Delaunay-like Triangulation of Smooth Orientable Submanifolds by L1-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 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.
Keywords
Cite
@article{arxiv.2203.06008,
title = {Delaunay-like Triangulation of Smooth Orientable Submanifolds by L1-Norm Minimization},
author = {Dominique Attali and André Lieutier},
journal= {arXiv preprint arXiv:2203.06008},
year = {2025}
}
Comments
67 pages, 13 figures