Implicit Manifold Reconstruction

Let P be a dense set of points sampled from an m-dimensional compact smooth manifold Σ 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 Σ. The Hausdorff distance between Σ and this homeomorphic approximation is at most eτ for any fixed τ < 2. Moreover, for every point x at distance eτ or less from Σ, the normal space of Sφ at x makes an O(e(τ-1)/2) angle with the normal space of Σ at the point nearest to x. The function φ has local support, which makes local homeomorphic reconstruction possible without a complete sampling.

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