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 in nite union of balls. We consider approximating the MAT of a three-dimensional object, and its complement, with a nite union of balls. Using this approximate MAT we de ne a new piecewise-linear approximation to the object surface, which we call the power crust. We assume that we are given as input a suÆciently dense sample of points from the object surface. We select a subset of the Voronoi balls of the sample, the polar balls, as the union of balls representation. We bound the geometric error of the union, and of the corresponding power crust, and show that both representations are topologically correct as well. Thus, our results provide a new algorithm for surface reconstruction from sample points. By construction, the power crust is always the boundary of a polyhedral solid, so we avoid the polygonization, holelling or manifold extraction steps used in previous algorithms. The union of balls representation and the power crust have corresponding piecewise-linear dual representations, which in some sense approximate the medial axis. We show a geometric relationship between these duals and the medial axis by proving that, as the sampling density goes to in nity, the set of poles, the centers of the polar balls, converges to the medial axis