Medial Axis Approximation with Bounded Error

A common approach to approximating the medial axis decides the presence of medial points in a region of nonzero size by analyzing the gradient of the distance transform at a finite number of locations in this region. In general, algorithms of this type do not guarantee completeness. In this paper, we consider a novel medial axis approximation algorithm of this type and present an analysis in the 2D case that reveals the geometric relationship between the quality of the medial axis approximation and the number and distribution of samples of the gradient of the distance transform. We use an extension of this algorithm to 3D to compute qualitatively accurate medial axes of polyhedra, as well as Voronoi diagrams of lines. Our results suggest that medial axis approximation algorithms based on sampling of the distance transform are theoretically well-motivated.