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<p><?tk 4?>In this paper, we study the shape reconstruction problem, when the shape we wish to reconstruct is an orientable smooth <i>d</i>-dimensional submanifold of the Euclidean space. Assuming we have as input a simplicial complex <i>K</i> that approximates the submanifold (such as the Čech complex or the Rips complex), we recast the problem of reconstructing the submanifold from <i>K</i> as an <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\ell _1\)</EquationSource> </InlineEquation>-norm minimization problem in which the optimization variable is a <i>d</i>-chain of <i>K</i> over the field <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> </InlineEquation>. Providing that <i>K</i>, 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\ell _1\)</EquationSource> </InlineEquation>-norm and the constraints are linear, the triangulation process can thus be implemented by linear programming.</p>

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

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Dominique Attali,
  • André Lieutier

摘要

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.