Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On estimating the perimeter using the alpha-shape

Ery Arias-Castro and Alberto Rodríguez-Casal

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We consider the problem of estimating the perimeter of a smooth domain in the plane based on a sample from the uniform distribution over the domain. We study the performance of the estimator defined as the perimeter of the alpha-shape of the sample. Some numerical experiments corroborate our theoretical findings.


Nous considérons le problème de l’estimation du périmètre d’un domaine à bord lisse dans le plan basé sur un échantillon tiré de la loi uniforme ayant pour support le domaine en question. Nous étudions la performance de l’estimateur défini par le périmètre de la forme-alpha (« alpha-shape ») de l’échantillon. Des expériences numériques confirment notre théorie.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1051-1068.

Received: 19 June 2015
Revised: 31 January 2016
Accepted: 13 February 2016
First available in Project Euclid: 21 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G99: None of the above, but in this section 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Perimeter estimation $\alpha$-shape $r$-convex hull Rolling condition Sets with positive reach


Arias-Castro, Ery; Rodríguez-Casal, Alberto. On estimating the perimeter using the alpha-shape. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1051--1068. doi:10.1214/16-AIHP747.

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