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 contraints are linear, the triangulation process can thus be implemented by linear programming.

翻译：本文研究形状重建问题，其中待重建形状为欧氏空间中的可定向光滑d维子流形。假设输入是逼近该子流形的单纯复形K（如Čech复形或Rips复形），我们将从K重建子流形的问题重新表述为L1范数最小化问题，其优化变量为K中的d链。在K满足特定合理条件的前提下，我们证明该最小化问题存在唯一解，该解三角剖分了子流形，并与另一篇配套论文中引入并研究的平坦Delaunay复形一致。由于目标函数为加权L1范数且约束条件为线性，该三角剖分过程可通过线性规划实现。