The Visual Computer

, 27:1005 | Cite as

Spectral feature selection for shape characterization and classification

  • S. Marini
  • G. Patané
  • M. Spagnuolo
  • B. Falcidieno
Original Article


This paper proposes a framework for selecting the Laplacian eigenvalues of 3D shapes that are more relevant for shape characterization and classification. We demonstrate the redundancy of the information coded by the shape spectrum and discuss the shape characterization capability of the selected eigenvalues. The feature selection methods used to demonstrate our claim are the AdaBoost algorithm and Support Vector Machine. The efficacy of the selection is shown by comparing the results of the selected eigenvalues on shape characterization and classification with those related to the first k eigenvalues, by varying k over the cardinality of the spectrum. Our experiments, which have been performed on 3D objects represented either as triangle meshes or point clouds, show that working directly with point clouds provides classification results that are comparable with respect to those related to surface-based representations. Finally, we discuss the stability of the computation of the Laplacian spectrum to matrix perturbations.


Shape characterization Feature selection Shape classification Point clouds Laplacian spectrum 


  1. 1.
    Adamson, A., Alexa, M.: Approximating and intersecting surfaces from points. In: Symp. on Geometry Processing, pp. 230–239 (2003) Google Scholar
  2. 2.
    Adamson, A., Alexa, M.: Ray tracing point set surfaces. In: IEEE Shape Modeling International, pp. 272–282 (2003) CrossRefGoogle Scholar
  3. 3.
    Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., Silva, C.T.: Point set surfaces. In: Proc. of Visualization, pp. 21–28 (2001) Google Scholar
  4. 4.
    Amenta, N., Kil, Y.J.: Defining point-set surfaces. In: ACM Siggraph, pp. 264–270 (2004) Google Scholar
  5. 5.
    Arya, S., Mount, D.M., Netanyahu, N.S., Silverman, R., Wu, A.Y.: An optimal algorithm for approximate nearest neighbor searching fixed dimensions. J. ACM 45(6), 891–923 (1998) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Attene, M., Patanè, G.: Hierarchical structure recovery of point-sampled surfaces. Comput. Graph. Forum 29, 1905–1920 (2010) CrossRefGoogle Scholar
  7. 7.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15(6), 1373–1396 (2003) MATHCrossRefGoogle Scholar
  8. 8.
    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) MathSciNetGoogle Scholar
  9. 9.
    Belkin, M., Sun, J., Wang, Y.: Constructing Laplace operator from point clouds in ℝd. In: Proc. of the Symposium on Discrete Algorithms, pp. 1031–1040 (2009) Google Scholar
  10. 10.
    Ben-Hur, A., Soon Ong, C., Sonnenburg, S., Schoelkopf, B., Raetsch, G.: Support vector machines and kernels for computational biology. PLoS Comput. Biol. 4(10), e1000173 (2008) CrossRefGoogle Scholar
  11. 11.
    Biasotti, S., Falcidieno, B., Frosini, P., Giorgi, D., Landi, C., Marini, S., Patanè, G., Spagnuolo, M.: 3D shape description and matching based on properties of real functions. In: Eurographics 2007—Tutorials, Prague, pp. 949–998 (2007) Google Scholar
  12. 12.
    Biasotti, S., Marini, S., Paraboschi, L.: Shape retrieval contest 2007 (SHREC07): Partial matching track. Technical Report 10/07 (2007) Google Scholar
  13. 13.
    Boser, B.E., Guyon, I.M., Vapnik, V.N.: A training algorithm for optimal margin classifiers. In: Proc. of the ACM Workshop on Computational Learning Theory, pp. 144–152 (1992) Google Scholar
  14. 14.
    Bronstein, A.M., Bronstein, M.M., Bustos, B., Castellani, U., Crisani, M., Falcidieno, B., Guibas, L.J., Murino, V., Kokkinos, I., Isipiran, I., Ovsjanikov, M., Patanè, G., Spagnuolo, M., Sun, J.: Shrec 2010: robust feature detection and description benchmark. In: Eurographics Workshop on 3D Object Retrieval (2010) Google Scholar
  15. 15.
    Bronstein, A.M., Bronstein, M.M., Castellani, U., Falcidieno, B., Fusiello, A., Godil, A., Guibas, L.J., Kokkinos, I., Lian, Z., Ovsjanikov, M., Patanè, G., Spagnuolo, M., Toldo, R.: Shrec 2010: robust large-scale shape retrieval benchmark. In: Eurographics Workshop on 3D Object Retrieval (2010) Google Scholar
  16. 16.
    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
  17. 17.
    Chen, D.-Y., Tian, X.-P., Shen, Y., Ouhyoung, M.: On visual similarity based 3D model retrieval. Comput. Graph. Forum 223–232 (2003) Google Scholar
  18. 18.
    Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997) MATHGoogle Scholar
  19. 19.
    Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Fleishman, S., Cohen-Or, D., Alexa, M., Silva, C.T.: Progressive point set surfaces. ACM Trans. Graph. 22(4), 997–1011 (2003) CrossRefGoogle Scholar
  21. 21.
    Freund, Y., Schapire, R.E.: A decision-theoretic generalization of on-line learning and an application to boosting. In: Proc. of Computational Learning Theory (1995) Google Scholar
  22. 22.
    Freund, Y., Schapire, R.E.: A short introduction to boosting. In: Proc. of the International Joint Conference on Artificial Intelligence, pp. 1401–1406 (1999) Google Scholar
  23. 23.
    Guyon, I., Gunn, S., Nikravesh, M., Zadeh, L. (eds.): Feature Extraction, Foundations and Applications. Studies in Fuzziness and Soft Computing, vol. 207. Springer, Berlin (2006) MATHGoogle Scholar
  24. 24.
    Guyon, I., Weston, J., Barnhill, S., Vapnik, V.: Gene selection for cancer classification using support vector machines. Mach. Learn. 46, 389–422 (2002) MATHCrossRefGoogle Scholar
  25. 25.
    Hou, S., Ramani, K.: A probability-based unified 3D shape search. In: Conference on Semantic and Digital Media Technologies. Lecture Notes in Computer Science, pp. 124–137 (2006) Google Scholar
  26. 26.
    Hou, S., Lou, K., Ramani, K.: SVM-based semantic clustering and retrieval of a 3D model database. J. Comput. Aided Des. Appl. 155–164 (2005) Google Scholar
  27. 27.
    Jain, V., Zhang, H.: A spectral approach to shape-based retrieval of articulated 3D models. Comput. Aided Des. 39, 398–407 (2007) CrossRefGoogle Scholar
  28. 28.
    Kalaiah, A., Varshney, A.: Modeling and rendering of points with local geometry. IEEE Trans. Vis. Comput. Graph. 9(1), 30–42 (2003) CrossRefGoogle Scholar
  29. 29.
    Lafon, S., Keller, Y., Coifman, R.R.: Data fusion and multicue data matching by diffusion maps. IEEE Trans. Pattern Anal. Mach. Intell. 28(11), 1784–1797 (2006) CrossRefGoogle Scholar
  30. 30.
    Laga, H., Nakajima, M.: A boosting approach to content-based 3D model retrieval. In: Proc. of Computer Graphics and Interactive Techniques, pp. 227–234 (2007) Google Scholar
  31. 31.
    Laga, H., Nakajima, M.: Supervised learning of salient 2d views of 3D models. J. Soc. Art. Sci. 7(4), 124–131 (2008) CrossRefGoogle Scholar
  32. 32.
    Lange, C., Polthier, K.: Anisotropic smoothing of point sets. Comput. Aided Geom. Des. 22(7), 680–692 (2005) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Levin, D.: Mesh-independent surface interpolation. Geom. Model. Sci. Vis. 3, 37–49 (2003) Google Scholar
  34. 34.
    Mahmoudi, M., Sapiro, G.: Three-dimensional point cloud recognition via distributions of geometric distances. Graph. Models 71, 22–31 (2009) CrossRefGoogle Scholar
  35. 35.
    Marini, S., Spagnuolo, M., Falcidieno, B.: Structural shape prototypes for the automatic classification of 3D objects. IEEE Comput. Graph. Appl. 27(4), 28–37 (2007) CrossRefGoogle Scholar
  36. 36.
    Marini, S., Patanè, G., Spagnuolo, M.: Falcidieno. Feature selection for enhanced spectral shape comparison. In: Eurographics Workshop on 3D Object Retrieval (2010) Google Scholar
  37. 37.
    Mateus, D., Cuzzolin, F., Horaud, R., Boyer, E.: Articulated shape matching using locally linear embedding and orthogonal alignment. In: IEEE International Conference on Computer Vision, pp. 1–8 (2007) CrossRefGoogle Scholar
  38. 38.
    Mederos, B., Velho, L., de Figueiredo, L.H.: Moving least squares multiresolution surface approximation. In: SibGrapi, pp. 19–26 (2003) Google Scholar
  39. 39.
    Mohar, B., Poljak, S.: Eigenvalues in combinatorial optimization. Comb. Graph-Theor. Probl. Linear Algebra 23(98), 107–151 (1993) MathSciNetGoogle Scholar
  40. 40.
    Ohbuchi, R., Kobayashi, J.: Unsupervised learning from a corpus for shape-based 3D model retrieval. In: Proc. of the Workshop on Multimedia Information Retrieval, pp. 163–172 (2006) CrossRefGoogle Scholar
  41. 41.
    Osada, R., Funkhouser, T., Chazelle, B., Dobkin, D.: Shape distributions. ACM Trans. Graph. 21(4), 807–832 (2002) CrossRefGoogle Scholar
  42. 42.
    Paraboschi, L., Giorgi, D., Biasotti, S.: Watertight models track. Technical Report 09/07 (2007) Google Scholar
  43. 43.
    Pauly, M., Gross, M.: Spectral processing of point-sampled geometry. In: ACM Siggraph, pp. 379–386 (2001) Google Scholar
  44. 44.
    Pauly, M., Gross, M.H., Kobbelt, L.: Efficient simplification of point-sampled surfaces. In: IEEE Visualization (2002) Google Scholar
  45. 45.
    Pauly, M., Kobbelt, L.P., Gross, M.: Point-based multiscale surface representation. ACM Trans. Graph. 25(2), 177–193 (2006) CrossRefGoogle Scholar
  46. 46.
    Pfister, H., Zwicker, M., van Baar, J., Gross, M.: Surfels: Surface elements as rendering primitives. In: ACM Siggraph, pp. 335–342 (2000) Google Scholar
  47. 47.
    Reuter, M., Wolter, F.-E., Peinecke, N.: Laplace–Beltrami spectra as Shape-DNA of surfaces and solids. Comput. Aided Des. 38(4), 342–366 (2006) CrossRefGoogle Scholar
  48. 48.
    Reuter, M., Biasotti, Giorgi, D., Patanè, G., Spagnuolo, M.: Discrete Laplace-Beltrami operators for shape analysis and segmentation. Comput. Graph. 33, 381–390 (2009) CrossRefGoogle Scholar
  49. 49.
    Rusinkiewicz, S., Levoy, M.: Qsplat: A multiresolution point rendering system for large meshes. In: ACM Siggraph, pp. 343–352 (2000) Google Scholar
  50. 50.
    Rustamov, R.M.: Laplace–Beltrami eigenfunctions for deformation invariant shape representation. In: Proc. of the Symposium on Geometry Processing, pp. 225–233 (2007) Google Scholar
  51. 51.
    Shilane, P., Funkhouser, T.: Selecting distinctive 3D shape descriptors for similarity retrieval. In: Proc. of Shape Modeling and Applications, p. 18 (2006) Google Scholar
  52. 52.
    Tangelder, J.W., Veltkamp, R.C.: A survey of content based 3D shape retrieval methods. Multimed. Tools Appl. 39(3), 441–471 (2008) CrossRefGoogle Scholar
  53. 53.
    Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000) CrossRefGoogle Scholar
  54. 54.
    Tieu, K., Viola, P.: Boosting image retrieval. Int. J. Comput. Vis. 56(1–2), 17–36 (2004) CrossRefGoogle Scholar
  55. 55.
    Vallet, B., Levy, B.: Spectral geometry processing with manifold harmonics. Comput. Graph. Forum 27(2) (2008) Google Scholar
  56. 56.
    Xie, H., Wang, J., Hua, J., Qin, H., Kaufman, A.: Piecewise C 1 continuous surface reconstruction of noisy point clouds via local implicit quadric regression. In: IEEE Visualization, p. 13 (2003) Google Scholar
  57. 57.
    Zwicker, M., Pfister, H., van Baar, J., Gross, M.: EWA splatting. IEEE Trans. Vis. Comput. Graph. 8(3), 223–238 (2002) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • S. Marini
    • 1
  • G. Patané
    • 1
  • M. Spagnuolo
    • 1
  • B. Falcidieno
    • 1
  1. 1.CNR-IMATIGenovaItaly

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