Convergence and Continuity Criteria for Discrete Approximations of the Continuous Planar Skeleton

Abstract

It has been proposed recently that the skeleton of a shape can be computed using the Voronoi diagram of a discrete sample set of the shape boundary. This method avoids many of the complications encountered when computing the skeleton directly from an image because it is based on a continuous-domain model for shapes. In order to make better use of this new approach, it is necessary to establish a bridge between the continuous domain skeleton and its approximation obtained from the discrete boundary sample set. In this paper, the skeleton and Voronoi diagram formulations are briefly reviewed and elaborated upon to establish criteria for the functions to be continuous. Then the new continuity results are related to the discrete sample set model in order to establish conditions under which the skeleton approximation converges to the exact continuous skeleton.