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
First Online:
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Accepted:

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 ℝ d . 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 
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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

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