Implicit Manifold Reconstruction

Let ${\cal M} \subset \mathbb{R}^d$ be a compact, smooth and boundaryless manifold with dimension $m$ and unit reach. We show how to construct a function $\varphi: \mathbb{R}^d \rightarrow \mathbb{R}^{d-m}$ from a uniform $(\varepsilon,\kappa)$-sample $P$ of $\cal M$ that offers several guarantees. Let $Z\varphi$ denote the zero set of $\varphi$. Let $\widehat{{\cal M}}$ denote the set of points at distance $\varepsilon$ or less from $\cal M$. There exists $\varepsilon0 \in (0,1)$ that decreases as $d$ increases such that if $\varepsilon \leq \varepsilon0$, the following guarantees hold. First, $Z\varphi \cap \widehat{\cal M}$ is a faithful approximation of $\cal M$ in the sense that $Z\varphi \cap \widehat{\cal M}$ is homeomorphic to $\cal M$, the Hausdorff distance between $Z\varphi \cap \widehat{\cal M}$ and $\cal M$ is $O(m^{{5/2}\varepsilon{2})$,} and the normal spaces at nearby points in $Z\varphi \cap \widehat{\cal M}$ and $\cal M$ make an angle $O(m^{2\sqrt{\kappa\varepsilon})$.} Second, $\varphi$ has local support; in particular, the value of $\varphi$ at a point is affected only by sample points in $P$ that lie within a distance of $O(m\varepsilon)$. Third, we give a projection operator that only uses sample points in $P$ at distance $O(m\varepsilon)$ from the initial point. The projection operator maps any initial point near $P$ onto $Z\varphi \cap \widehat{\cal M}$ in the limit by repeated applications.