Sub-sampling for Efficient Spectral Mesh Processing

  • Rong Liu
  • Varun Jain
  • Hao Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4035)


In this paper, we apply Nyström method, a sub-sampling and reconstruction technique, to speed up spectral mesh processing. We first relate this method to Kernel Principal Component Analysis (KPCA). This enables us to derive a novel measure in the form of a matrix trace, based soly on sampled data, to quantify the quality of Nyström approximation. The measure is efficient to compute, well-grounded in the context of KPCA, and leads directly to a greedy sampling scheme via trace maximization. On the other hand, analyses show that it also motivates the use of the max-min farthest point sampling, which is a more efficient alternative. We demonstrate the effectiveness of Nyström method with farthest point sampling, compared with random sampling, using two applications: mesh segmentation and mesh correspondence.


Point Sampling Quality Measure Spectral Cluster Kernel Principal Component Analysis Matrix Trace 
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.


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  1. 1.
    Ng, A.Y., Jordan, M.I., Weiss, Y.: On spectral clustering: analysis and an algorithm. In: NIPS, pp. 857–864 (2002)Google Scholar
  2. 2.
    Weiss, Y.: Segmentation using eigenvectors: a unifying view. In: ICCV, pp. 975–982 (1999)Google Scholar
  3. 3.
    Caelli, T., Kosinov, S.: An eigenspace projection clustering method for inexact graph matching. IEEE Trans. on PAMI 26(4), 515–519 (2004)Google Scholar
  4. 4.
    Carcassoni, M., Hancock, E.R.: Spectral correspondence for point pattern matching. Pattern Recognition 36, 193–204 (2003)zbMATHCrossRefGoogle Scholar
  5. 5.
    Karni, Z., Gotsman, C.: Spectral compression of mesh geometry. In: SIGGRAPH 2000, pp. 279–286 (2000)Google Scholar
  6. 6.
    Liu, R., Zhang, H.: Segmentation of 3d meshes through spectral clustering. In: Proc. Pacific Graphics, pp. 298–305 (2004)Google Scholar
  7. 7.
    Jain, V., Zhang, H.: Robust 3d shape correspondence in the spectral domain. In: Shape Modeling International (to appear, 2006)Google Scholar
  8. 8.
    Isenburg, M., Lindstrom, P.: Streaming meshes. IEEE Visualization, 231–238 (2005)Google Scholar
  9. 9.
    Zhang, H., Liu, R.: Mesh segmentation via recursive and visually salient spectral cuts. In: Vision, Modeling, and Visualization (2005)Google Scholar
  10. 10.
    Kolluri, R., Shewchuk, J.R., O’Brien, J.F.: Spectral surface reconstruction from noisy point clouds. In: Eurographics SGP, pp. 11–21 (2004)Google Scholar
  11. 11.
    Zigelman, G., Kimmel, R., Kiryati, N.: Texture mapping using surface flattening via multidimensional scaling. IEEE TVCG 8(2), 198–207 (2002)Google Scholar
  12. 12.
    Press, W., Tekolsky, S., Vetterling, W., Flannery, B.: Numerical Recipies in C. Cambridge University Press, Cambridge (1992)Google Scholar
  13. 13.
    Fowlkes, C., Belongie, S., Chung, F., Malik, J.: Spectral grouping using the nyström method. IEEE Trans. PAMI 26, 214–225 (2004)Google Scholar
  14. 14.
    de Silva, V., Tenenbaum, B.: Sparse multidimensional scaling using landmark points. Technical report, Stanford University (2004)Google Scholar
  15. 15.
    Sorkine, O.: Laplacian mesh processing. In: Eurographics State-of-the-Art Report (2005)Google Scholar
  16. 16.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. In: CVPR, pp. 731–737 (1997)Google Scholar
  17. 17.
    Harel, D., Koren, Y.: A fast multi-scale method for drawing large graphs. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 183–196. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Williams, C., Seeger, M.: Using the nyström method to speed up kernel machines. In: Advances in Neural Information Processing Systems, pp. 682–688 (2001)Google Scholar
  19. 19.
    Scholkopf, B., Smola, A., Muller, K.R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation 10, 1299–1319 (1998)CrossRefGoogle Scholar
  20. 20.
    Kraevoy, V., Sheffer, A.: Cross-parameterization and compatible remeshing of 3d models. In: ACM SIGGRAPH (2004)Google Scholar
  21. 21.
    Shapiro, L.S., Brady, J.M.: Feature based correspondence: An eigenvector approach. Image and Vision Computing 10(5), 283–288 (1992)CrossRefGoogle Scholar
  22. 22.
    Garland, M.: Qslim simplification software — qslim 2.1Google Scholar
  23. 23.
    Shamir, A.: A formalization of boundary mesh segmentation. In: Proc. 2nd Int’l Symposium on 3D Data Processing, Visualization, and Transmission, pp. 82–89 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Rong Liu
    • 1
  • Varun Jain
    • 1
  • Hao Zhang
    • 1
  1. 1.GrUVi Lab, School of Computing SciencesSFUCanada

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