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

TL;DR

This paper introduces a consistent estimator for the Vorob'ev expectation of a random compact set, enabling practical mean shape estimation in applications like image analysis, with error control under mild boundary regularity assumptions.

Contribution

It proposes a novel, practical estimator for the Vorob'ev expectation of random sets, with proven consistency and error control, applicable to real-world data.

Findings

Estimator is consistent and easy to implement.

Discretization errors are controlled under mild boundary regularity.

Application demonstrated on cosmological data.

Abstract

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 $\E_V(X)$, which is closely linked to quantile sets. In this paper, we propose a consistent and ready to use estimator of $\E_V(X)$ 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.