Blanche Buet: "Geometric Inference in the Framework of Varifolds"
We propose to model both regular and discrete surfaces using the concept of varifolds. Varifolds were introduced by Almgren in the study of minimal surfaces. Varifolds provide a suitable framework for the analysis of variational geometric problems: it is possible to associate a varifold structure with regular surfaces as well as with discrete analogs (such as triangulated surfaces or point clouds). This allows for a unified framework equipped with tools from varifold theory: a topology and distances for comparing two surfaces, even if one is regular and the other discrete, as well as a distributional notion of mean curvature. These ingredients, combined with a regularization step, enable the definition of a discrete curvature with convergence properties with respect to the varifold topology, typically valid when a sequence of discrete surfaces approaches a regular surface. However, the theoretical convergence rates obtained are difficult to evaluate numerically for a given sequence of point clouds, for example. We therefore propose to reconsider the question by assuming that a point cloud of N points is a realization of an i.i.d. sample (X_1, ... , X_N) drawn from a distribution supported on a surface or a d-rectifiable set S. This reformulates estimation problems (such as curvature estimation) into a statistical framework, where we aim to obtain explicit mean convergence rates (i.e., in expectation) as a function of the number of points N.