Advertisement

Deformable Shape Retrieval by Learning Diffusion Kernels

  • Yonathan Aflalo
  • Alexander M. Bronstein
  • Michael M. Bronstein
  • Ron Kimmel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6667)

Abstract

In classical signal processing, it is common to analyze and process signals in the frequency domain, by representing the signal in the Fourier basis, and filtering it by applying a transfer function on the Fourier coefficients. In some applications, it is possible to design an optimal filter. A classical example is the Wiener filter that achieves a minimum mean squared error estimate for signal denoising. Here, we adopt similar concepts to construct optimal diffusion geometric shape descriptors. The analogy of Fourier basis are the eigenfunctions of the Laplace-Beltrami operator, in which many geometric constructions such as diffusion metrics, can be represented. By designing a filter of the Laplace-Beltrami eigenvalues, it is theoretically possible to achieve invariance to different shape transformations, like scaling. Given a set of shape classes with different transformations, we learn the optimal filter by minimizing the ratio between knowingly similar and knowingly dissimilar diffusion distances it induces. The output of the proposed framework is a filter that is optimally tuned to handle transformations that characterize the training set.

Keywords

Heat Kernel Retrieval Performance Fourier Basis Shape Class Spectral Distance 
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.
    Bérard, P., Besson, G., Gallot, S.: Embedding riemannian manifolds by their heat kernel. Geometric and Functional Analysis 4(4), 373–398 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bronstein, A.M., Bronstein, M.M., Bustos, B., Castellani, U., Crisani, M., Falcidieno, B., Guibas, L.J., Sipiran, I., Kokkinos, I., Murino, V., Ovsjanikov, M., Patané, G., Spagnuolo, M., Sun, J.: SHREC 2010: robust feature detection and description benchmark. In: Proc. 3DOR (2010)Google Scholar
  3. 3.
    Bronstein, A.M., Bronstein, M.M., Castellani, U., Falcidieno, B., Fusiello, A., Godil, A., Guibas, L.J., Kokkinos, I., Lian, Z., Ovsjanikov, M., Patané, G., Spagnuolo, M., Toldo, R.: Shrec 2010: robust large-scale shape retrieval benchmark. In: Proc. 3DOR (2010)Google Scholar
  4. 4.
    Bronstein, A.M., Bronstein, M.M., Ovsjanikov, M., Guibas, L.J.: Shape google: a computer vision approach to invariant shape retrieval. In: Proc. NORDIA (2009)Google Scholar
  5. 5.
    Bronstein, M.M., Bronstein, A.M.: Shape recognition with spectral distances. Trans. PAMI (2010) (to appear)Google Scholar
  6. 6.
    Bronstein, M.M., Kokkinos, I.: Scale-invariant heat kernel signatures for non-rigid shape recognition. In: Proc. CVPR (2010)Google Scholar
  7. 7.
    Coifman, R.R., Lafon, S.: Diffusion maps. Applied and Computational Harmonic Analysis 21, 5–30 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling, vol. 1 (2005)Google Scholar
  9. 9.
    Lévy, B.: Laplace-Beltrami eigenfunctions towards an algorithm that “understands” geometry. In: Proc. Shape Modeling and Applications (2006)Google Scholar
  10. 10.
    Mahmoudi, M., Sapiro, G.: Three-dimensional point cloud recognition via distributions of geometric distances. Graphical Models 71(1), 22–31 (2009)CrossRefGoogle Scholar
  11. 11.
    Mateus, D., Horaud, R.P., Knossow, D., Cuzzolin, F., Boyer, E.: Articulated shape matching using laplacian eigenfunctions and unsupervised point registration. In: Proc. CVPR (June 2008)Google Scholar
  12. 12.
    Meyer, M., Desbrun, M., Schroder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Visualization and Mathematics III, pp. 35–57 (2003)Google Scholar
  13. 13.
    Ovsjanikov, M., Sun, J., Guibas, L.J.: Global intrinsic symmetries of shapes. Computer Graphics Forum 27, 1341–1348 (2008)CrossRefGoogle Scholar
  14. 14.
    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2(1), 15–36 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    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
  16. 16.
    Rustamov, R.M.: Laplace-Beltrami eigenfunctions for deformation invariant shape representation. In: Proc. SGP, pp. 225–233 (2007)Google Scholar
  17. 17.
    Spira, A., Sochen, N., Kimmel, R.: Geometric filters, diffusion flows, and kernels in image processing. In: Handbook of Computational Geometry for Pattern Recognition, Computer Vision, Neurocomputing and Robotics. Springer, Heidelberg (2005)Google Scholar
  18. 18.
    Sun, J., Ovsjanikov, M., Guibas, L.J.: A concise and provably informative multi-scale signature based on heat diffusion. In: Proc. SGP (2009)Google Scholar
  19. 19.
    Wardetzky, M., Mathur, S., Kälberer, F., Grinspun, E.: Discrete Laplace operators: no free lunch. In: Conf. Computer Graphics and Interactive Techniques (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yonathan Aflalo
    • 1
  • Alexander M. Bronstein
    • 2
  • Michael M. Bronstein
    • 3
  • Ron Kimmel
    • 1
  1. 1.Technion, Israel Institute of TechnologyHaifaIsrael
  2. 2.Dept. of Electrical EngineeringTel Aviv UniversityIsrael
  3. 3.Inst. of Computational Science, Faculty of InformaticsUniversità della Svizzera ItalianaLuganoSwitzerland

Personalised recommendations