Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees

Given a smooth compact codimension one submanifold S of Rk and a compact approximation K of S, we prove that it is possible to reconstruct S and to approximate the medial axis of S with topological guarantees using unions of balls centered on K. We consider two notions of noisy-approximation that generalize sampling conditions introduced by Amenta et al. and Dey et al. The first one generalizes uniform sampling based on the minimum value of the local feature size. The second one generalizes non-uniform sampling based on the local feature size function of S. The density and noise of the approximation are bounded by a constant times the local feature size function. This constant does not depend on the surface S. Our results are based upon critical point theory for distance functions. For the two approximation conditions, we prove that the connected components of the boundary of unions of balls centered on K are isotopic to S. We consider using both balls of uniform radius and also balls whose radii vary with the local level of detail of the manifold. For the first approximation condition, we prove that a subset (known as the λ-medial axis) of the medial axis of Rk∖K is homotopy equivalent to the medial axis of S. Our results generalize to smooth compact submanifolds S of Rk of any codimension.