Delaunay-like Triangulation of Smooth Orientable Submanifolds by L1-Norm Minimization

TL;DR

This work tackles reconstructing a compact orientable smooth $d$-dimensional submanifold $\mathcal{M}$ from a finite point sample by recasting triangulation as a weighted $\ell_1$-minimization over $d$-chains in a simplicial complex $K$. The authors define a Delaunay energy, expressible as a weighted sum of per-simplex weights $\omega(\sigma)$, and show that under geometric sampling conditions the unique minimizer corresponds to a faithful triangulation encoded by the Delloc complex, equivalently the flat Delaunay complex. The optimization problem is convex and amenable to linear programming, enabling practical computation of triangulations that faithfully triangulate $\mathcal{M}$. The paper also provides a pathway to realistic algorithms via localized load constraints, tangent-space approximations, and perturbation schemes that satisfy safety conditions, with extensions to higher-dimensional sampling contexts and potential homology-based variants.

Abstract

In this paper, we study the shape reconstruction problem, when the shape we wish to reconstruct is an orientable smooth d-dimensional submanifold of the Euclidean space. Assuming we have as input a simplicial complex K that approximates the submanifold (such as the Cech complex or the Rips complex), we recast the problem of reconstucting the submanifold from K as a L1-norm minimization problem in which the optimization variable is a d-chain of K. Providing that K satisfies certain reasonable conditions, we prove that the considered minimization problem has a unique solution which triangulates the submanifold and coincides with the flat Delaunay complex introduced and studied in a companion paper. Since the objective is a weighted L1-norm and the constraints are linear, the triangulation process can thus be implemented by linear programming.

Figures (13)

• Figure 1: (a) A finite set of (black) points $P$ that sample a (blue) curve $\mathcal{C}\xspace$. (b) The graph $K$ is a proximity graph constructed from $P$ by connecting every pair of points that are less than $2r$ apart. Each edge in $K$ is arbitrarily oriented. (c) The segment $[a,b]$ intersects both $\mathcal{C}\xspace$ and $K$ transversally with $b$ in the inside region and $a$ in the outside region. (d) A 1-cycle of $K$ whose flux through $[a,b]$ equals 2.
• Figure 2: (a) A 1-cycle is said to be normalized if its flux through $[a,b]$ is equal to 1. (b) A normalized cycle that minimizes the length. (c) A normalized cycle that minimizes the sum of the edge lengths squared. (d) A normalized cycle that minimizes the sum of the edge lengths cubed, also called the Delaunay energy (up to a multiplicative constant). Our reconstruction method returns the support of that cycle.
• Figure 3: Left: the Delaunay weight of $\sigma$ can be depicted as the $(d+1)$-volume of the blue region between the lifted geometric simplex $\operatorname{conv}\hat{\sigma}$ and the paraboloid $\mathscr{P}\xspace$ (see Lemma \ref{['lemma:interpreting-Delaunay-weights']}). Right: the Delaunay weight of $\sigma$ is also the $(d+1)$-volume of the blue region lying below the graph of $-\operatorname{Power}_\sigma$ and above $\operatorname{conv}\sigma$.
• Figure 4: Two triangulations of six points (black dots) in the plane. For each triangulation $T$, the Delaunay energy is the volume between the convex hull of the points and the piecewise parabolic surface $\mathcal{W}_T$. The surface $\mathcal{W}_T$ is lowest and therefore the Delaunay energy of $T$ is smallest when $T$ is the Delaunay complex as is the case on the right.
• Figure 5: Left: a $d$-dimensional submanifold $\mathcal{M}\xspace$ (for $d=1$) and a noisy sample $P$ of $\mathcal{M}\xspace$. Right: a simplicial complex $K$ whose vertex set is $P$.

Theorems & Definitions (81)

• Definition 1: Delaunay simplex
• Definition 2: Delaunay complex
• Definition 3: General position
• Definition 4: Triangulation
• Theorem 1
• Theorem 2
• Definition 5: Delaunay weight
• Lemma 3
• proof
• Lemma 4