567-3_4     Document Type: Conference Paper

References (24)

1. Fletcher, P., Lu, C., Pizer, S., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging 23, 995-1005 (2004)
2. Sommer, S., Lauze, F., Nielsen, M.: The differential of the exponential map, jacobi fields, and exact principal geodesic analysis (2010) (submitted)
3. Fletcher, P.T., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Processing 87, 250-262 (2007)
4. Fletcher, P.T., Joshi, S.: Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors. In: Sonka, M., Kakadiaris, I.A., Kybic, J. (eds.) CVAMIA/MMBIA 2004. LNCS, vol. 3117, pp. 87-98. Springer, Heidelberg (2004)
5. Pennec, X., Fillard, P., Ayache, N.: A riemannian framework for tensor computing. Int. J. Comput. Vision 66, 41-66 (2006)
6. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. International Journal of Computer Vision 22, 61-79 (1995)
7. Pennec, X., Guttmann, C., Thirion, J.: Feature-based registration of medical images: Estimation and validation of the pose accuracy. In: Wells, W.M., Colchester, A.C.F., Delp, S.L. (eds.) MICCAI 1998. LNCS, vol. 1496, pp. 1107-1114. Springer, Heidelberg (1998) (Pubitemid 128152761)
8. Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81-121 (1984)
9. Sminchisescu, C., Jepson, A.: Generative modeling for continuous Non-Linearly embedded visual inference. In: ICML, pp. 759-766 (2004)
10. Hauberg, S., Sommer, S., Pedersen, K.S.: Gaussian-like spatial priors for articulated tracking. In: Daniilidis, K. (ed.) ECCV 2010, Part I. LNCS, vol. 6311, pp. 425-437. Springer, Heidelberg (2010)
11. Karcher, H.: Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics 30, 509-541 (1977)
12. Pennec, X.: Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements. J. Math. Imaging Vis. 25, 127-154 (2006)
13. Huckemann, S., Hotz, T., Munk, A.: Intrinsic shape analysis: Geodesic PCA for riemannian manifolds modulo isometric lie group actions. Statistica Sinica 20, 1-100 (2010)
14. do Carmo, M.P.: Riemannian geometry. Mathematics: Theory & Applications. Birkhauser, Boston (1992)
15. Lee, J.M.: Riemannian manifolds. Graduate Texts in Mathematics, vol. 176. Springer, New York (1997); An introduction to curvature
16. Dedieu, J., Nowicki, D.: Symplectic methods for the approximation of the exponential map and the newton iteration on riemannian submanifolds. Journal of Complexity 21, 487-501 (2005) (Pubitemid 40963110)
17. Noakes, L.: A global algorithm for geodesics. Journal of the Australian Mathematical Society 64, 37-50 (1998)
18. Klassen, E., Srivastava, A.: Geodesics between 3D closed curves using Path-Straightening. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3951, pp. 95-106. Springer, Heidelberg (2006) (Pubitemid 44029771)
19. Schmidt, F., Clausen, M., Cremers, D.: Shape matching by variational computation of geodesics on a manifold. In: Pattern Recognition, pp. 142-151. Springer, Berlin (2006)
20. Sommer, S., Tatu, A., Chen, C., Jørgensen, D., de Bruijne, M., Loog, M., Nielsen, M., Lauze, F.: Bicycle chain shape models. In: MMBIA/CVPR 2009, pp. 157-163 (2009)
21. Huckemann, S., Ziezold, H.: Principal component analysis for riemannian manifolds, with an application to triangular shape spaces. Advances in Applied Probability 38, 299-319 (2006) (Pubitemid 44057292)
22. Fletcher, P., Lu, C., Joshi, S.: Statistics of shape via principal geodesic analysis on lie groups. In: CVPR 2003, vol. 1, p. I-95 - I-101 (2003)
23. Wu, J., Smith, W., Hancock, E.: Weighted principal geodesic analysis for facial gender classification. In: Progress in Pattern Recognition, Image Analysis and Applications, pp. 331-339. Springer, Berlin (2008)
24. Said, S., Courty, N., Bihan, N.L., Sangwine, S.: Exact principal geodesic analysis for data on so(3). In: EUSIPCO 2007 (2007)