Implicit Manifold Reconstruction

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 , 1) that decreases as d increases such that if ε≤ ε, 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 / 2}ε²) , and the normal spaces at nearby points in Z_φ∩ M^ and M make an angle O(m2κε). 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.