Surface and length estimation based on Crofton´s formula

Catherine Aaron (1) , Alejandro Cholaquidis (2) , Ricardo Fraiman (2)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 CMAT - Centro de Matemática [Montevideo]

We study the problem of estimating the surface area of the boundary of a sufficiently smooth set when the available information is only a set of points (random or not) that becomes dense (with respect to Hausdorff distance) in the set or the trajectory of a reflected diffusion. We obtain consistency results in this general setup, and we derive rates of convergence for the iid case or when the data corresponds to the trajectory of a reflected Brownian motion. We propose an algorithm based on Crofton's formula, which estimates the number of intersections of random lines with the boundary of the set by counting, in a suitable way (given by the proposed algorithm), the number of intersections with the boundary of two different estimators: the Devroye-Wise esti-mator and the α-convex hull of the data. As a by-product, our results also cover the convex case, for any dimension.

Domaines: Statistiques [math.ST]

Catherine Aaron, Alejandro Cholaquidis, Ricardo Fraiman. Surface and length estimation based on Crofton´s formula. 2021. ⟨hal-02907297v2⟩