Advertisement

Shape Matching by Variational Computation of Geodesics on a Manifold

  • Frank R. Schmidt
  • Michael Clausen
  • Daniel Cremers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4174)

Abstract

Klassen et al. [9] recently developed a theoretical formulation to model shape dissimilarities by means of geodesics on appropriate spaces. They used the local geometry of an infinite dimensional manifold to measure the distance dist(A,B) between two given shapes A and B. A key limitation of their approach is that the computation of distances developed in the above work is inherently unstable, the computed distances are in general not symmetric, and the computation times are typically very large. In this paper, we revisit the shooting method of Klassen et al. for their angle-oriented representation. We revisit explicit expressions for the underlying space and we propose a gradient descent algorithm to compute geodesics. In contrast to the shooting method, the proposed variational method is numerically stable, it is by definition symmetric, and it is up to 1000 times faster.

Keywords

Variational Method Variational Computation Exponential Mapping Shooting Method Geodesic Equation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bergtholdt, M., Schnörr, C.: Shape Priors and Online Appearance Learning for Variational Segmentation and Object Recognition in Static Scenes. In: Kropatsch, W.G., Sablatnig, R., Hanbury, A. (eds.) DAGM 2005. LNCS, vol. 3663, pp. 342–350. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. In: Grundlehren der Mathematischen Wissenschaften, vol. 315. Springer, Heidelberg (1997)Google Scholar
  3. 3.
    Chern, S., Chen, W., Lam, K.S.: Lectures on Differential Geometry. World Scientific, Singapore (1999)MATHGoogle Scholar
  4. 4.
    Cremers, D., Kohlberger, T., Schnörr, C.: Shape statistics in kernel space for variational image segmentation. Pattern Recognition 36(9), 1929–1943 (2003)MATHCrossRefGoogle Scholar
  5. 5.
    Cremers, D., Soatto, S.: A pseudo-distance for shape priors in level set segmentation. In: Paragios, N. (ed.) IEEE 2nd Int. Workshop on Variational, Geometric and Level Set Methods, Nice, pp. 169–176 (2003)Google Scholar
  6. 6.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces, 503 pages. Prentice-Hall, Englewood Cliffs (1976)MATHGoogle Scholar
  7. 7.
    Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, Chichester (1998)MATHGoogle Scholar
  8. 8.
    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
  9. 9.
    Klassen, E., Srivastava, A., Mio, W., Joshi, S.H.: Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans. Pattern Anal. Mach. Intell. 26(3), 372–383 (2003)CrossRefGoogle Scholar
  10. 10.
    Marques, J.S., Abrantes, A.J.: Shape alignment – optimal initial point and pose estimation. Pattern Recognition Letters 18(1), 49–53 (1997)CrossRefGoogle Scholar
  11. 11.
    Michor, P., Mumford, D.: Riemannian geometries on spaces of plane curves. J. of the European Math. Society (2003)Google Scholar
  12. 12.
    Mokhtarian, F., Abbasi, S., Kittler, J.: Efficient and robust retrieval by shape content through curvature scale space (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Frank R. Schmidt
    • 1
  • Michael Clausen
    • 1
  • Daniel Cremers
    • 1
  1. 1.Department of Computer ScienceUniversity of BonnBonnGermany

Personalised recommendations