Manifold Valued Statistics, Exact Principal Geodesic Analysis and the Effect of Linear Approximations

  • Stefan Sommer
  • François Lauze
  • Søren Hauberg
  • Mads Nielsen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6316)


Manifolds are widely used to model non-linearity arising in a range of computer vision applications. This paper treats statistics on manifolds and the loss of accuracy occurring when linearizing the manifold prior to performing statistical operations. Using recent advances in manifold computations, we present a comparison between the non-linear analog of Principal Component Analysis, Principal Geodesic Analysis, in its linearized form and its exact counterpart that uses true intrinsic distances. We give examples of datasets for which the linearized version provides good approximations and for which it does not. Indicators for the differences between the two versions are then developed and applied to two examples of manifold valued data: outlines of vertebrae from a study of vertebral fractures and spacial coordinates of human skeleton end-effectors acquired using a stereo camera and tracking software.


Vertebral Fracture Tangent Space Hand Position Stereo Camera Human Skeleton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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)CrossRefGoogle Scholar
  2. 2.
    Sommer, S., Lauze, F., Nielsen, M.: The differential of the exponential map, jacobi fields, and exact principal geodesic analysis (2010) (submitted)Google Scholar
  3. 3.
    Fletcher, P.T., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Processing 87, 250–262 (2007)zbMATHCrossRefGoogle Scholar
  4. 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)Google Scholar
  5. 5.
    Pennec, X., Fillard, P., Ayache, N.: A riemannian framework for tensor computing. Int. J. Comput. Vision 66, 41–66 (2006)CrossRefGoogle Scholar
  6. 6.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. International Journal of Computer Vision 22, 61–79 (1995)CrossRefGoogle Scholar
  7. 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)Google Scholar
  8. 8.
    Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Sminchisescu, C., Jepson, A.: Generative modeling for continuous Non-Linearly embedded visual inference. In: ICML, pp. 759–766 (2004)Google Scholar
  10. 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)Google Scholar
  11. 11.
    Karcher, H.: Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics 30, 509–541 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pennec, X.: Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements. J. Math. Imaging Vis. 25, 127–154 (2006)CrossRefMathSciNetGoogle Scholar
  13. 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)zbMATHMathSciNetGoogle Scholar
  14. 14.
    do Carmo, M.P.: Riemannian geometry. Mathematics: Theory & Applications. Birkhauser, Boston (1992)Google Scholar
  15. 15.
    Lee, J.M.: Riemannian manifolds. Graduate Texts in Mathematics, vol. 176. Springer, New York (1997); An introduction to curvatureGoogle Scholar
  16. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Noakes, L.: A global algorithm for geodesics. Journal of the Australian Mathematical Society 64, 37–50 (1998)CrossRefMathSciNetGoogle Scholar
  18. 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)CrossRefGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 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)Google Scholar
  21. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 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)Google Scholar
  23. 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)CrossRefGoogle Scholar
  24. 24.
    Said, S., Courty, N., Bihan, N.L., Sangwine, S.: Exact principal geodesic analysis for data on so(3). In: EUSIPCO 2007 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Stefan Sommer
    • 1
  • François Lauze
    • 1
  • Søren Hauberg
    • 1
  • Mads Nielsen
    • 1
    • 2
  1. 1.Dept. of Computer ScienceUniv. of CopenhagenDenmark
  2. 2.Nordic Bioscience ImagingHerlevDenmark

Personalised recommendations