Geodesics in Shape Space via Variational Time Discretization

  • Benedikt Wirth
  • Leah Bar
  • Martin Rumpf
  • Guillermo Sapiro
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5681)


A variational approach to defining geodesics in the space of implicitly described shapes is introduced in this paper. The proposed framework is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 1-1 property of the induced flow in shape space. For decreasing time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the shape space. Considering shapes as boundary contours, the proposed shape metric is identical to a physical dissipation in a viscous fluid model of optimal transportation. If the pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, for decreasing time step size one obtains an additional optical flow term controlling the transport of the shape by the underlying motion field. The implementation of the proposed approach is based on a level set representation of shapes, which allows topological transitions along the geodesic path. For the spatial discretization a finite element approximation is employed both for the pairwise deformations and for the level set representation. The numerical relaxation of the energy is performed via an efficient multi–scale procedure in space and time. Examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach.


Rigid Body Motion Shape Space Boundary Contour Geodesic Path Optimal Transportation 
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.


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  1. 1.
    Zhu, L., Yang, Y., Haker, S., Allen, T.: An image morphing technique based on optimal mass preserving mapping. IEEE T. Image Process 16(6), 1481–1495 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Fuchs, M., Jüttler, B., Scherzer, O., Yang, H.: Shape metrics based on elastic deformations. Journal of Mathematical Imaging and Vision (to appear, 2009)Google Scholar
  3. 3.
    Miller, M.I., Younes, L.: Group actions, homeomorphisms and matching: a general framework. International Journal of Computer Vision 41(1-2), 61–84 (2001)CrossRefzbMATHGoogle Scholar
  4. 4.
    Schmidt, F.R., Clausen, M., Cremers, D.: Shape matching by variational computation of geodesics on a manifold. In: Franke, K., Müller, K.-R., Nickolay, B., Schäfer, R. (eds.) DAGM 2006. LNCS, vol. 4174, pp. 142–151. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Zhao, H.K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comp. Phys. 127, 179–195 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Luckhaus, S., Sturzenhecker, T.: Implicit time discretization for the mean curvature flow equation. Calc. Var. 3, 253–271 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ciarlet, P.G.: Three-dimensional elasticity. Elsevier Science Publisers B. V., Amsterdam (1988)zbMATHGoogle Scholar
  8. 8.
    Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Found. Comput. Math. 5, 313–347 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bronstein, A., Bronstein, M., Kimmel, R.: Numerical Geometry of Non-Rigid Shapes. Monographs in Computer Science. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  11. 11.
    Charpiat, G., Faugeras, O., Keriven, R.: Approximations of shape metrics and application to shape warping and empirical shape statistics. Foundations of Computational Mathematics 5(1), 1–58 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eckstein, I., Pons, J., Tong, Y., Kuo, C., Desbrun, M.: Generalized surface flows for mesh processing. In: Eurographics Symposium on Geometry Processing (2007)Google Scholar
  13. 13.
    Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8, 1–48 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Younes, L.: Computable elastic distances between shapes. SIAM J. Appl. Math. 58, 565–586 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Miller, M.I., Younes, L.: Group actions, homeomorphisms and matching: a general framework. Technical report, John Hopkins University, Maryland (1999)Google Scholar
  16. 16.
    Klassen, E., Srivastava, A., Mio, W., Joshi, S.: Analysis of planar shapes using geodesic paths on shape spaces. IEEE T. Pattern Anal. 26(3), 372–383 (2004)CrossRefGoogle Scholar
  17. 17.
    Dupuis, D., Grenander, U., Miller, M.: Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics 56, 587–600 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Miller, M., Trouvé, A., Younes, L.: On the metrics and Euler-Lagrange equations of computational anatomy. Ann. Rev. Biomed. Eng. 4, 375–405 (2002)CrossRefGoogle Scholar
  19. 19.
    Sundaramoorthi, G., Yezzi, A., Mennucci, A.: Sobolev active contours. International Journal of Computer Vision 73(3), 345–366 (2007)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kilian, M., Mitra, N.J., Pottmann, H.: Geometric modeling in shape space. ACM Transactions on Graphics 26(64), 1–8 (2007)Google Scholar
  21. 21.
    Droske, M., Rumpf, M.: Multi scale joint segmentation and registration of image morphology. IEEE Trans. Pattern Anal. 29(12), 2181–2194 (2007)CrossRefGoogle Scholar
  22. 22.
    Ball, J.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh 88A, 315–328 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Charpiat, G., Maurel, P., Pons, J.P., Keriven, R., Faugeras, O.: Generalized gradients: Priors on minimization flows. Int. J. Comput. Vision 73(3), 325–344 (2007)CrossRefGoogle Scholar
  24. 24.
    Kornprobst, P., Deriche, R., Aubert, G.: Image sequence analysis via partial differential equations. Journal of Mathematical Imaging and Vision 11, 5–26 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Black, M.J., Anandan, P.: A framework for the robust estimation of optical flow. In: Fourth International Conference on Computer Vision, ICCV 1993, pp. 231–236 (1993)Google Scholar
  26. 26.
    Kapur, T., Yezzi, L., Zöllei, L.: A variational framework for joint segmentation and registration. In: IEEE CVPR - MMBIA, pp. 44–51 (2001)Google Scholar
  27. 27.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Transactions on Image Processing 10(2), 266–277 (2001)CrossRefzbMATHGoogle Scholar
  28. 28.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. International Journal of Computer Vision 22(1), 61–79 (1997)CrossRefzbMATHGoogle Scholar
  29. 29.
    Bornemann, F., Deuflhard, P.: The cascadic multigrid method for elliptic problems. Num. Math. 75(2), 135–152 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Benedikt Wirth
    • 1
  • Leah Bar
    • 2
  • Martin Rumpf
    • 1
  • Guillermo Sapiro
    • 2
  1. 1.Institute for Numerical SimulationUniversity of BonnGermany
  2. 2.Department of Electrical and Computer EngineeringUniversity of MinnesotaMinneapolisU.S.A.

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