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

TL;DR

This paper presents a method for reconstructing smooth orientable submanifolds using L1-norm minimization on chains within a simplicial complex, resulting in a triangulation that aligns with the flat Delaunay complex, implementable via linear programming.

Contribution

It introduces a novel L1-norm minimization approach for shape reconstruction that guarantees a unique triangulation matching the flat Delaunay complex.

Findings

The minimization problem has a unique solution under certain conditions.

The solution triangulates the submanifold and matches the flat Delaunay complex.

Reconstruction can be efficiently implemented using linear programming.

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.