A Correspondence-Less Approach to Matching of Deformable Shapes

  • Jonathan Pokrass
  • Alexander M. Bronstein
  • Michael M. Bronstein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6667)

Abstract

Finding a match between partially available deformable shapes is a challenging problem with numerous applications. The problem is usually approached by computing local descriptors on a pair of shapes and then establishing a point-wise correspondence between the two. In this paper, we introduce an alternative correspondence-less approach to matching fragments to an entire shape undergoing a non-rigid deformation. We use diffusion geometric descriptors and optimize over the integration domains on which the integral descriptors of the two parts match. The problem is regularized using the Mumford-Shah functional. We show an efficient discretization based on the Ambrosio-Tortorelli approximation generalized to triangular meshes. Experiments demonstrating the success of the proposed method are presented.

Keywords

deformable shapes partial matching partial correspondence partial similarity diffusion geometry Laplace-Beltrami operator shape descriptors heat kernel signature Mumford-Shah regularization 

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References

  1. 1.
    Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via-convergence. Comm. Pure Appl. Math. 43(8), 999–1036 (1990)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bronstein, A.M., Bronstein, M.: Not only size matters: regularized partial matching of nonrigid shapes. In: Prof. NORDIA (2008)Google Scholar
  3. 3.
    Bronstein, A.M., Bronstein, M.M.: Regularized partial matching of rigid shapes. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part II. LNCS, vol. 5303, pp. 143–154. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Bronstein, A.M., Bronstein, M.M., Bruckstein, A.M., Kimmel, R.: Partial similarity of objects, or how to compare a centaur to a horse. IJCV 84(2), 163–183 (2009)CrossRefGoogle Scholar
  5. 5.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proc. National Academy of Science (PNAS) 103(5), 1168–1172 (2006)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Numerical geometry of non-rigid shapes. Springer-Verlag New York Inc., Secaucus (2008)MATHGoogle Scholar
  7. 7.
    Bronstein, A.M., Bronstein, M.M., Bustos, B., Castellani, U., Crisani, M., Falcidieno, B., Guibas, L.J., Kokkinos, I., Murino, V., Ovsjanikov, M., et al.: SHREC 2010: robust feature detection and description benchmark. In: Proc. 3DOR (2010)Google Scholar
  8. 8.
    Bronstein, A.M., Bronstein, M.M., Castellani, U., Dubrovina, A., Guibas, L.J., Horaud, R.P., Kimmel, R., Knossow, D., von Lavante, E., Mateus, D., et al.: SHREC 2010: robust correspondence benchmark. In: Eurographics Workshop on 3D Object Retrieval (2010)Google Scholar
  9. 9.
    Bronstein, M.M., Kokkinos, I.: Scale-invariant heat kernel signatures for non-rigid shape recognition. In: Proc. CVPR (2010)Google Scholar
  10. 10.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Processing 10(2), 266–277 (2001)CrossRefMATHGoogle Scholar
  11. 11.
    Clarenz, U., Rumpf, M., Telea, A.: Robust feature detection and local classification for surfaces based on moment analysis. Trans. Visualization and Computer Graphics 10(5), 516–524 (2004)CrossRefGoogle Scholar
  12. 12.
    Domokos, C., Kato, Z.: Affine puzzle: Realigning deformed object fragments without correspondences. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010. LNCS, vol. 6312, pp. 777–790. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling, vol. 1 (2005)Google Scholar
  14. 14.
    Gromov, M.: Structures Métriques Pour les Variétés Riemanniennes. Textes Mathématiques, vol. (1) (1981)Google Scholar
  15. 15.
    Jacobs, D., Weinshall, D., Gdalyahu, Y.: Class representation and image retrieval with non-metric distances. Trans. PAMI 22(6), 583–600 (2000)CrossRefGoogle Scholar
  16. 16.
    Johnson, A.E., Hebert, M.: Using spin images for efficient object recognition in cluttered 3D scenes. Trans. PAMI 21(5), 433–449 (1999)CrossRefGoogle Scholar
  17. 17.
    Lévy, B.: Laplace-Beltrami eigenfunctions towards an algorithm that “understands” geometry. In: Proc. Shape Modeling and Applications (2006)Google Scholar
  18. 18.
    Manay, S., Hong, B.W., Yezzi, A.J., Soatto, S.: Integral invariant signatures. LNCS, pp. 87–99 (2004)Google Scholar
  19. 19.
    Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations of Computational Mathematics 5, 313–346 (2005)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Meyer, M., Desbrun, M., Schroder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. Visualization and Mathematics III, 35–57 (2003)Google Scholar
  21. 21.
    Mitra, N.J., Guibas, L.J., Giesen, J., Pauly, M.: Probabilistic fingerprints for shapes. In: Proc. SGP (2006)Google Scholar
  22. 22.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Communications on pure and applied mathematics 42(5), 577–685 (1989)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Ovsjanikov, M., Bronstein, A.M., Guibas, L.J., Bronstein, M.M.: Shape Google: a computer vision approach to invariant shape retrieval. In: Proc. NORDIA, Citeseer (2009)Google Scholar
  24. 24.
    Pauly, M., Keiser, R., Gross, M.: Multi-scale feature extraction on point-sampled surfaces. In: Computer Graphics Forum, vol. 22, pp. 281–289 (2003)Google Scholar
  25. 25.
    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2(1), 15–36 (1993)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Reuter, M., Wolter, F.-E., Peinecke, N.: Laplace-spectra as fingerprints for shape matching. In: Proc. ACM Symp. Solid and Physical Modeling, pp. 101–106 (2005)Google Scholar
  27. 27.
    Sipiran, I., Bustos, B.: A robust 3D interest points detector based on Harris operator. In: Proc. 3DOR, pp. 7–14. Eurographics (2010)Google Scholar
  28. 28.
    Sivic, J., Zisserman, A.: Video Google: a text retrieval approach to object matching in videos. In: Proc. CVPR (2003)Google Scholar
  29. 29.
    Sun, J., Ovsjanikov, M., Guibas, L.: A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion. In: Computer Graphics Forum, vol. 28, pp. 1383–1392 (2009)Google Scholar
  30. 30.
    Toldo, R., Castellani, U., Fusiello, A.: Visual vocabulary signature for 3D object retrieval and partial matching. In: Proc. 3DOR (2009)Google Scholar
  31. 31.
    Wardetzky, M., Mathur, S., Kälberer, F., Grinspun, E.: Discrete Laplace operators: no free lunch. In: Conf. Computer Graphics and Interactive Techniques (2008)Google Scholar
  32. 32.
    Zaharescu, A., Boyer, E., Varanasi, K., Horaud, R.: Surface feature detection and description with applications to mesh matching. In: Proc. CVPR (2009)Google Scholar
  33. 33.
    Zhang, C., Chen, T.: Efficient feature extraction for 2D/3D objects in mesh representation. In: Proc. ICIP, vol. 3 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jonathan Pokrass
    • 1
  • Alexander M. Bronstein
    • 1
  • Michael M. Bronstein
    • 2
  1. 1.Dept. of Electrical EngineeringTel Aviv UniversityIsrael
  2. 2.Inst. of Computational Science, Faculty of InformaticsUniversità della Svizzera ItalianaLuganoSwitzerland

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