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Persistent Homology for the Quantitative Evaluation of Architectural Features in Prostate Cancer Histology.

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A Faithful Discretization of Verbose Directional Transforms.

Brittany Terese Fasy1,2, Samuel Micka3, David L Millman1,4

  • 1School of Computing, Montana State U, Bozeman, Montana USA.

Discrete & Computational Geometry
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Summary
This summary is machine-generated.

This study introduces the first explicit construction for discretizing shape transforms like persistent homology, using finite sets of directions. These discretizations faithfully represent shapes and are stable under perturbations.

Keywords:
Betti functionsDirectional transformsEuler characteristic curvesImmersed simplicial complexesPersistence diagramsReconstructionShape representation

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Area of Science:

  • Topology
  • Computational Geometry
  • Data Analysis

Background:

  • Persistent homology, Betti function, and Euler characteristic transforms represent shapes using multisets.
  • These representations are parameterized by directions in ambient space.
  • Existing methods can be verbose and computationally intensive.

Purpose of the Study:

  • To develop the first explicit construction of finite sets of directions for discretizing shape transforms.
  • To demonstrate that these discretizations faithfully represent the underlying shape.
  • To ensure the method is stable and does not require restrictive assumptions.

Main Methods:

  • Developed explicit constructions for discretizing persistent homology, Betti function, and Euler characteristic transforms.
  • Utilized finite sets of directions for parameterization.
  • Analyzed the faithfulness and stability of the discretizations.

Main Results:

  • Successfully constructed finite discretizations for verbose shape transforms.
  • Proved that these discretizations accurately represent the original shape.
  • Demonstrated stability with respect to perturbations and independence from immersion restrictions beyond general position.

Conclusions:

  • The proposed discretization method offers a practical approach to shape representation using topological transforms.
  • This work advances computational topology by providing stable and faithful discretizations.
  • The method's efficiency is exponential in shape dimension but generalizable.