# Implicit Manifold Reconstruction

arXiv: https://arxiv.org/abs/1904.03764 · HTML: https://mlsanity.com/paper/1904.03764/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Apr 2019.

## Built on by (most cited first, 1 of 1)

- [High-Fidelity Surface Splatting-Based 3D Reconstruction from Multi-View Images](https://mlsanity.com/paper/2605.07254/) (2026)

## Abstract (arXiv)

Let cal M subset ℝd be a compact, smooth and boundaryless manifold with dimension m and unit reach. We show how to construct a function varphi: ℝd rightarrow ℝd-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 widehatcal M denote the set of points at distance varepsilon or less from cal M. There exists varepsilon0 ∈ (0,1) that decreases as d increases such that if varepsilon ≤ varepsilon0, the following guarantees hold. First, Z_varphi cap widehatcal M is a faithful approximation of cal M in the sense that Z_varphi cap widehatcal M is homeomorphic to cal M, the Hausdorff distance between Z_varphi cap widehatcal M and cal M is O(m5/2varepsilon2), and the normal spaces at nearby points in Z_varphi cap widehatcal M and cal M make an angle O(m2√kappavarepsilon). 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(mvarepsilon). Third, we give a projection operator that only uses sample points in P at distance O(mvarepsilon) from the initial point. The projection operator maps any initial point near P onto Z_varphi cap widehatcal M in the limit by repeated applications.