Discrete & Computational Geometry

, Volume 41, Issue 3, pp 461–479 | Cite as

A Sampling Theory for Compact Sets in Euclidean Space

  • Frédéric Chazal
  • David Cohen-Steiner
  • André Lieutier


We introduce a parameterized notion of feature size that interpolates between the minimum of the local feature size and the recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to noisy samplings of general compact sets in euclidean space. These conditions are sufficient to ensure the topological correctness of a reconstruction given by an offset of the sampling. Our approach also yields new stability results for medial axes, critical points, and critical values of distance functions.


Distance function Sampling Surface and manifold reconstruction 


  1. 1.
    Amenta, N., Bern, M.: Surface reconstruction by Voronoi filtering. Discrete Comput. Geom. 22(4), 481–504 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Amenta, N., Choi, S., Dey, T., Leekha, N.: A simple algorithm for homeomorphic surface reconstruction. Int. J. Comput. Geom. Appl. 12(1, 2), 125–141 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boissonnat, J.D., Oudot, S.: Provably good sampling and meshing of Lipschitz surfaces. In: Graphical Models (GMOD), vol. 67, pp. 405–451 (2005) Google Scholar
  4. 4.
    Chaine, R.: A geometric convection approach of 3-D reconstruction. In: Proc. 1st Symposium on Geometry Processing, pp. 218–229 (2003) Google Scholar
  5. 5.
    Chazal, F., Lieutier, A.: The λ-medial axis. In: Graphical Models (GMOD), vol. 67(4), pp. 304–331 (2005) Google Scholar
  6. 6.
    Chazal, F., Lieutier, A.: Weak feature size and persistent homology: computing homology of solids in ℝn from noisy data samples. In: Proc. 21st ACM Sympos. Comput. Geom., pp. 255–262 (2005) Google Scholar
  7. 7.
    Chazal, F., Lieutier, A.: Topology guaranteeing manifold reconstruction using distance function to noisy data. In: Proc. 22st ACM Sympos. Comput. Geom., pp. 255–262 (2006) Google Scholar
  8. 8.
    Chazal, F., Soufflet, R.: Stability and finiteness properties of medial axis and skeleton. J. Dyn. Control Syst. 10(2), 149–170 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cheeger, J.: Critical points of distance functions and applications to geometry. In: Geometric Topology: Recent Developments, Montecatini Terme, 1990. Springer Lecture Notes, vol. 1504, pp. 1–38 (1991) Google Scholar
  10. 10.
    Clarke, F.H.: Optimization and NonSmooth Analysis. Wiley, New York (1983) zbMATHGoogle Scholar
  11. 11.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. In: Proc. 21st ACM Sympos. Comput. Geom., pp. 263–271 (2005) Google Scholar
  12. 12.
    Dey, T.K., Goswami, S.: Provable surface reconstruction from noisy samples. In: Proc. 20th ACM Sympos. Comput. Geom., pp. 330–339 (2004) Google Scholar
  13. 13.
    Dey, T.K., Wenger, R.: Reconstructing curves with sharp corners. In: Proc. 16th ACM Sympos. Comput. Geom., pp. 233–241 (2000) Google Scholar
  14. 14.
    Dey, T.K., Zhao, W.: Approximate medial axis as a Voronoi subcomplex. In: Proc. 7th ACM Sympos. Solid Modeling Applications, pp. 356–366 (2002) Google Scholar
  15. 15.
    Dey, T.K., Giesen, J., Ramos, E.A., Sadri, B.: Critical points of the distance to an epsilon-sampling of a surface and flow-complex-based surface reconstruction. In: Proc. 21st ACM Sympos. Comput. Geom., pp. 218–227 (2005) Google Scholar
  16. 16.
    Edelsbrunner, H.: Surface reconstruction by wrapping finite point sets in space. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Ricky Pollack and Eli Goodman Festschrift, pp. 379–404. Springer, New York (2002) Google Scholar
  17. 17.
    Edelsbrunner, H., Mücke, E.P.: Three-dimensional alpha shapes. ACM Trans. Graph. 13, 43–72 (1994) zbMATHCrossRefGoogle Scholar
  18. 18.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–419 (1959) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Foskey, M., Lin, M.C., Manocha, D.: Efficient computation of a simplified medial axis. In: ACM Symposium on Solid Modeling and Applications, pp. 96–107 (2003) Google Scholar
  20. 20.
    Giesen, J., John, M.: The flow complex: A data structure for geometric modeling. In: Proc. 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 285–294 (2003) Google Scholar
  21. 21.
    Grove, K.: Critical point theory for distance functions. In: Proc. of Symp. in Pure Mathematics, vol. 54, Part 3 (1993) Google Scholar
  22. 22.
    Lieutier, A.: Any open bounded subset of ℝn has the same homotopy type as its medial axis. Comput.-Aided Des. 36, 1029–1046 (2004). Elsevier CrossRefGoogle Scholar
  23. 23.
    Mederos, B., Amenta, N., Velho, L., de Figueiredo, L.H.: Surface reconstruction from noisy point clouds. In: Proc. of Geometry Processing 2005 (Eurographics/ ACM SIGGRAPH), pp. 53–62 Google Scholar
  24. 24.
    Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Preprint available at
  25. 25.
    Robins, V., Meiss, J.D., Bradley, L.: Computing connectedness: Disconnectedness and discreteness. Physica D 139, 276–300 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Ruppert, J.: A Delaunay refinement algorithm for quality 2-dimensional mesh generation. J. Algorithms 18(3), 548–585 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Starck, J.L., Martínez, V.J., Donoho, D.L., Levi, O., Querre, P., Saar, E.: Analysis of the spatial distribution of galaxies by multiscale methods. arXiv:astro-ph/0406425 v1 (18 Jun. 04)

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Frédéric Chazal
    • 1
  • David Cohen-Steiner
    • 2
  • André Lieutier
    • 3
  1. 1.INRIA FutursOrsay CedexFrance
  2. 2.INRIA Sophia-AntipolisSophia-Antipolis CedexFrance
  3. 3.Dassault Systèmes and LMC/IMAGGrenobleFrance

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