Surface and length estimation based on Crofton´s formula - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2021

Surface and length estimation based on Crofton´s formula

Résumé

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.
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Dates et versions

hal-02907297 , version 1 (27-07-2020)
hal-02907297 , version 2 (02-02-2021)
hal-02907297 , version 3 (13-07-2021)
hal-02907297 , version 4 (28-10-2021)
hal-02907297 , version 5 (24-03-2022)

Identifiants

  • HAL Id : hal-02907297 , version 3

Citer

Catherine Aaron, Alejandro Cholaquidis, Ricardo Fraiman. Surface and length estimation based on Crofton´s formula. 2021. ⟨hal-02907297v3⟩
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