Level sets estimation and Vorob'ev expectation of random compact sets

The issue of a “mean shape” of a random set X often arises, in particular in image analysis and pattern detection. There is no canonical definition but one possible approach is the so-called Vorob'ev expectation EV(X), which is closely linked to level or quantile sets. In this paper, we propose a consistent and ready to use estimator of EV(X) built from independent copies of X with spatial discretisation. The control of discretisation errors is handled with a mild regularity assumption on the boundary of X. Several examples are developed and an application to cosmological data is presented.