Delaunay-like Triangulation of Smooth Orientable Submanifolds by L1-Norm Minimization
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909542815629312 |
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| author | Attali, Dominique Lieutier, André |
| author_facet | Attali, Dominique Lieutier, André |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_06008 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Delaunay-like Triangulation of Smooth Orientable Submanifolds by L1-Norm Minimization Attali, Dominique Lieutier, André Computational Geometry 65D18, 57Q05, 55U05, 51F99 F.2.2; I.3.5 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. |
| title | Delaunay-like Triangulation of Smooth Orientable Submanifolds by L1-Norm Minimization |
| topic | Computational Geometry 65D18, 57Q05, 55U05, 51F99 F.2.2; I.3.5 |
| url | https://arxiv.org/abs/2203.06008 |