Multi-scale and Adaptive CS-RBFs for Shape Reconstruction from Clouds of Points

  • Yutaka Ohtake
  • Alexander Belyaev
  • Hans-Peter Seidel
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


We describe a multi-scale approach for interpolation and approximation of a point set surface by compactly supported radial basis functions. Given a set of points scattered over a surface, we first use down-sampling to construct a point set hierarchy. Then starting from the coarsest level, for each level of the hierarchy, we use compactly supported RBFs to approximate the set of points at the level as an offset of the RBF approximation computed at the previous level. A simple RBF centre reduction scheme combined with the multi-scale approach accelerates the latter and allows us to achieve high quality approximations using relatively small number of RBF centres.

We also develop an adaptive RBF fitting prsocedure for which the RBF centres are randomly chosen from the set of points of the level. The randomness is controlled by the density of points and geometric characteristic of the set. The support size of the RBF we use to approximate the point set at a vicinity of a point depends on the local density of the set at that point. Thus parts with complex geometry are approximated by dense RBFs with small supports.

Numerical experiments demonstrate high speed and good performance of the proposed methods in processing irregularly sampled and/or incomplete data.


Radial Basis Function Implicit Surface Support Size Radial Basis Function Interpolation Scattered Data Interpolation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Alexa. Hierarchical partition of unity approximation. Technical report, TU Darmstadt, August 2002.Google Scholar
  2. 2.
    I. Babuška and J. M. Melenk. The partition of unity method. International Journal of Numerical Methods in Engineering, 40:727–758, 1997.CrossRefGoogle Scholar
  3. 3.
    T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl. Meshless methods: An overview and recent developments. Computer Methods in Applied Mechanics and Engineering, 139:3–47, 1996.CrossRefGoogle Scholar
  4. 4.
    D. Buhmann, M. Radial Basis Functions: Theory and Implementations. Cambridge University Press, 2003.Google Scholar
  5. 5.
    J. C. Carr, R. K. Beatson, J. B. Cherrie, T. J. Mitchell, W. R. Fright, B. C. McCallum, and T. R. Evans. Reconstruction and representation of 3D objects with radial basis functions. In Proc. ACM SIGGRAPH, pages 67–76, August 2001.Google Scholar
  6. 6.
    M. S. Floater and A. A. Iske. Multistep scattered data interpolation using compactly supported radial basis functions. Journal of Comp. Appl. Math., 73:65–78, 1996.MathSciNetCrossRefGoogle Scholar
  7. 7.
    S. Haykin. Neural Networks: A Comprehensive Foundation. Macmillan College Publishing Company, Inc., 1994.Google Scholar
  8. 8.
    A. Iske. Scattered data modelling using radial basis functions. In A. Iske, E. Quak, and M. S. Floater, editors, Tutorials on Multiresolution in Geometric Modelling, pages 205–242. Springer, 2002.Google Scholar
  9. 9.
    A. Iske and J. Levesley. Multilevel scattered data approximation by adaptive domain decomposition. Technical Report TUM-M0208, Technische Universität München, July 2002.Google Scholar
  10. 10.
    D. Lazzaro and L. B. Montefusco. Radial basis functions for multivariate interpolation of large scattered data sets. Journal of Computational and Applied Mathematics, 140:521–536, 2002.MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. K. Lodha and R. Franke. Scattered data techniques for surfaces. In H. Hagen, G. Nielson, and F. Post, editors, Proceedings of Dagstuhl Conference on Scientific Visualization, pages 182–222. IEEE Computer Society Press, 1999.Google Scholar
  12. 12.
    M. Mitzenmacher, A. Richa, and R. Sitaraman. The power of two random choices: A survey of techniques and results. In Handbook of Randomized Computing, Chapter 9. Kluwer, 2001.Google Scholar
  13. 13.
    Y. Ohtake, A. Belyaev, M. Alexa, G. Turk, and H.-P. Seidel. Multi-level partition of unity implicits. ACM Transactions on Graphics, 22(3):463–470, July 2003. Proc. ACM SIGGRAPH 2003.CrossRefGoogle Scholar
  14. 14.
    Y. Ohtake, A. G. Belyaev, and H.-P. Seidel. A multi-scale approach to 3D scattered data interpolation with compactly supported basis functions. In Shape Modeling International 2003, pages 153–161, Seoul, Korea, May 2003.Google Scholar
  15. 15.
    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, 1993.Google Scholar
  16. 16.
    V. V. Savchenko, A. A. Pasko, O. G. Okunev, and T. L. Kunii. Function representation of solids reconstructed from scattered surface points and contours. Computer Graphics Forum, 14(4):181–188, 1995.CrossRefGoogle Scholar
  17. 17.
    G. Taubin. Estimation of planar curves, surfaces and nonplanar space curves defined by implicit equations, with applications to edge and range image segmentation. IEEE Trans. on Pattern Analysis and Machine Intelligence, 13(11):1115–1138, 1991.CrossRefGoogle Scholar
  18. 18.
    I. Tobor, P. Reuter, and C. Schlick. Efficient reconstruction of large scattered geometric datasets using the partition of unity and radial basis functions. In The 12-th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision (WSCG'04), February 2004.Google Scholar
  19. 19.
    G. Turk and J. O'Brien. Shape transformation using variational implicit surfaces. In Proc. ACM SIGGRAPH, pages 335–342, August 1999.Google Scholar
  20. 20.
    G. Turk and J. O'Brien. Modelling with implicit surfaces that interpolate. ACM Transactions on Graphics, 21(4):855–873, October 2002.CrossRefGoogle Scholar
  21. 21.
    H. Wendland. Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree. Advances in Computational Mathematics, 4:389–396, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    H. Wendland. Fast evaluation of radial basis functions: Methods based on partition of unity. In L. Schumaker and J. Stöckler, editors, Approximation Theory X: Wavelets, Splines, and Applications, pages 473–483. Vanderbilt University Press, Nashville, 2002.Google Scholar
  23. 23.
    J. Wu and L. P. Kobbelt. Fast mesh decimation by multiple-choice techniques. In Vision, Modeling, Visualization 2002 Proceedings, pages 241–248, Erlangen, Germany, November 2002.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Yutaka Ohtake
    • 1
  • Alexander Belyaev
    • 1
  • Hans-Peter Seidel
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations