When alpha-complexes collapse onto codimension-1 submanifolds

TL;DR

The work tackles recovering a smooth codimension-one manifold $\mathcal{M}$ from a finite sample by operating on the $\alpha$-complex with vertical collapses. It introduces NaiveSquash (with knowledge of $\mathcal{M}$) and PracticalSquash (no prior knowledge) and proves they can produce a triangulation of $\mathcal{M}$ under explicit sampling conditions, including a quantitative bound on triangle–manifold alignment. A key result is an angle bound $\sin \angle \mathrm{Aff}(abc), \mathbf{T}_{a}\mathcal{M} \le \frac{\sqrt{3}\,\rho}{\mathcal{R}}$ and corresponding sampling ratios $\varepsilon/\mathcal{R}$ that ensure correctness in $d=3$, with $\varepsilon/\mathcal{R} \le 0.225$ for NaiveSquash and $\le 0.178$ for PracticalSquash; the restricted Delaunay complex is shown to be generically homeomorphic to $\mathcal{M}$ under $\varepsilon/\mathcal{R} \le 0.225$. The approach yields a modular, geometry-driven pathway to robust surface reconstruction and improves conditioning bounds for 3D settings, while the proofs decompose into independent geometric components via vertical convexity and skin projections.

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 $α$-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 $α$-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.

Figures (6)

• Figure 1: Points sampling a surface in $\mathbb{R}^3$ (a) with the corresponding $\alpha$-complex, where tetrahedra are highlighted (b). Applying Practical Squash with parameter $\alpha$ outputs (c).
• Figure 2: Left: $P$ is such that neither $P^{\oplus \alpha}$ nor $\operatorname{Del}(P,\alpha)$ are vertically convex relative to a horizontal line. Right: Decomposition of $P^{\oplus \alpha} \setminus \boldsymbol{\lvert}\operatorname{Del}(P,\alpha)\boldsymbol{\rvert}^\circ$ in joins as described in edelsbrunner2011alpha.
• Figure 3: A simplicial complex $K$ vertically convex relative to the curve $\mathcal{M}$. Each segment $\mathbf{N}_{m}\mathcal{M}\xspace \cap B(m,r)$ (in dashed orange) intersects $\boldsymbol{\lvert}K\boldsymbol{\rvert}$ in a line segment, as highlighted (blue) for the point $m$ (represented by a black square). Lemma \ref{['lemma:upper-and-lower-skins-homeomorphic-to-manifold']} shows that each of the two skins of $\boldsymbol{\lvert}K\boldsymbol{\rvert}$, depicted in green and pink according to the labeling arrows, is homeomorphic to $\mathcal{M}$.
• Figure 4: Upper (smooth green edges) and lower (dotted pink edge) facets of a $2$-simplex $\sigma\subseteq\mathbb{R}\xspace^2$.
• Figure 5: Schematic drawings of $K$ in blue (smooth filled areas). Top row: the edge $\tau$ is free but not vertically free relative to $\mathcal{M}$ and collapsing $\tau$ does not preserve the vertical convexity of $K$. Bottom row: the vertex $\tau$ is free from above relative to $\mathcal{M}$, so that collapsing $\tau$ preserves the vertical convexity of $K$ (Lemma \ref{['lemma:invariant']}). The $(d-1)$-simplices of $K$ that disappear with $\tau$ are precisely the upper facets of $\sigma$ (smooth edges, in green).

Theorems & Definitions (12)

• Definition 1: Vertical convexity
• Definition 2: Vertical simplex
• Definition 3: Non-vertical skeleton
• Lemma 3
• Lemma 3
• Lemma 3
• Lemma 3
• Definition 4: Vertically free simplices
• Remark 5
• Definition 6: Vertical collapse