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Détection de structure géométrique dans les nuages de points

Quentin Mérigot 1 
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : This thesis deals with the general question of geometric inference. Given an object that is only known through finite sampling, what conditions are required on the sampling in order to be able to estimate correctly some of its topological or geometric properties ? Topological estimation is by now quite well understood. Most existing approaches rely on the notion of distance function. We use the distance function in order to estimate a notion of curvature due to Federer, that is defined for a rather general class of non-smooth objects. We study the stability of an approximate version of these measures when the unknown object is replaced by a discrete approximation; we also deal with the practical computation of these measures in the discrete setting. An anisotropic notion of these curvature measures can be used to robustly estimate the locus and the direction of sharp edges of a piecewise smooth surface from a point-cloud sampling. Theoretical results required to study some regularity properties of the distance function, such as the volume of the medial axis. A drawback of distance-based methods is their extreme sensibility to outliers. In order to overcome this problem we propose to leave the purely geometric setting, and replace compact sets with measures (as in Lebesgue theory). We introduce a notion of distance function to a measure, which is robust to Wasserstein perturbations -- hence in particular to the addition of outliers. This distance function shares any regularity and stability properties with the usual distance function; this allows to extend many existing geometric inference theorems to this new setting.
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Submitted on : Tuesday, September 21, 2010 - 5:18:26 PM
Last modification on : Thursday, January 20, 2022 - 5:26:32 PM
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  • HAL Id : tel-00443038, version 2



Quentin Mérigot. Détection de structure géométrique dans les nuages de points. Mathématiques [math]. Université Nice Sophia Antipolis, 2009. Français. ⟨tel-00443038v2⟩



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