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Surface and length estimation based on Crofton´s formula

Abstract : 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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https://hal.archives-ouvertes.fr/hal-02907297
Contributor : Catherine Aaron <>
Submitted on : Monday, July 27, 2020 - 2:13:26 PM
Last modification on : Wednesday, July 29, 2020 - 4:03:05 AM

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  • HAL Id : hal-02907297, version 1

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Catherine Aaron, Alejandro Cholaquidis, Ricardo Fraiman. Surface and length estimation based on Crofton´s formula. 2020. ⟨hal-02907297⟩

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