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

Rahmani, Saïd; Pinoli, Jean-Charles; Debayle, Johan

doi:10.15559/16-VMSTA70

Modern Stochastics: Theory and Applications

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

Volume 3, Issue 4 (2016), pp. 325–364

Saïd Rahmani Jean-Charles Pinoli Johan Debayle

https://doi.org/10.15559/16-VMSTA70

Pub. online: 3 January 2017 Type: Research Article

Open Access

Received
24 October 2016

Revised
7 December 2016

Accepted
14 December 2016

Published
3 January 2017

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.

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Keywords

Zonotopes random closed set the Feret diameter polygonal approximation

MSC2010

60Dxx 52A22

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