Topology guaranteeing manifold reconstruction using distance function to noisy data

Abstract

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 & al. and Dey & al. 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. Our results allow to consider balls of different radii. For the first approximation condition, we also 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. We obtain similar results for smooth compact submanifolds S of Rk of any codimension.