Advertisement

Sparse Models for Intrinsic Shape Correspondence

  • Jonathan Pokrass
  • Alexander M. BronsteinEmail author
  • Michael M. Bronstein
  • Pablo Sprechmann
  • Guillermo Sapiro
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

We present a novel sparse modeling approach to non-rigid shape matching using only the ability to detect repeatable regions. As the input to our algorithm, we are given only two sets of regions in two shapes; no descriptors are provided so the correspondence between the regions is not know, nor do we know how many regions correspond in the two shapes. We show that even with such scarce information, it is possible to establish very accurate correspondence between the shapes by using methods from the field of sparse modeling, being this, the first non-trivial use of sparse models in shape correspondence. We formulate the problem of permuted sparse coding, in which we solve simultaneously for an unknown permutation ordering the regions on two shapes and for an unknown correspondence in functional representation. We also propose a robust variant capable of handling incomplete matches. Numerically, the problem is solved efficiently by alternating the solution of a linear assignment and a sparse coding problem. The proposed methods are evaluated qualitatively and quantitatively on standard benchmarks containing both synthetic and scanned objects.

Keywords

Sparse Code Iterative Close Point Sparse Modeling Proximal Operator Pursuit Problem 
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.

Notes

Acknowledgements

Work partially supported by GIF, ISF, and BSF. M.B. is supported by the ERC Starting grant No. 307047. A.B. is supported by the ERC Starting Grant No. 335491.

References

  1. 1.
    Aflalo, Y., Bronstein, A., Kimmel, R.: On convex relaxation of graph isomorphism. Proc. Nat. Acad. Sci. 112 (10), 2942–2947 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Anguelov, D., Srinivasan, P., Koller, D., Thrun, S., Rodgers, J., Davis, J.: Scape: shape completion and animation of people. In: Proceedings of the SIGGRAPH Conference, Los Angeles (2005)CrossRefGoogle Scholar
  3. 3.
    Aubry, M., Schlickewei, U., Cremers, D.: The wave kernel signature: a quantum mechanical approach to shape analysis. In: Proceeding of Workshop on Dynamic Shape Capture and Analysis, Barcelona (2011)Google Scholar
  4. 4.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Img. Sci. 2, 183–202 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Besl, P.J., McKay, N.D.: A method for registration of 3D shapes. Trans. PAMI 14, 239–256 (1992)CrossRefGoogle Scholar
  6. 6.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. PNAS 103 (5), 1168–1172 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Numerical Geometry of Non-rigid Shapes. Springer, New York (2008)zbMATHGoogle Scholar
  8. 8.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R., Mahmoudi, M., Sapiro, G.: A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching. IJCV 89 (2–3), 266–286 (2010)CrossRefGoogle Scholar
  9. 9.
    Bronstein, M.M., Bustos, B., Darom, T., Horaud, R., Hotz, I., Keller, Y., Keustermans, J., Kovnatsky, A., Litman, R., Reininghaus, J., Sipiran, I., Smeets, D., Suetens, P., Vandermeulen, D., Zaharescu, A., Zobel, V., Boyer, E., Bronstein, A.M.: Shrec 2011: robust feature detection and description benchmark. In: EUROGRAPHICS Workshop on 3D Object Retrieval (3DOR), Llandudno (2011)Google Scholar
  10. 10.
    Chen, Y., Medioni, G.: Object modeling by registration of multiple range images. In: Proceeding of Conference on Robotics and Automation, Sacramento (1991)CrossRefGoogle Scholar
  11. 11.
    Digne, J., Morel, J.M., Audfray, N., Mehdi-Souzani, C.: The level set tree on meshes. In: Proceeding 3DPVT, Paris (2010)Google Scholar
  12. 12.
    Elad, M.: Sparse and redundant representations: from theory to applications in signal and image processing. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Elad, A., Kimmel, R.: Bending invariant representations for surfaces. In: Proceedings of CVPR, Colorado, pp. 168–174 (2001)Google Scholar
  14. 14.
    Gebal, K., Bærentzen, J.A., Aanæs, H., Larsen, R.: Shape analysis using the auto diffusion function. Comput. Graph. Forum 28 (5), 1405–1413 (2009)CrossRefGoogle Scholar
  15. 15.
    Golovinskiy, A., Funkhouser, T.: Consistent segmentation of 3d models. Comput. Graph. 33 (3), 262–269 (2009)CrossRefGoogle Scholar
  16. 16.
    Huang, Q., Koltun, V., Guibas, L.: Joint shape segmentation with linear programming. TOG 30, 125 (2011)Google Scholar
  17. 17.
    Kaick, O.V., Zhang, H., Hamarneh, G., Cohen-Or, D.: A survey on shape correspondence. Comput. Graph. Forum 20, 1–23 (2010)Google Scholar
  18. 18.
    Kim, V.G., Lipman, Y., Funkhouser, T.: Blended intrinsic maps. TOG 30 (4), 79 (2011)CrossRefGoogle Scholar
  19. 19.
    Kuhn, H.W.: The Hungarian method for the assignment problem. Nav. Res. Logist. Quart. 2, 83–97 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lipman, Y., Funkhouser, T.: Mobius voting for surface correspondence. ACM Trans. Graph. (Proc. SIGGRAPH) 28 (3), 72 (2009)Google Scholar
  21. 21.
    Litman, R., Bronstein, A.M., Bronstein, M.M.: Diffusion-geometric maximally stable component detection in deformable shapes. Comput. Graph. 35 (3), 549–560 (2011)CrossRefGoogle Scholar
  22. 22.
    Mateus, D., Horaud, R., Knossow, D., Cuzzolin, F., Boyer, E.: Articulated shape matching using Laplacian eigenfunctions and unsupervised point registration. In: Proceeding CVPR, Anchorage (2008)CrossRefGoogle Scholar
  23. 23.
    Memoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Found. Comput. Math. 5 (3), 313–347 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nesterov, Y.: Gradient methods for minimizing composite objective function. In: CORE Discussion Paper 2007/76, Center for Operations Research and Econometrics (CORE). Catholic University of Louvain, Louvain-la-Neuve (2007)Google Scholar
  25. 25.
    Nguyen, A., Ben-Chen, M., Welnicka, K., Ye, Y., Guibas, L.: An optimization approach to improving collections of shape maps. Comput. Graph. Forum 30, 1481–1491 (2011)CrossRefGoogle Scholar
  26. 26.
    Ovsjanikov, M., Ben-Chen, M., Solomon, J., Butscher, A., Guibas, L.: Functional maps: a flexible representation of maps between shapes. TOG 31 (4), 129–139 (2012)CrossRefGoogle Scholar
  27. 27.
    Ovsjanikov, M., Mérigot, Q., Mémoli, F., Guibas, L.: One point isometric matching with the heat kernel. Comput. Graph. Forum 29, 1555–1564 (2010)CrossRefGoogle Scholar
  28. 28.
    Pokrass, J., Bronstein, A.M., Bronstein, M.M.: A correspondence-less approach to matching of deformable shapes. In: Proceeding SSVM, Ein-Gedi (2011)Google Scholar
  29. 29.
    Raviv, D., Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Symmetries of non-rigid shapes. In: Proceeding of Workshop on Non-rigid Registration and Tracking Through Learning (NRTL), Stony Brook (2005)Google Scholar
  30. 30.
    Rustamov, R.M.: Laplace-Beltrami eigenfunctions for deformation invariant shape representation. In: Proceeding of SGP, Barcelona, pp. 225–233 (2007)Google Scholar
  31. 31.
    Sahillioglu, Y., Yemez, Y.: Coarse-to-fine combinatorial matching for dense isometric shape correspondence. Comput. Graph. Forum 32, 177–189 (2012)CrossRefGoogle Scholar
  32. 32.
    Sprechmann, P., Bronstein, A.M., Sapiro, G.: Learning efficient structured sparse models. In: Proceedings of ICML, Edinburgh (2012)Google Scholar
  33. 33.
    Sun, J., Ovsjanikov, M., Guibas, L.J.: A concise and provably informative multi-scale signature based on heat diffusion. In: Proceedings of SGP, Berlin (2009)Google Scholar
  34. 34.
    Tevs, A., Berner, A., Wand, M., Ihrke, I., Seidel, H.P.: Intrinsic shape matching by planned landmark sampling. Comput. Graph. Forum 30, 543–552 (2011)CrossRefGoogle Scholar
  35. 35.
    Tibshirani, R.: Regression shrinkage and selection via the LASSO. J. R. Stat. Soc. Ser. B 58 (1), 267–288 (1996)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Van Kaick, O., Tagliasacchi, A., Sidi, O., Zhang, H., Cohen, D.-Or, Wolf, L., Hamarneh, G.: Prior knowledge for part correspondence. Comput. Graph. Forum 30, 553–562 (2011)Google Scholar
  37. 37.
    Zaharescu, A., Boyer, E., Varanasi, K., Horaud, R.: Surface feature detection and description with applications to mesh matching. In: Proceedings of CVPR, Miami (2009)CrossRefGoogle Scholar
  38. 38.
    Zeng, Y., Wang, C., Wang, Y., Gu, X., Samaras, D., Paragios, N.: Dense non-rigid surface registration using high-order graph matching. In: Proceedings of CVPR, San Francisco (2010)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Jonathan Pokrass
    • 1
  • Alexander M. Bronstein
    • 1
    Email author
  • Michael M. Bronstein
    • 2
  • Pablo Sprechmann
    • 3
  • Guillermo Sapiro
    • 3
  1. 1.School of Electrical EngineeringTel Aviv UniversityTel AvivIsrael
  2. 2.Faculty of InformaticsInstitute of Computational Science University of LuganoLuganoSwitzerland
  3. 3.School of Electrical and Computer EngineeringDuke UniversityDurhamUSA

Personalised recommendations