The wrapping hull and a unified framework for volume estimation

TL;DR

This paper introduces a unified framework for estimating the volume of various sets in Euclidean space using a new geometric construct called the wrapping hull, applicable to broad classes of sets with minimal assumptions.

Contribution

It proposes the wrapping hull as a generalized convex hull, providing a consistent, nearly unbiased volume estimator for intersection-stable set classes without boundary restrictions.

Findings

The wrapping hull is a sufficient and complete statistic.

The volume estimator is consistent and nearly unbiased.

Numerical examples confirm the framework's effectiveness.

Abstract

This paper develops a unified framework for estimating the volume of a set in $\mathbb{R}^d$ based on observations of points uniformly distributed over the set. The framework applies to all classes of sets satisfying one simple axiom: a class is assumed to be intersection stable. No further hypotheses on the boundary of the set are imposed; in particular, the convex sets and the so-called weakly-convex sets are covered by the framework. The approach rests upon a homogeneous Poisson point process model. We introduce the so-called wrapping hull, a generalization of the convex hull, and prove that it is a sufficient and complete statistic. The proposed estimator of the volume is simply the volume of the wrapping hull scaled with an appropriate factor. It is shown to be consistent for all classes of sets satisfying the axiom and mimics an unbiased estimator with uniformly minimal variance. The construction and proofs hinge upon an interplay between probabilistic and geometric arguments. The tractability of the framework is numerically confirmed in a variety of examples.