Advertisement

Foundations of Computational Mathematics

, Volume 8, Issue 4, pp 409–425 | Cite as

Meshfree Thinning of 3D Point Clouds

  • Nira Dyn
  • Armin IskeEmail author
  • Holger Wendland
Article

Abstract

An efficient data reduction scheme for the simplification of a surface given by a large set X of 3D point-samples is proposed. The data reduction relies on a recursive point removal algorithm, termed thinning, which outputs a data hierarchy of point-samples for multiresolution surface approximation. The thinning algorithm works with a point removal criterion, which measures the significances of the points in their local neighbourhoods, and which removes a least significant point at each step. For any point x in the current point set YX, its significance reflects the approximation quality of a local surface reconstructed from neighbouring points in Y. The local surface reconstruction is done over an estimated tangent plane at x by using radial basis functions. The approximation quality of the surface reconstruction around x is measured by using its maximal deviation from the given point-samples X in a local neighbourhood of x. The resulting thinning algorithm is meshfree, i.e., its performance is solely based upon the geometry of the input 3D surface point-samples, and so it does not require any further topological information, such as point connectivities. Computational details of the thinning algorithm and the required data structures for efficient implementation are explained and its complexity is discussed. Two examples are presented for illustration.

Keywords

Surface simplification Thinning algorithms Meshfree methods Approximation by radial basis functions 3D point cloud 

AMS Subject Classifications

65D07 65D15 65D17 65D18 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Alexa, J. Behr, D. Cohen-Or, S. Fleishman, D. Levin, and C. T. Silva, Point set surfaces, IEEE Vis. (2001), 21–28. Google Scholar
  2. 2.
    M. Alexa, J. Behr, D. Cohen-Or, S. Fleishman, D. Levin, and C. T. Silva, Computing and rendering point set surfaces, IEEE Trans. Vis. Comput. Graph. 9(1) (2003), 3–15. CrossRefGoogle Scholar
  3. 3.
    D. Brodsky and B. Watson, Model simplification through refinement, in Proceedings of Graphics Interface (Montreal, 2000), pp. 221–228. A.K. Peters, New York, 2000. Google Scholar
  4. 4.
    M. D. Buhmann, Radial Basis Functions, Cambridge University Press, Cambridge, 2003. zbMATHGoogle Scholar
  5. 5.
    T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms (2nd edn.), MIT Press, Cambridge, 2001. zbMATHGoogle Scholar
  6. 6.
    L. Demaret, N. Dyn, M. S. Floater, and A. Iske, Adaptive thinning for terrain modelling and image compression, in Advances in Multiresolution for Geometric Modelling (N. A. Dodgson, M. S. Floater, M. A. Sabin, eds.), pp. 321–340, Springer, Berlin, 2005. Google Scholar
  7. 7.
    J. Duchon, Splines minimizing rotation-invariant semi-norms in Sobolev spaces, in Constructive Theory of Functions of Several Variables (W. Schempp, K. Zeller, eds.), pp. 85–100, Springer, Berlin, 1977. CrossRefGoogle Scholar
  8. 8.
    N. Dyn, M. S. Floater, and A. Iske, Adaptive thinning for bivariate scattered data, J. Comput. Appl. Math. 145(2) (2002), 505–517. zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    S. Fleishman, D. Cohen-Or, M. Alexa, and C. T. Silva, Progressive point set surfaces, ACM Trans. Graph. 22(4) (2003), 997–1011. CrossRefGoogle Scholar
  10. 10.
    L. De Floriani and P. Magillo, Multiresolution mesh representation: models and data structures, in Tutorials on Multiresolution in Geometric Modelling (A. Iske, E. Quak, M. S. Floater, eds.), pp. 363–417, Springer, Berlin, 2002. Google Scholar
  11. 11.
    F. G. Friedlander and M. S. Joshi, Introduction to the Theory of Distributions (2nd edn.), Cambridge University Press, Cambridge, 1999. Google Scholar
  12. 12.
    M. Garland and P. Heckbert, Surface simplification using quadratic error metrics, in SIGGRAPH’97. ACM, Washington, 1997. Google Scholar
  13. 13.
    H. Hoppe, Surface reconstruction from unorganized points, Ph.D. thesis, University of Washington, 1994. Google Scholar
  14. 14.
    H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, Surface reconstruction from unorganized points, in SIGGRAPH’92, vol. 26, pp. 71–78, ACM, Washington, 1992. Google Scholar
  15. 15.
    A. Iske, Multiresolution Methods in Scattered Data Modelling, Springer, Berlin, 2004. zbMATHGoogle Scholar
  16. 16.
    D. Levin, The approximation power of moving least-squares, Math. Comput. 67 (1998), 1517–1531. zbMATHCrossRefGoogle Scholar
  17. 17.
    F. J. Narcowich, J. D. Ward, and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comput. 74 (2005), 643–763. MathSciNetGoogle Scholar
  18. 18.
    F. J. Narcowich, J. D. Ward, and H. Wendland, Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions, Constr. Approx. 24 (2006), 175–186. zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    M. Pauly, M. Gross, and L. P. Kobbelt, Efficient simplification of point-sampled surfaces, in Proceedings of the Conference on Visualization’02, pp. 163–170, IEEE, Washington, 2002. Google Scholar
  20. 20.
    J. Rossignac and P. Borrel, Multi-resolution 3d approximations for rendering complex scenes, in Modeling in Computer Graphics: Methods and Application (B. Falcidieno, T.L. Kunii, eds.), pp. 455–465, Springer, Berlin, 1993. Google Scholar
  21. 21.
    E. Shaffer and M. Garland Efficient adaptive simplification of massive meshes, in Proceedings of the conference on Visualization’01, pp. 127–134, IEEE, Washington, 2001. Google Scholar
  22. 22.
    Stanford University, The Stanford 3d scanning repository, 2005, Stanford Computer Graphics Laboratory, University of Stanford, http://graphics.stanford.edu/data/3Dscanrep/.
  23. 23.
    G. Turk, Re-tiling polygonal surfaces, in SIGGRAPH’92, ACM, Washington, 1992. Google Scholar
  24. 24.
    G. Wahba, Spline models for observational data, in CBMS-NSF, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1990. Google Scholar
  25. 25.
    H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), 389–396. zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    H. Wendland, Local polynomial reproduction and moving least squares approximation, IMA J. Numer. Anal. 21 (2001), 285–300. zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2005. zbMATHGoogle Scholar
  28. 28.
    H. Wendland and C. Rieger, Approximate interpolation with applications to selecting smoothing parameters, Numer. Math. 101 (2005), 643–662. CrossRefMathSciNetGoogle Scholar

Copyright information

© SFoCM 2007

Authors and Affiliations

  1. 1.Department of MathematicsTel-Aviv UniversityTel-AvivIsrael
  2. 2.Department of MathematicsUniversity of HamburgHamburgGermany
  3. 3.Department of MathematicsUniversity of SussexBrightonUK

Personalised recommendations