Foundations of Computational Mathematics

, Volume 11, Issue 6, pp 733–751 | Cite as

Geometric Inference for Probability Measures

  • Frédéric Chazal
  • David Cohen-Steiner
  • Quentin Mérigot
Article

Abstract

Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (e.g., Betti numbers, normals) of this subset from the approximating point cloud data. It appears that the study of distance functions allows one to address many of these questions successfully. However, one of the main limitations of this framework is that it does not cope well with outliers or with background noise. In this paper, we show how to extend the framework of distance functions to overcome this problem. Replacing compact subsets by measures, we introduce a notion of distance function to a probability distribution in ℝd. These functions share many properties with classical distance functions, which make them suitable for inference purposes. In particular, by considering appropriate level sets of these distance functions, we show that it is possible to reconstruct offsets of sampled shapes with topological guarantees even in the presence of outliers. Moreover, in settings where empirical measures are considered, these functions can be easily evaluated, making them of particular practical interest.

Keywords

Geometric inference Computational topology Optimal transportation Nearest neighbor Surface reconstruction 

Mathematics Subject Classification (2000)

62G05 62-07 28A33 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Amenta, S. Choi, T.K. Dey, N. Leekha, A simple algorithm for homeomorphic surface reconstruction, Int. J. Comput. Geom. Appl. 12(1–2), 125–141 (2002). MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    F. Bolley, A. Guillin, C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces, Probab. Theory Relat. 137(3), 541–593 (2007). MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    F. Chazal, A. Lieutier, Stability and computation of topological invariants of solids in ℝn, Discrete Comput. Geom. 37(4), 601–617 (2007). MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    F. Chazal, A. Lieutier, Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees, Comput. Geom. Theor. Appl. 40(2), 156–170 (2008). MathSciNetMATHGoogle Scholar
  5. 5.
    F. Chazal, S.Y. Oudot, Towards persistence-based reconstruction in Euclidean spaces, in Proc. 24th ACM Sympos. Comput. Geom. (2008), pp. 232–241. Google Scholar
  6. 6.
    F. Chazal, D. Cohen-Steiner, A. Lieutier, B. Thibert, Stability of curvature measures, Comput. Graph. Forum 28, 1485–1496 (2008) (proc. SGP 2009). CrossRefGoogle Scholar
  7. 7.
    F. Chazal, D. Cohen-Steiner, A. Lieutier, A sampling theory for compact sets in Euclidean space, Discrete Comput. Geom. 41(3), 461–479 (2009). MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    F. Chazal, D. Cohen-Steiner, A. Lieutier, Normal cone approximation and offset shape isotopy, Comput. Geom. Theor. Appl. 42(6-7), 566–581 (2009). MathSciNetMATHGoogle Scholar
  9. 9.
    F. Chazal, D. Cohen-Steiner, Q. Mérigot, Boundary measures for geometric inference, Found. Comput. Math. 10, 221–240 (2010). MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983). MATHGoogle Scholar
  11. 11.
    D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37(1), 103–120 (2007). MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    V. de Silva, G. Carlsson, Topological estimation using witness complexes, in Symposium on Point-Based Graphics, ETH, Zürich, Switzerland (2004). Google Scholar
  13. 13.
    H. Edelsbrunner, The union of balls and its dual shape, Discrete Comput. Geom. 13, 415–440 (1995). MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    H. Edelsbrunner, J. Harer, Computational Topology. An Introduction (American Mathematical Society, Providence, 2010). MATHGoogle Scholar
  15. 15.
    H. Federer, Curvature measures, Trans. Am. Math. Soc. 93, 418–491 (1959). MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry (Springer, Berlin, 1990). MATHGoogle Scholar
  17. 17.
    K. Grove, Critical point theory for distance functions, in Proc. of Symposia in Pure Mathematics, vol. 54 (1993). Google Scholar
  18. 18.
    A. Lieutier, Any open bounded subset of ℝn has the same homotopy type as its medial axis, Comput. Aided Geom. Des. 36(11), 1029–1046 (2004). Google Scholar
  19. 19.
    Q. Mérigot, M. Ovsjanikov, L. Guibas, Robust Voronoi-based curvature and feature estimation, in Proc. SIAM/ACM Joint Conference on Geom. and Phys. Modeling (2009), pp. 1–12. CrossRefGoogle Scholar
  20. 20.
    P. Niyogi, S. Smale, S. Weinberger, A topological view of unsupervised learning from noisy data. Preprint (2008). Google Scholar
  21. 21.
    P. Niyogi, S. Smale, S. Weinberger, Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom. 39(1), 419–441 (2008). MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    S. Peleg, M. Werman, H. Rom, A unified approach to the change of resolution: space and gray-level, IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 739–742 (1989). CrossRefGoogle Scholar
  23. 23.
    A. Petrunin, Semiconcave functions in Alexandrov’s geometry, in Surveys in differential geometry, vol. XI (International Press, Somerville, 2007), pp. 137–201. Google Scholar
  24. 24.
    V. Robins, Towards computing homology from finite approximations, Topol. Proc. 24, 503–532 (1999). MathSciNetMATHGoogle Scholar
  25. 25.
    Y. Rubner, C. Tomasi, L.J. Guibas, The earth mover’s distance as a metric for image retrieval, Int. J. Comput. Vis. 40(2), 99–121 (2000). MATHCrossRefGoogle Scholar
  26. 26.
    C. Villani, Topics in Optimal Transportation (American Mathematical Society, Providence, 2003). MATHGoogle Scholar

Copyright information

© SFoCM 2011

Authors and Affiliations

  • Frédéric Chazal
    • 1
  • David Cohen-Steiner
    • 2
  • Quentin Mérigot
    • 3
  1. 1.INRIA SaclaySaclayFrance
  2. 2.INRIA Sophia-AntipolisNiceFrance
  3. 3.Laboratoire Jean KuntzmannCNRS/Université de Grenoble IGrenobleFrance

Personalised recommendations