Voronoi-based estimation of Minkowski tensors from finite point samples

TL;DR

This paper introduces a Voronoi-based estimator for Minkowski tensors that approximates geometric quantities of objects from digital images, converging to true values as image resolution increases.

Contribution

It provides a novel estimator using Voronoi decompositions based on a generalized Steiner formula for sets of positive reach.

Findings

Estimator converges to true Minkowski tensors with increasing resolution

Uses Voronoi cells for efficient computation

Applicable to digital images of real-world objects

Abstract

Intrinsic volumes and Minkowski tensors have been used to describe the geometry of real world objects. This paper presents an estimator that allows to approximate these quantities from digital images. It is based on a generalized Steiner formula for Minkowski tensors of sets of positive reach. When the resolution goes to infinity, the estimator converges to the true value if the underlying object is a set of positive reach. The underlying algorithm is based on a simple expression in terms of the cells of a Voronoi decomposition associated with the image.