Implicit Manifold Reconstruction
Siu-Wing Cheng, Man-Kwun Chiu · 2013
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 ∊τ for any fixed τ < 2. Moreover, for every point × at distance ∊τ or less from Σ, the normal space of Sφ at × makes an O(∊(τ–1)/2) angle with the normal space of Σ at the point nearest to ×. The function φ has local support, which makes local homeomorphic reconstruction possible without a complete sampling.