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DOI: 10.4230/LIPIcs.SEA.2024.22

Efficient Computation of Topological Integral Transforms

Authors: Vadim Lebovici, Steve Oudot, and Hugo Passe

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract

Topological integral transforms have found many applications in shape analysis, from prediction of clinical outcomes in brain cancer to analysis of barley seeds. Using Euler characteristic as a measure, these objects record rich geometric information on weighted polytopal complexes. While some implementations exist, they only enable discretized representations of the transforms, and they do not handle weighted complexes (such as for instance images). Moreover, recent hybrid transforms lack an implementation. In this paper, we introduce eucalc, a novel implementation of three topological integral transforms - the Euler characteristic transform, the Radon transform, and hybrid transforms - for weighted cubical complexes. Leveraging piecewise linear Morse theory and Euler calculus, the algorithms significantly reduce computational complexity by focusing on critical points. Our software provides exact representations of transforms, handles both binary and grayscale images, and supports multi-core processing. It is publicly available as a C++ library with a Python wrapper. We present mathematical foundations, implementation details, and experimental evaluations, demonstrating eucalc’s efficiency.

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Vadim Lebovici, Steve Oudot, and Hugo Passe. Efficient Computation of Topological Integral Transforms. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lebovici_et_al:LIPIcs.SEA.2024.22,
  author =	{Lebovici, Vadim and Oudot, Steve and Passe, Hugo},
  title =	{{Efficient Computation of Topological Integral Transforms}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{22:1--22:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.22},
  URN =		{urn:nbn:de:0030-drops-203878},
  doi =		{10.4230/LIPIcs.SEA.2024.22},
  annote =	{Keywords: Topological data analysis, Euler calculus, Topological integral transform, Euler characteristic transform, Hybrid transforms}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2020.22

On Rectangle-Decomposable 2-Parameter Persistence Modules

Authors: Magnus Bakke Botnan, Vadim Lebovici, and Steve Oudot

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract

This paper addresses two questions: (1) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (2) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: (a) a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; (b) algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules. Our algorithms are backed up by a new structure theorem, whereby a 2-parameter persistence module is rectangle-decomposable if, and only if, its restrictions to squares are. This local condition is key to the efficiency of our algorithms, and it generalizes previous conditions from the class of block-decomposable modules to the larger one of rectangle-decomposable modules. It also admits an algebraic formulation that turns out to be a weaker version of the one for block-decomposability. Our analysis focuses on the case of modules indexed over finite grids, the more general cases are left as future work.

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Magnus Bakke Botnan, Vadim Lebovici, and Steve Oudot. On Rectangle-Decomposable 2-Parameter Persistence Modules. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{botnan_et_al:LIPIcs.SoCG.2020.22,
  author =	{Botnan, Magnus Bakke and Lebovici, Vadim and Oudot, Steve},
  title =	{{On Rectangle-Decomposable 2-Parameter Persistence Modules}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{22:1--22:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.22},
  URN =		{urn:nbn:de:0030-drops-121802},
  doi =		{10.4230/LIPIcs.SoCG.2020.22},
  annote =	{Keywords: topological data analysis, multiparameter persistence, rank invariant}
}

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Documents authored by Lebovici, Vadim

Efficient Computation of Topological Integral Transforms

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On Rectangle-Decomposable 2-Parameter Persistence Modules

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