Description of the symmetric convex random closed sets as zonotopes from their Feret diameters

TL;DR

This paper demonstrates that 2-D symmetric convex random closed sets can be accurately approximated by random zonotopes using Feret diameters, linking their face lengths to Feret diameter moments.

Contribution

It introduces a method to approximate symmetric convex RACS with zonotopes based on Feret diameters, establishing a connection between face length moments and Feret diameter moments.

Findings

Symmetric convex RACS can be approximated by zonotopes in Hausdorff distance.

Feret diameter moments relate to zonotope face length moments.

Approximation accuracy can be arbitrarily improved.

Abstract

In this paper, the 2-D random closed sets (RACS) are studied by means of the Feret diameter, also known as the caliper diameter. More specifically, it is shown that a 2-D symmetric convex RACS can be approximated as precisely as we want by some random zonotopes (polytopes formed by the Minkowski sum of line segments) in terms of the Hausdorff distance. Such an approximation is fully defined from the Feret diameter of the 2-D convex RACS. Particularly, the moments of the random vector representing the face lengths of the zonotope approximation are related to the moments of the Feret diameter random process of the RACS.