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<p>The persistent homology transform, Betti function transform, and Euler characteristic transform represent a shape with a multiset of persistence diagrams, Betti functions, or Euler characteristic functions, respectively, parameterized by the sphere of directions in the ambient space. In this work, we give the first explicit construction of finite sets of directions discretizing the verbose variants of these transforms and show that such discretizations faithfully represent the underlying shape. Our discretization, while exponential in the dimension of the shape, does not depend on any restrictions on the particular immersion beyond general position, and is stable with respect to various perturbations.</p>

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A Faithful Discretization of Verbose Directional Transforms

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月25日

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  • Brittany Terese Fasy,
  • Samuel Micka,
  • David L. Millman,
  • Anna Schenfisch,
  • Lucia Williams

摘要

The persistent homology transform, Betti function transform, and Euler characteristic transform represent a shape with a multiset of persistence diagrams, Betti functions, or Euler characteristic functions, respectively, parameterized by the sphere of directions in the ambient space. In this work, we give the first explicit construction of finite sets of directions discretizing the verbose variants of these transforms and show that such discretizations faithfully represent the underlying shape. Our discretization, while exponential in the dimension of the shape, does not depend on any restrictions on the particular immersion beyond general position, and is stable with respect to various perturbations.