When alpha-complexes collapse onto codimension-1 submanifolds

TL;DR

This paper introduces the Naive Squash algorithm for collapsing alpha-complexes onto codimension-1 submanifolds, enabling triangulation of unknown smooth surfaces from point samples with weaker conditions than previous methods.

Contribution

The paper presents a new collapse technique called Naive Squash that simplifies alpha-complexes onto smooth surfaces without requiring prior knowledge of the surface.

Findings

Naive Squash correctly triangulates the surface under certain sampling conditions.

The method provides a bound on the angles of triangles in the alpha-complex.

It demonstrates that the restricted Delaunay complex triangulates the surface under weaker conditions.

Abstract

Given a finite set of points $P$ sampling an unknown smooth surface $\mathcal{M} \subseteq \mathbb{R}^3$, our goal is to triangulate $\mathcal{M}$ based solely on $P$. Assuming $\mathcal{M}$ is a smooth orientable submanifold of codimension 1 in $\mathbb{R}^d$, we introduce a simple algorithm, Naive Squash, which simplifies the $\alpha$-complex of $P$ by repeatedly applying a new type of collapse called vertical relative to $\mathcal{M}$. Naive Squash also has a practical version that does not require knowledge of $\mathcal{M}$. We establish conditions under which both the naive and practical Squash algorithms output a triangulation of $\mathcal{M}$. We provide a bound on the angle formed by triangles in the $\alpha$-complex with $\mathcal{M}$, yielding sampling conditions on $P$ that are competitive with existing literature for smooth surfaces embedded in $\mathbb{R}^3$, while offering a more compartmentalized proof. As a by-product, we obtain that the restricted Delaunay complex of $P$ triangulates $\mathcal{M}$ when $\mathcal{M}$ is a smooth surface in $\mathbb{R}^3$ under weaker conditions than existing ones.