Détection de structure géométrique dans les nuages de points. (Geometric structure detection in point clouds)

In this short chapter, we introduce some of the objects that will be used throughout the thesis: the distance functions, projection functions and medial axes. We survey some known result concerning the local regularity of medial axis (eg. Hausdorff dimension, rectifiability), a few of which come from the semi-concavity of the distance function. The compilation of these results complements the survey on the stability and computation of medial axes by Attali, Boissonnat and Edelsbrunner. In the second part of the chapter, we study the (d − 1)-volume and the covering numbers of the medial axis of a compact set. In general, this volume is infinite; however, the (d − 1)-volume and covering numbers of a filtered medial axis (the μ-medial axis) that is at distance greater than ε from the compact set will be explicitely bounded. The behaviour of the bound we obtain with respect to μ, ε and the covering numbers of K is optimal. Let us stress that bounding the covering numbers is much stronger than bounding the (d − 1)-volume, since we cannot disregard the lower-dimensional parts of the medial axis. This result will be used in the next chapter to give a first stability statement for projection functions.