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 ${\text{E}}_V(X)$, which is closely linked to quantile sets. In this paper, we propose a consistent and ready to use estimator of ${\text{E}}_V(X)$ built from independent copies of $X$ with spatial discretization. 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 are developed and an application to cosmological data is presented.