Skip to main content
Log in

Interpolated eigenfunctions for volumetric shape processing

  • Original Article
  • Published:
The Visual Computer Aims and scope Submit manuscript

Abstract

This paper introduces a set of volumetric functions suitable for geometric processing of volumes. We start with Laplace–Beltrami eigenfunctions on the bounding surface and interpolate them into the interior using barycentric coordinates. The interpolated eigenfunctions: (1) can be computed efficiently by using the boundary mesh only; (2) can be seen as a shape-aware generalization of barycentric coordinates; (3) can be used for efficiently representing volumetric functions; (4) can be naturally plugged into existing spectral embedding constructions such as the diffusion embedding to provide their volumetric counterparts. Using the interior diffusion embedding, we define the interior Heat Kernel Signature (iHKS) and examine its performance for the task of volumetric point correspondence. We show that the three main qualities of the surface Heat Kernel Signature—being informative, multiscale, and insensitive to pose—are inherited by this volumetric construction. Next, we construct a bag of features based shape descriptor that aggregates the iHKS signatures over the volume of a shape, and evaluate its performance on a public shape retrieval benchmark. We find that while, theoretically, strict isometry invariance requires concentrating on the intrinsic surface properties alone, yet, practically, pose insensitive shape retrieval can be achieved using volumetric information.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anguelov, D., Srinivasan, P., Koller, D., Thrun, S., Rodgers, J., Davis, J.: Scape: shape completion and animation of people. ACM Trans. Graph. 24(3), 408–416 (2005)

    Article  Google Scholar 

  2. Belyaev, A.: On transfinite barycentric coordinates. In: SGP, pp. 89–99 (2006)

    Google Scholar 

  3. Ben-Chen, M., Gotsman, C.: On the optimality of spectral compression of mesh data. ACM Trans. Graph. 24, 60–80 (2005). http://doi.acm.org/10.1145/1037957.1037961

    Article  Google Scholar 

  4. Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  5. Botsch, M., Sorkine, O.: On linear variational surface deformation methods. IEEE Trans. Vis. Comput. Graph. 14(1), 213–230 (2008)

    Article  Google Scholar 

  6. Bronstein, A.M., Bronstein, M.M., Guibas, L.J., Ovsjanikov, M.: Shape google: Geometric words and expressions for invariant shape retrieval. ACM Trans. Graph. 30, 1–20 (2011). http://doi.acm.org/10.1145/1899404.1899405

    Article  Google Scholar 

  7. Bustos, B., Keim, D.A., Saupe, D., Schreck, T., Vranić, D.V.: Feature-based similarity search in 3d object databases. ACM Comput. Surv. 37(4), 345–387 (2005)

    Article  Google Scholar 

  8. Coifman, R.R., Lafon, S.: Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions. Appl. Comput. Harmon. Anal. 21, 31–52 (2006). doi:10.1016/j.acha.2005.07.005

    Article  MathSciNet  MATH  Google Scholar 

  9. Coifman, R.R., Lafon, S., Lee, A.B., Maggioni, M., Nadler, B., Warner, F., Zucker, S.W.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps. Proc. Natl. Acad. Sci. USA 102(21), 7426–7431 (2005). http://www.pnas.org/cgi/content/abstract/102/21/7432

    Article  Google Scholar 

  10. Gal, R., Shamir, A., Cohen-Or, D.: Pose-oblivious shape signature. IEEE Trans. Vis. Comput. Graph. 13(2), 261–271 (2007). http:doi.ieeecomputersociety.org/10.1109/TVCG.2007.45

    Article  Google Scholar 

  11. Garland, M., Zhou, Y.: Quadric-based simplification in any dimension. ACM Trans. Graph. 24(2), 209–239 (2005)

    Article  Google Scholar 

  12. Iyer, N., Jayanti, S., Lou, K., Kalyanaraman, Y., Ramani, K.: Three-dimensional shape searching: state-of-the-art review and future trends. Comput. Aided Des. 37(5), 509–530 (2005)

    Article  Google Scholar 

  13. Joshi, P., Meyer, M., DeRose, T., Green, B., Sanocki, T.: Harmonic coordinates for character articulation. In: TOG (SIGGRAPH), p. 71 (2007)

    Google Scholar 

  14. Ju, T., Schaefer, S., Warren, J.: Mean value coordinates for closed triangular meshes. In: TOG (SIGGRAPH), pp. 561–566 (2005)

    Google Scholar 

  15. Lévy, B.: Laplace–Beltrami eigenfunctions: Towards an algorithm that understands geometry. In: Shape Modeling International (2006)

    Google Scholar 

  16. Lian, Z., Godil, A., Bustos, B., Daoudi, M., Hermans, J., Kawamura, S., Kurita, Y., Lavoue, G., Nguyen, H.V., Ohbuchi, R., Ohkita, Y., Ohishi, Y., Porikli, F., Reuter, M., Sipiran, I., Smeets, D., Suetens, P., Tabia, H., Vandermeulen, D.: SHREC ’11 track: shape retrieval on non-rigid 3d watertight meshes, pp. 79–88. doi:10.2312/3DOR/3DOR11/079-088. http://diglib.eg.org/EG/DL/WS/3DOR/3DOR11/079-088.pdf

  17. Ling, H., Jacobs, D.: Shape classification using the inner-distance. IEEE Trans. Pattern Anal. Mach. Intell. 29(2), 286–299 (2007). doi:10.1109/TPAMI.2007.41

    Article  Google Scholar 

  18. Lipman, Y., Rustamov, R.M., Funkhouser, T.A.: Biharmonic distance. ACM Trans. Graph. 29, 1–11 (2010). http://doi.acm.org/10.1145/1805964.1805971

    Google Scholar 

  19. Liu, Y.S., Fang, Y., Ramani, K.: Idss: deformation invariant signatures for molecular shape comparison. BMC Bioinform. 10(1), 157 (2009). doi:10.1186/1471-2105-10-157. http://www.biomedcentral.com/1471-2105/10/157

    Article  Google Scholar 

  20. Mémoli, F.: A spectral notion of Gromov–Wasserstein distances and related methods. Appl. Comput. Harmon. Anal. 30, 363–401 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Meyer, M., Desbrun, M., Schröder, P., Barr, A.: Discrete differential geometry operators for triangulated 2-manifolds. In: Proceedings of Visual Mathematics (2002)

    Google Scholar 

  22. Min, P.: Binvox. http://www.google.com/search?q=binvox

  23. Nooruddin, F.S., Turk, G.: Simplification and repair of polygonal models using volumetric techniques. IEEE Trans. Vis. Comput. Graph. 9(2), 191–205 (2003)

    Article  Google Scholar 

  24. Ovsjanikov, M., Sun, J., Guibas, L.: Global intrinsic symmetries of shapes. In: Eurographics Symposium on Geometry Processing (SGP) (2008)

    Google Scholar 

  25. Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)

    MathSciNet  MATH  Google Scholar 

  26. Raviv, D., Bronstein, M.M., Bronstein, A.M., Kimmel, R.: Volumetric heat kernel signatures. In: Proceedings of the ACM Workshop on 3D Object Retrieval, 3DOR ’10, pp. 39–44. ACM, New York (2010). http://doi.acm.org/10.1145/1877808.1877817

    Chapter  Google Scholar 

  27. Reuter, M., Wolter, F.E., Peinecke, N.: Laplace-spectra as fingerprints for shape matching. In: Solid and Physical Modeling, pp. 101–106 (2005)

    Google Scholar 

  28. Reuter, M., Wolter, F.E., Shenton, M., Niethammer, M.: Laplace-Beltrami eigenvalues and topological features of eigenfunctions for statistical shape analysis. Comput. Aided Des. 41(10), 739–755 (2009). doi:10.1016/j.cad.2009.02.007

    Article  Google Scholar 

  29. Rustamov, R.: Laplace–Beltrami eigenfunctions for deformation invariant shape representation. In: Symposium on Geometry Processing (2007)

    Google Scholar 

  30. Rustamov, R.: On manifold learning and mesh editing. Tech. rep. (2008)

  31. Rustamov, R., Lipman, Y., Funkhouser, T.: Interior distance using barycentric coordinates. Comput. Graph. Forum (Symposium on Geometry Processing) 28(5) (2009)

  32. Shen, Y., Ma, L., Liu, H.: An mls-based cartoon deformation. Vis. Comput. 26, 1229–1239 (2010). doi:10.1007/s00371-009-0404-7

    Article  Google Scholar 

  33. Sun, J., Ovsjanikov, M., Guibas, L.: A concise and provably informative multi-scale signature based on heat diffusion. In: Proceedings of the Symposium on Geometry Processing, SGP ’09, pp. 1383–1392. Eurographics Association, Aire-la-Ville (2009). http://portal.acm.org/citation.cfm?id=1735603.1735621

    Google Scholar 

  34. Tangelder, J., Veltkamp, R.: A survey of content based 3d shape retrieval methods. Multimed. Tools Appl. 39, 441–471 (2008)

    Article  Google Scholar 

  35. Yu, Y., Zhou, K., Xu, D., Shi, X., Bao, H., Guo, B., Shum, H.Y.: Mesh editing with Poisson-based gradient field manipulation. In: TOG (SIGGRAPH), pp. 644–651 (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Raif M. Rustamov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rustamov, R.M. Interpolated eigenfunctions for volumetric shape processing. Vis Comput 27, 951–961 (2011). https://doi.org/10.1007/s00371-011-0629-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00371-011-0629-0

Keywords

Navigation