Computation of the Metric Average of 2D Sets with Piecewise Linear Boundaries

Evgeny Lipovetsky,Shay Kels,Nira Dyn

doi:10.3390/a3030265

Evgeny Lipovetsky, Shay Kels + Show 1 more

Open Access

https://doi.org/10.3390/a3030265

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Journal: Algorithms	Publication Date: Jul 26, 2010
Citations: 6	License type: CC BY 4.0

Affiliation: PTC (United States), Tel Aviv University

Abstract

The metric average is a binary operation between sets in Rn which is used in the approximation of set-valued functions. We introduce an algorithm that applies tools of computational geometry to the computation of the metric average of 2D sets with piecewise linear boundaries.

Highlights

Approximation of set-valued functions has various potential applications in optimization, mathematical economics, control, robotics and more
The definition of the metric average of two sets with piecewise linear boundaries requires at least one operation for each point in the two sets, our algorithm reduces the computation to boundaries of cells of the partition of the two sets, obtained by the segment Voronoi diagram
The geometric “artifacts” produced by the metric average are explained by the fact that the projection point on a set may move discontinuously for points near the boundaries between faces of the Voronoi diagram induced by the boundary of the set

Summary

Introduction

Approximation of set-valued functions has various potential applications in optimization, mathematical economics, control, robotics and more. Since a compact 2D set with boundary consisting of closed curves can be linearly approximated by a 2D set with piecewise linear boundaries, our algorithm provides a computational method for approximation of set-valued functions with 2D images from a finite number of samples. The definition of the metric average of two sets with piecewise linear boundaries requires at least one operation for each point in the two sets, our algorithm reduces the computation to boundaries of cells of the partition of the two sets, obtained by the segment Voronoi diagram. Our algorithm is based on geometric objects such as segment Voronoi diagrams, arrangements of conic arcs and conic polygons with holes. In particular we review the main geometric concepts relevant to our work such as the metric average, segment Voronoi diagrams, planar arrangements and conic polygons with holes. We assume that the diagram is bounded by a rectangular frame, which is large enough not to influence the computation of the metric average

The Algorithm for Simple Polygons

Complexity Bounds

Without loss of generality assume that

Conclusions