Implicit manifold reconstruction

Let P be a dense set of points sampled from an Tridimensional compact smooth manifold ∑ in Rd. We show how to construct an implicit function φ : Rd→Rd-m from P so that the zero-set Sφ of φ contains a homeomorphic approximation of ∑. The Hausdorff distance between ∑ and this homeomorphic approximation is at most ετfor any fixed τ < 2. Moreover, for every point x at distance ε τ or less from ∑, the normal space of Sφ at x makes an O(ε(τ-1)/2)angle with the normal space of E at the point nearest to x. The functionøhas local support, which makes local homeomorphic reconstruction possible without a complete sampling. Copyright © 2014 by the Society for Industrial and Applied Mathematics