Geometric Inference for Probability Measures |
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Authors: | Frédéric Chazal David Cohen-Steiner Quentin Mérigot |
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Institution: | 1.INRIA Saclay,Saclay,France;2.INRIA Sophia-Antipolis,Nice,France;3.Laboratoire Jean Kuntzmann,CNRS/Université de Grenoble I,Grenoble,France |
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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. |
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Keywords: | |
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