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arXiv · 2203.06008

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

Dominique Attali·André Lieutier
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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.

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Dominique Attali, André Lieutier. 2025-03-19. Delaunay-like Triangulation of Smooth Orientable Submanifolds by L1-Norm Minimization. https://arxiv.org/abs/2203.06008

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