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Anisotropic Laplace-Beltrami Operators for Shape Analysis

  • Mathieu AndreuxEmail author
  • Emanuele Rodolà
  • Mathieu Aubry
  • Daniel Cremers
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8928)

Abstract

This paper introduces an anisotropic Laplace-Beltrami operator for shape analysis. While keeping useful properties of the standard Laplace-Beltrami operator, it introduces variability in the directions of principal curvature, giving rise to a more intuitive and semantically meaningful diffusion process. Although the benefits of anisotropic diffusion have already been noted in the area of mesh processing (e.g. surface regularization), focusing on the Laplacian itself, rather than on the diffusion process it induces, opens the possibility to effectively replace the omnipresent Laplace-Beltrami operator in many shape analysis methods. After providing a mathematical formulation and analysis of this new operator, we derive a practical implementation on discrete meshes. Further, we demonstrate the effectiveness of our new operator when employed in conjunction with different methods for shape segmentation and matching.

Keywords

Shape analysis Anisotropic diffusion Curvature Non-rigid matching Segmentation Laplace-Beltrami operator 

References

  1. 1.
    Aubry, M., Schlickewei, U., Cremers, D.: The wave kernel signature: A quantum mechanical approach to shape analysis. In: ICCV Workshops, pp. 1626–1633 (2011)Google Scholar
  2. 2.
    Black, M.J., Sapiro, G., Marimont, D.H., Heeger, D.: Robust anisotropic diffusion. Trans. Img. Proc. 7(3), 421–432 (1998)CrossRefGoogle Scholar
  3. 3.
    Boucher, M., Evans, A., Siddiqi, K.: Anisotropic Diffusion of Tensor Fields for Fold Shape Analysis on Surfaces. In: Székely, G., Hahn, H.K. (eds.) IPMI 2011. LNCS, vol. 6801, pp. 271–282. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  4. 4.
    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., Ovsjanikov, M., Sharma, A.: Shrec 2010: robust correspondence benchmark. In: Proc. EUROGRAPHICS Workshop on 3D Object Retrieval, EG 3DOR 2010 (2010)Google Scholar
  5. 5.
    Bronstein, A., Bronstein, M., Kimmel, R.: Numerical Geometry of Non-Rigid Shapes, 1st edn. Springer Publishing Company, Incorporated (2008)zbMATHGoogle Scholar
  6. 6.
    Clarenz, U., Diewald, U., Rumpf, M.: Anisotropic geometric diffusion in surface processing. In: Proc. of the Conference on Visualization 2000, VIS 2000, pp. 397–405 (2000)Google Scholar
  7. 7.
    Cohen-Steiner, D., Morvan, J.M.: Restricted delaunay triangulations and normal cycle. In: Proc. of the Nineteenth Annual Symposium on Computational Geometry, SCG 2003, pp. 312–321 (2003)Google Scholar
  8. 8.
    Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: Proc. of the 26th Annual Conference on Computer Graphics and Interactive Techniques. pp. 317–324. SIGGRAPH ’99 (1999)Google Scholar
  9. 9.
    Fillard, P., Arsigny, V., Ayache, N., Pennec, X.: A Riemannian Framework for the Processing of Tensor-Valued Images. In: Fogh Olsen, O., Florack, L.M.J., Kuijper, A. (eds.) DSSCV 2005. LNCS, vol. 3753, pp. 112–123. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  10. 10.
    de Goes, F., Liu, B., Budninskiy, M., Tong, Y., Desbrun, M.: Discrete 2-tensor fields on triangulations. Computer Graphics Forum 33(5) (2014)Google Scholar
  11. 11.
    Kim, K., Tompkin, J., Theobalt, C.: Curvature-aware regularization on Riemannian submanifolds. In: Proc. of the IEEE International Conference on Computer Vision, ICCV 2013, pp. 881–888 (2013)Google Scholar
  12. 12.
    Kovnatsky, A., Raviv, D., Bronstein, M.M., Bronstein, A.M., Kimmel, R.: Geometric and photometric data fusion in non-rigid shape analysis. Numerical Mathematics: Theory, Methods and Applications (NM-TMA) 6(1), 199–222 (2013)Google Scholar
  13. 13.
    Litman, R., Bronstein, A.: Learning spectral descriptors for deformable shape correspondence. IEEE Transactions on Pattern Analysis and Machine Intelligence 36(1), 171–180 (2014)CrossRefGoogle Scholar
  14. 14.
    Litman, R., Bronstein, A.M., Bronstein, M.M.: Diffusion-geometric maximally stable component detection in deformable shapes. Computers & Graphics 35(3), 549–560 (2011)CrossRefGoogle Scholar
  15. 15.
    Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. Visualization and Mathematics III, pp. 35–57 (2003)Google Scholar
  16. 16.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)CrossRefGoogle Scholar
  17. 17.
    Peyré, G.: Toolbox graph - a toolbox to process graph and triangulated meshes. (2008) https://www.ceremade.dauphine.fr~peyre/matlab/graph/content.html
  18. 18.
    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2(1), 15–36 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Pokrass, J., Bronstein, A.M., Bronstein, M.M., Sprechmann, P., Sapiro, G.: Sparse modeling of intrinsic correspondences. Computer Graphics Forum 32(2pt. 4), 459–468 (2013)Google Scholar
  20. 20.
    Reuter, M., Biasotti, S., Giorgi, D., Patanè, G., Spagnuolo, M.: Discrete laplace-beltrami operators for shape analysis and segmentation. Computers & Graphics 33(3), 381–390 (2009)CrossRefGoogle Scholar
  21. 21.
    Rodolà, E., Rota Bulò, S., Cremers, D.: Robust region detection via consensus segmentation of deformable shapes. Computer Graphics Forum 33(5) (2014)Google Scholar
  22. 22.
    Rodolà, E., Rota Bulò, S., Windheuser, T., Vestner, M., Cremers, D.: Dense non-rigid shape correspondence using random forests. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2014)Google Scholar
  23. 23.
    Rustamov, R.M.: Laplace-beltrami eigenfunctions for deformation invariant shape representation. In: Proc. of the Fifth Eurographics Symposium on Geometry Processing, SGP 2007, pp. 225–233 (2007)Google Scholar
  24. 24.
    Sun, J., Ovsjanikov, M., Guibas, L.: A concise and provably informative multi-scale signature based on heat diffusion. In: Proc. of the Symposium on Geometry Processing, SGP 2009, pp. 1383–1392 (2009)Google Scholar
  25. 25.
    Tasdizen, T., Whitaker, R., Burchard, P., Osher, S.: Geometric surface smoothing via anisotropic diffusion of normals. In: Proc. of the Conference on Visualization 2002, VIS 2002, pp. 125–132 (2002)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Mathieu Andreux
    • 1
    Email author
  • Emanuele Rodolà
    • 2
  • Mathieu Aubry
    • 3
  • Daniel Cremers
    • 2
  1. 1.École polytechniquePalaiseauFrance
  2. 2.Technische Universität MünchenMunichGermany
  3. 3.Université Paris Est LIGM - École des Ponts ParistechChamps-sur-MarneFrance

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