Skip to main content
Log in

A versatile framework for shape description

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

Abstract

We present a shape description framework that generates a multitude of shape descriptors through a variety of design and continuum of parameter choices. Our parameter is a surface mesh, referred to as the template, which is supplied at run time, and allows generating different shape descriptors for the same model. Our framework extracts a numerical shape descriptor by computing a selected function on the model mesh, mapping (transferring) it to the template, expanding the mapped function in terms of a basis on the template, and collecting the expansion coefficients into a vector. We investigate possible design choices for the steps in the framework, and introduce novel approaches that provide further freedom in generating a multitude of previously unknown descriptors. We show that our approach is a generalization of the way some of the existing numerical descriptors are defined, and that for appropriate template choices one is able to reproduce some of the well-known descriptors. Finally, we show empirically that design and parameter choices have non-trivial effects on the descriptor’s performance, and that better retrieval results can be obtained by combining descriptors obtained via different templates.

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.

Similar content being viewed by others

References

  1. Belkin, M., Niyogi, P., Sindhwani, V.: Manifold regularization: a geometric framework for learning from labeled and unlabeled examples. J. Mach. Learn. Res. 7, 2399–2434 (2006)

    MathSciNet  Google Scholar 

  2. Biasotti, S., De Floriani, L., Falcidieno, B., Frosini, P., Giorgi, D., Landi, C., Papaleo, L., Spagnuolo, M.: Describing shapes by geometrical–topological properties of real functions. ACM Comput. Surv. 40(4), 1–87 (2008)

    Article  Google Scholar 

  3. Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proc. Natl. Acad. Sci. 103, 1168–1172 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bronstein, A., Bronstein, M., Kimmel, R., Mahmoudi, M., Sapiro, G.: A Gromov–Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching. Int. J. Comput. Vis.

  5. 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 

  6. Chazal, F., Cohen-Steiner, D., Guibas, L.J., Mémoli, F., Oudot, S.Y.: Gromov–Hausdorff stable signatures for shapes using persistence. In: Computer Graphics Forum (Proc. SGP 2009) (2009)

  7. Chen, D.-Y., Tian, X.-P., Shen, Y.-T., Ouhyoung, M.: On visual similarity based 3d model retrieval. Comput. Graph. Forum 22(3), 223–232 (2003)

    Article  Google Scholar 

  8. Coifman, R.R., Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21(1), 53–94 (2006)

    Article  MATH  MathSciNet  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. 102(21), 7426–7431 (2005)

    Article  Google Scholar 

  10. Floater, M.S.: Mean value coordinates. Comput. Aided Geom. Des. 20(1), 19–27 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Floater, M.S., Kós, G., Reimers, M.: Mean value coordinates in 3d. Comput. Aided Geom. Des. 22(7), 623–631 (2005)

    Article  MATH  Google Scholar 

  12. Frosini, P., Giorgi, D., Landi, C., Marini, S., Patanè, G., Spagnuolo, M., Biasotti, S., Falcidieno, B.: 3d shape description and matching based on properties of real functions. In: Eurographics 2007 Tutorial Notes, pp. 1025–1074. The Eurographics Association, Geneve (2007)

    Google Scholar 

  13. Garland, M., Heckbert, P.S.: Surface simplification using quadric error metrics. In: SIGGRAPH’97: Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques, pp. 209–216 (1997)

  14. Geng, X., Liu, W., Liu, H.: Force field based expression for 3d shape retrieval. In: HCI (2). Lecture Notes in Computer Science, vol. 4551, pp. 587–596. Springer, Berlin (2007)

    Google Scholar 

  15. Gibbs, J.W.: Fourier series. Nature 59, 200 and 606 (1899)

  16. Giorgi, D., Biasotti, S., Paraboschi, L.: Watertight models track. Technical Report IMATI-CNR-GE 9/07 (September 2007)

  17. Gordon, W.J., Wixom, J.A.: Shepard’s method of “metric interpolation” to bivariate and multivariate interpolation. Math. Comput. (1978)

  18. Heczko, M., Keim, D., Saupe, D., Vranic, D.V.: Methods for similarity search of 3d objects. Datenbank-Spektrum Z. Datenbanktechnol. 2(2), 54–63 (2002) (in German)

    Google Scholar 

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

    Article  Google Scholar 

  20. Jain, V., Zhang, H.: Robust 3D shape correspondence in the spectral domain. In: Shape Modeling International (2006)

  21. Jolliffe, I.T.: Principal Component Analysis, 2nd edn. Springer, Berlin (2002)

    MATH  Google Scholar 

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

  23. Kazhdan, M., Funkhouser, T., Rusinkiewicz, S.: Rotation invariant spherical harmonic representation of 3D shape descriptors. In: Symposium on Geometry Processing (2003)

  24. Laga, H., Takahashi, H., Nakajima, M.: Spherical wavelet descriptors for content-based 3D model retrieval. In: SMI, p. 15 (2006)

  25. Mademlis, A., Daras, P., Tzovaras, D., Strintzis, M.G.: 3d object retrieval based on resulting fields. In: Workshop on 3D Object Retrieval (April 2008)

  26. Mademlis, A., Daras, P., Tzovaras, D., Strintzis, M.G.: Using ellipsoidal harmonics for 3d shape representation. In: INTERMEDIA Workshop on Hypermedia 3D Internet (2008)

  27. Mémoli, F.: On the use of Gromov–Hausdorff distances for shape comparison. In: Symposium on Point Based Graphics, Prague, pp. 81–90. Eurographics Association, Geneve (2007)

    Google Scholar 

  28. Mémoli, F., Sapiro, G.: Comparing point clouds. In: SGP’04: Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp. 32–40. ACM, New York (2004)

    Chapter  Google Scholar 

  29. Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Found. Comput. Math. 5(3), 313–347 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

  31. Osada, R., Funkhouser, T., Chazelle, B., Dobkin, D.: Shape distributions. ACM Trans. Graph. 21(4), 807–832 (2002)

    Article  Google Scholar 

  32. Paquet, E., Rioux, M.: Nefertiti: a query by content system for three-dimensional model and image databases management. Image Vis. Comput. 17(2), 157–166 (1999)

    Article  Google Scholar 

  33. Rustamov, R.M.: On mesh editing, manifold learning, and diffusion wavelets, In: Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII, pp. 307–321. Springer, Berlin (2009)

    Google Scholar 

  34. Rustamov, R.M.: Template based shape descriptor. In: 3DOR, pp. 1–7 (2009)

  35. Saupe, D., Vranic, D.V.: 3d model retrieval with spherical harmonics and moments. In: Proceedings of the 23rd DAGM-Symposium on Pattern Recognition, pp. 392–397. Springer, London (2001)

    Google Scholar 

  36. Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM National Conference, pp. 517–524. ACM, New York (1968)

    Chapter  Google Scholar 

  37. Shilane, P., Min, P., Kazhdan, M., Funkhouser, T.: The Princeton shape benchmark. In: Proc. Shape Modeling International, Genoa, Italy, pp. 167–178. IEEE Computer Society, Los Alamitos (2004)

    Google Scholar 

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

    Article  Google Scholar 

  39. Taubin, G.: A signal processing approach to fair surface design. In: SIGGRAPH, pp. 351–358 (1995)

  40. Vallet, B., Lévy, B.: Manifold harmonics. In: Computer Graphics Forum (Proceedings Eurographics) (2008)

  41. Vranić, D.V., Saupe, D.: 3d model retrieval. In: Proceedings of the Spring Conference on Computer Graphics and its Applications, pp. 89–93 (2000)

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. A versatile framework for shape description. Vis Comput 26, 1245–1256 (2010). https://doi.org/10.1007/s00371-010-0518-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00371-010-0518-y

Keywords

Navigation