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Open Access Publications from the University of California

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 infinite union of balls. We consider approximating the MAT of a three-dimensional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewise-linear approximation to the object surface, which we call the power crust. We assume that we are given as imput a sufficiently 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 represenation. We bound the geometric error of the union, and of the corresponding power crust, and show that both representation and 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, hole-filling or manifold extraction steps used in previous algorithms. The union of balls representation and the power crust have corresponding piecewise-linear dual representation, which is 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 into infinity, the set of poles, the center of the polar balls, converge to the medial axis.

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