The power crust, unions of balls, and the medial axis transform

The medial axis transform (or MAT) is a representation of an object as an i nfinite union of balls. We consider approximating the MAT of a three-dimens ional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewise-linear approximation to the objec t surface, which we call the power crust. We assume that we are given as input a sufficiently dense sample of points from the object surface. We select a subset of the Voronoi balls of the sam ple, the polar balls, as the union of balls representation. We bound the ge ometric error of the union, and of the corresponding power crust, and show that both representations are topologically correct as well. Thus, our resu lts provide a new algorithm for surface reconstruction from sample points. By construction, the power crust is always the boundary of a polyhedral sol id, so we avoid the polygonization, hole-filling or manifold extraction ste ps used in previous algorithms. The union of balls representation and the power crust have corresponding pi ecewise-linear dual representations, which in some sense approximate the me dial axis. We show a geometric relationship between these duals and the med ial axis by proving that, as the sampling density goes to infinity, the set of poles, the centers of the polar balls, converges to the medial axis. (C ) 2001 Elsevier Science B.V All rights reserved.