Implicit Manifold Reconstruction
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Let M⊂R^d be a compact, smooth and boundaryless manifold with dimension m and unit reach. We show how to construct a function φ: R^d →R^d-m from a uniform (ε,κ)-sample P of M that offers several guarantees. Let Z_φ denote the zero set of φ. Let M denote the set of points at distance ε or less from M. There exists ε_0 ∈ (0,1) that decreases as d increases such that if ε≤ε_0, the following guarantees hold. First, Z_φ∩ M is a faithful approximation of M in the sense that Z_φ∩ M is homeomorphic to M, the Hausdorff distance between Z_φ∩ M and M is O(m^5/2ε^2), and the normal spaces at nearby points in Z_φ∩ M and M make an angle O(m^2√(κε)). Second, φ has local support; in particular, the value of φ at a point is affected only by sample points in P that lie within a distance of O(mε). Third, we give a projection operator that only uses sample points in P at distance O(mε) from the initial point. The projection operator maps any initial point near P onto Z_φ∩ M in the limit by repeated applications.
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