The Visual Computer

, 27:951 | Cite as

Interpolated eigenfunctions for volumetric shape processing

Original Article


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.


Volumetric shape processing Laplace–Beltrami eigenfunctions Barycentric interpolation Heat kernel signature 


  1. 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) CrossRefGoogle Scholar
  2. 2.
    Belyaev, A.: On transfinite barycentric coordinates. In: SGP, pp. 89–99 (2006) Google Scholar
  3. 3.
    Ben-Chen, M., Gotsman, C.: On the optimality of spectral compression of mesh data. ACM Trans. Graph. 24, 60–80 (2005). CrossRefGoogle Scholar
  4. 4.
    Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003) MATHCrossRefGoogle Scholar
  5. 5.
    Botsch, M., Sorkine, O.: On linear variational surface deformation methods. IEEE Trans. Vis. Comput. Graph. 14(1), 213–230 (2008) CrossRefGoogle Scholar
  6. 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). CrossRefGoogle Scholar
  7. 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) CrossRefGoogle Scholar
  8. 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 MathSciNetMATHCrossRefGoogle Scholar
  9. 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). CrossRefGoogle Scholar
  10. 10.
    Gal, R., Shamir, A., Cohen-Or, D.: Pose-oblivious shape signature. IEEE Trans. Vis. Comput. Graph. 13(2), 261–271 (2007). CrossRefGoogle Scholar
  11. 11.
    Garland, M., Zhou, Y.: Quadric-based simplification in any dimension. ACM Trans. Graph. 24(2), 209–239 (2005) CrossRefGoogle Scholar
  12. 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) CrossRefGoogle Scholar
  13. 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. 14.
    Ju, T., Schaefer, S., Warren, J.: Mean value coordinates for closed triangular meshes. In: TOG (SIGGRAPH), pp. 561–566 (2005) Google Scholar
  15. 15.
    Lévy, B.: Laplace–Beltrami eigenfunctions: Towards an algorithm that understands geometry. In: Shape Modeling International (2006) Google Scholar
  16. 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.
  17. 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 CrossRefGoogle Scholar
  18. 18.
    Lipman, Y., Rustamov, R.M., Funkhouser, T.A.: Biharmonic distance. ACM Trans. Graph. 29, 1–11 (2010). Google Scholar
  19. 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. CrossRefGoogle Scholar
  20. 20.
    Mémoli, F.: A spectral notion of Gromov–Wasserstein distances and related methods. Appl. Comput. Harmon. Anal. 30, 363–401 (2011) MathSciNetMATHCrossRefGoogle Scholar
  21. 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. 22.
  23. 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) CrossRefGoogle Scholar
  24. 24.
    Ovsjanikov, M., Sun, J., Guibas, L.: Global intrinsic symmetries of shapes. In: Eurographics Symposium on Geometry Processing (SGP) (2008) Google Scholar
  25. 25.
    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993) MathSciNetMATHGoogle Scholar
  26. 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). CrossRefGoogle Scholar
  27. 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. 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 CrossRefGoogle Scholar
  29. 29.
    Rustamov, R.: Laplace–Beltrami eigenfunctions for deformation invariant shape representation. In: Symposium on Geometry Processing (2007) Google Scholar
  30. 30.
    Rustamov, R.: On manifold learning and mesh editing. Tech. rep. (2008) Google Scholar
  31. 31.
    Rustamov, R., Lipman, Y., Funkhouser, T.: Interior distance using barycentric coordinates. Comput. Graph. Forum (Symposium on Geometry Processing) 28(5) (2009) Google Scholar
  32. 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 CrossRefGoogle Scholar
  33. 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). Google Scholar
  34. 34.
    Tangelder, J., Veltkamp, R.: A survey of content based 3d shape retrieval methods. Multimed. Tools Appl. 39, 441–471 (2008) CrossRefGoogle Scholar
  35. 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

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Drew UniversityMadisonUSA

Personalised recommendations