Flat Delaunay Complexes for Homeomorphic Manifold Reconstruction

TL;DR

This paper introduces the flat Delaunay complex (FDC), a new method for reconstructing smooth manifolds from point clouds, proving it is homeomorphic to the original manifold under certain conditions, even with noisy data.

Contribution

The paper presents the flat Delaunay complex (FDC), a novel variant of the tangential Delaunay complex, with a proof of homeomorphism to the manifold and a noise-robust perturbation scheme.

Findings

FDC is homeomorphic to the underlying manifold under certain sampling conditions

The construction remains valid even with noisy data points

Provides a framework for a variational approach to manifold reconstruction

Abstract

Given a smooth submanifold of the Euclidean space, a finite point cloud and a scale parameter, we introduce a construction which we call the flat Delaunay complex (FDC). This is a variant of the tangential Delaunay complex (TDC) introduced by Boissonnat et al.. Building on their work, we provide a short and direct proof that when the point cloud samples sufficiently nicely the submanifold and is sufficiently safe (a notion which we define in the paper), our construction is homeomorphic to the submanifold. Because the proof works even when data points are noisy, this allows us to propose a perturbation scheme that takes as input a point cloud sufficiently nice and returns a point cloud which in addition is sufficiently safe. Equally importantly, our construction provides the framework underlying a variational formulation of the reconstruction problem which we present in a companion paper.