Approximation of Vorob'ev expectation for random closed sets

Philippe Heinrich (1) , Radu Stoica (1) , Viet Chi Tran (1, 2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

Random sets appear in many applications, in particular in image analysis. The issue of a "mean shape" often arises since there is no canonical definition. In this paper, we propose a consistent and ready to use estimator for the Vorob'ev expectation of a random set $X$. It is a kind of mean closely linked to quantile-like quantities and built from independent copies of $X$ with spatial discretization. The convergence is established through the Strong Law of Large Numbers of Kovyazin. The control of discretization errors is handled with a mild regularity assumption on the boundary of $X$: a not too large 'box counting' dimension. Some examples, including Boolean models, are studied.

Mots clés: Stochastic geometry; random closed sets; Vorob'ev expectation; quantile sets

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Philippe Heinrich, Radu Stoica, Viet Chi Tran. Approximation of Vorob'ev expectation for random closed sets. 2010. ⟨hal-00495449v1⟩