Discrete geometric inference

Keywords :

• discrete geometrie
• surfaces analysis and reconstruction
• triangulation
• voxel

The purpose of geometric inference is to answer the following problem : Given a geometric object that is only known through an approximation, can we get a robust estimation of its properties? We consider in this thesis the case where the approximation is a point cloud or a digital set in a finite dimensional Euclidean space. We first show a stability result for a normal estimator based on the principal component analysis, as well as a result of multigrid convergence of an estimator of the Voronoi covariance measure, which uses covariance matrices of Voronoi cells. As most of geometric inference results, these two last results use the robustness of the distance function to a compact set. However, the presence of a single outlier is sufficient to make the assumptions of these results not satisfied. The distance to a measure is a generalized distance function introduced recently, that is robust to outliers. In this work, we generalize the Voronoi Covariance Measure to generalized distance functions and we show that this estimator applied to the distance to a measure is robust to outliers. We deduce a very robust normal estimator. We present experiments showing the robustness of our approach for normals, curvatures, curvature directions and sharp features estimation. These results are favorably compared to the state of the art.

Directors:

• Mr Boris Thibert (Maître de Conférence - Université Joseph Fourier )
• Mr Jacques-Olivier Lachaud (Professeur - Université de Savoie (LAMA) )

Raporteurs:

• Mr Rémy Malgouyres (Professeur - Université d'Auvergne (LIMOS) )
• Mr Pierre Alliez (Directeur de Recherche - INRIA-Sophia Antipolis )

Examinators:

• Mr Simon Masnou (Professeur - Université Lyon 1 )