Manifold Statistics for Essential Matrices

  • Gijs Dubbelman
  • Leo Dorst
  • Henk Pijls
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7573)


Riemannian geometry allows for the generalization of statistics designed for Euclidean vector spaces to Riemannian manifolds. It has recently gained popularity within computer vision as many relevant parameter spaces have such a Riemannian manifold structure. Approaches which exploit this have been shown to exhibit improved efficiency and accuracy. The Riemannian logarithmic and exponential mappings are at the core of these approaches.

In this contribution we review recently proposed Riemannian mappings for essential matrices and prove that they lead to sub-optimal manifold statistics. We introduce correct Riemannian mappings by utilizing a multiple-geodesic approach and show experimentally that they provide optimal statistics.


Riemannian Manifold Tangent Vector Riemannian Geometry Riemannian Mapping Unit Quaternion 
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.


  1. 1.
    Karcher, H.: Riemannian Center of Mass and Mollifier Smoothing. Communications on Pure and Applied Mathematics 30, 509–541 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Pennec, X., Ayache, N.: Uniform Distribution, Distance and Expectation Problems for Geometric Features Processing. Journal of Mathematical Imaging and Vision 9, 49–67 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Pennec, X.: Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. Journal of Mathematical Imaging and Vision 25, 127–154 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Subbarao, R., Meer, P.: Nonlinear Mean Shift for Clustering over Analytic Manifolds. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 1168–1175 (2006)Google Scholar
  5. 5.
    Subbarao, R., Meer, P.: Nonlinear Mean Shift over Riemannian Manifolds. International Journal of Computer Vision 84, 1–20 (2009)CrossRefGoogle Scholar
  6. 6.
    Subbarao, R., Genc, Y., Meer, P.: Nonlinear Mean Shift for Robust Pose Estimation. In: IEEE Workshop on Applications of Computer Vision, vol. 6 (2007)Google Scholar
  7. 7.
    Subbarao, R., Genc, Y., Meer, P.: Robust Unambiguous Parametrization of the Essential Manifold. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2008)Google Scholar
  8. 8.
    Tuzel, O., Subbarao, R., Meer, P.: Simultaneous Multiple 3D Motion Estimation via Mode Finding on Lie Groups. In: Proceedings of the Tenth IEEE International Conference on Computer Vision, vol. 1, pp. 18–25 (2005)Google Scholar
  9. 9.
    Costa, J., Hero, A.O.: Learning I. In: Statistics and Analysis of Shapes, pp. 231–252. Birkhauser, Cambridge (2006)Google Scholar
  10. 10.
    Fletcher, P.T., Joshi, S.: Riemannian Geometry for the Statistical Analysis of Diffusion Tensor Data. Signal Processing 87, 250–262 (2007)zbMATHCrossRefGoogle Scholar
  11. 11.
    Pennec, X., Guttmann, C.R.G., 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)CrossRefGoogle Scholar
  12. 12.
    Begelfor, E., Werman, M.: Affine Invariance Revisited. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 2087–2094 (2006)Google Scholar
  13. 13.
    Goh, A., Vidal, R.: Clustering and Dimensionality Reduction on Riemannian Manifolds. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 1–7 (2008)Google Scholar
  14. 14.
    Tuzel, O., Porikli, F., Meer, P.: Pedestrian Detection via Classification on Riemannian Manifolds. IEEE Transactions on Pattern Analysis and Machine Intelligence 30, 1713–1727 (2008)CrossRefGoogle Scholar
  15. 15.
    Helmke, U., Hüper, K., Lee, P.Y., Moore, J.: Essential Matrix Estimation Using Gauss-Newton Iterations on a Manifold. International Journal of Computer Vision 74, 117–136 (2007)CrossRefGoogle Scholar
  16. 16.
    Kǒsecká, J., Ma, Y., Sastry, S.S.: Optimization Criteria, Sensitivity and Robustness of Motion and Structure Estimation. In: Triggs, B., Zisserman, A., Szeliski, R. (eds.) ICCV-WS 1999. LNCS, vol. 1883, pp. 166–182. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. 17.
    Ma, Y., Košecká, J., Sastry, S.: Optimization Criteria and Geometric Algorithms for Motion and Structure Estimation. International Journal of Computer Vision 44, 219–249 (2001)zbMATHCrossRefGoogle Scholar
  18. 18.
    Fischler, M.A., Bolles, R.C.: Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM 24, 381–395 (1981)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Raguram, R., Frahm, J., Pollefeys, M.: Exploiting Uncertainty in Random Sample Consensus. In: International Conference on Computer Vision (2009)Google Scholar
  20. 20.
    Chum, O., Matas, J., Kittler, J.: Locally Optimized RANSAC. In: Michaelis, B., Krell, G. (eds.) DAGM 2003. LNCS, vol. 2781, pp. 236–243. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  21. 21.
    do Carmo, M.P.: Riemannian Geometry, 1st edn. Mathematics: Theory and Applications. Birkhäuser (1992)Google Scholar
  22. 22.
    Cullen, H.F.: Introduction to General Topology, Heath, Boston (1967)Google Scholar
  23. 23.
    Longuet-Higgins, H.C.: A Computer Algorithm for Reconstructing a Scene from Two Projections. Nature 293, 133–135 (1981)CrossRefGoogle Scholar
  24. 24.
    Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press (2004) ISBN: 0521540518 Google Scholar
  25. 25.
    Park, F.C.: Distance Metrics on the Rigid-Body Motions with Applications to Mechanism Design. Transactions of the ASME 117, 48–54 (1995)CrossRefGoogle Scholar
  26. 26.
    Eisenberg, M., Guy, R.: A Proof of the Hairy Ball Theorem. The American Mathematical Monthly 86, 571–574 (1979)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Selig, J.M.: Geometrical Methods in Robotics. Springer (1996)Google Scholar
  28. 28.
    Strecha, C., von Hansen, W., van Gool, L., Fua, P., Thoennessen, U.: On Benchmarking Camera Calibration and Multi-View Stereo for High Resolution Imagery. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Gijs Dubbelman
    • 1
  • Leo Dorst
    • 2
  • Henk Pijls
    • 2
  1. 1.The Robotics InstituteCarnegie Mellon UniversityPittsburghUSA
  2. 2.Faculty of ScienceUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations