Distributed boundary tracking using alpha and Delaunay-Cech shapes

TL;DR

This paper presents a distributed algorithm for computing alpha shapes of point sets in the plane, useful in sensor networks, and introduces a new geometric shape called Delaunay-Cech shape that is topologically equivalent to alpha shapes.

Contribution

The paper develops a distributed algorithm for alpha shape computation and introduces the Delaunay-Cech shape, a new geometric object with topological equivalence to alpha shapes.

Findings

Distributed algorithm successfully computes alpha shapes from pairwise distances.

Delaunay-Cech shape is topologically equivalent to alpha shapes.

Applicable to sensor networks with limited distance information.

Abstract

For a given point set $S$ in a plane, we develop a distributed algorithm to compute the $\alpha-$shape of $S$. $\alpha-$shapes are well known geometric objects which generalize the idea of a convex hull, and provide a good definition for the shape of $S$. We assume that the distances between pairs of points which are closer than a certain distance $r>0$ are provided, and we show constructively that this information is sufficient to compute the alpha shapes for a range of parameters, where the range depends on $r$. Such distributed algorithms are very useful in domains such as sensor networks, where each point represents a sensing node, the location of which is not necessarily known. We also introduce a new geometric object called the Delaunay-\v{C}ech shape, which is geometrically more appropriate than an $\alpha-$shape for some cases, and show that it is topologically equivalent to $\alpha-$shapes.