Quentin Mérigot > Publications

Geometric inference for measures based on distance functions

Preprint

Frédéric Chazal, David Cohen-Steiner, Quentin Mérigot

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 (Betti numbers, curvatures,...) of this subset from the approximating point cloud data. In recent years, it appeared that the study of distance functions allows 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 nor 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 the d-dimensional Euclidean space. These functions share many properties with classical distance functions, which makes them suitable for inference purposes. In particular, by considering appropriate level sets of these distance functions, it is possible to associate in a robust way topological and geometric features to a probability measure. We also discuss connections between our approach and non parametric density estimation as well as mean-shift clustering.

On the left, a point cloud sampled on a mechanical part to which 10% of outliers have been added -- the outliers are uniformly distributed in a box enclosing the original point cloud. On the right, the reconstruction of an isosurface of the distance function d_{μ_C,m_0} to the uniform probability measure on this point cloud (obtained by using the CGAL Surface Mesher).