Delaunay-Like Triangulation of Smooth Orientable Submanifolds by \(\ell _1\)-Norm Minimization

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 Čech complex or the Rips complex), we recast the problem of reconstructing the submanifold from K as an \(\ell _1\)-norm minimization problem in which the optimization variable is a d-chain of K over the field \(\mathbb {R}\). 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 (Attali and Lieutier in Flat Delaunay complexes for homeomorphic manifold reconstruction, 2022). Since the objective is a weighted \(\ell _1\)-norm and the constraints are linear, the triangulation process can thus be implemented by linear programming.