3 Search Results for "Curry, Justin"

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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.

Cite as

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.2024.44

Stability and Approximations for Decorated Reeb Spaces

Authors: Justin Curry, Washington Mio, Tom Needham, Osman Berat Okutan, and Florian Russold

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract

Given a map f:X → M from a topological space X to a metric space M, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when M = ℝ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of f. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.

Cite as

Justin Curry, Washington Mio, Tom Needham, Osman Berat Okutan, and Florian Russold. Stability and Approximations for Decorated Reeb Spaces. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 44:1-44:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{curry_et_al:LIPIcs.SoCG.2024.44,
  author =	{Curry, Justin and Mio, Washington and Needham, Tom and Okutan, Osman Berat and Russold, Florian},
  title =	{{Stability and Approximations for Decorated Reeb Spaces}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{44:1--44:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.44},
  URN =		{urn:nbn:de:0030-drops-199891},
  doi =		{10.4230/LIPIcs.SoCG.2024.44},
  annote =	{Keywords: Reeb spaces, Gromov-Hausdorff distance, Persistent homology}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2022.24

The Universal 𝓁^p-Metric on Merge Trees

Authors: Robert Cardona, Justin Curry, Tung Lam, and Michael Lesnick

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract

Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an 𝓁^p-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the p-Wasserstein distance between the associated barcodes. For each p ∈ [1,∞], we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the p = ∞ case, this gives a novel proof of universality for the interleaving distance on merge trees.

Cite as

Robert Cardona, Justin Curry, Tung Lam, and Michael Lesnick. The Universal 𝓁^p-Metric on Merge Trees. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cardona_et_al:LIPIcs.SoCG.2022.24,
  author =	{Cardona, Robert and Curry, Justin and Lam, Tung and Lesnick, Michael},
  title =	{{The Universal 𝓁^p-Metric on Merge Trees}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.24},
  URN =		{urn:nbn:de:0030-drops-160325},
  doi =		{10.4230/LIPIcs.SoCG.2022.24},
  annote =	{Keywords: merge trees, hierarchical clustering, persistent homology, Wasserstein distances, interleavings}
}

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3 Mathematics of computing → Algebraic topology
2 Theory of computation → Computational geometry
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