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Level sets estimation and Vorobév expectation of random compact sets

Abstract

The issue of a "mean shape" of a random set XX often arises, in particular in image analysis and pattern detection. There is no canonical definition but one possible approach is the so-called Vorobév expectation \EV(X)\E_V(X), which is closely linked to quantile sets. In this paper, we propose a consistent and ready to use estimator of \EV(X)\E_V(X) built from independent copies of XX with spatial discretization. The control of discretization errors is handled with a mild regularity assumption on the boundary of XX: a not too large 'box counting' dimension. Some examples are developed and an application to cosmological data is presented.

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