Efficient Computation of Image Persistence

Authors Ulrich Bauer , Maximilian Schmahl

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Author Details

Ulrich Bauer
  • Department of Mathematics, TUM School of Computation, Information and Technology, and Munich Data Science Institute, Technical University of Munich, Germany
  • www.ulrich-bauer.org
Maximilian Schmahl
  • Universität Heidelberg, Germany

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Ulrich Bauer and Maximilian Schmahl. Efficient Computation of Image Persistence. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We present an algorithm for computing the barcode of the image of a morphism in persistent homology induced by an inclusion of filtered finite-dimensional chain complexes. The algorithm makes use of the clearing optimization and can be applied to inclusion-induced maps in persistent absolute homology and persistent relative cohomology for filtrations of pairs of simplicial complexes. The clearing optimization works particularly well in the context of relative cohomology, and using previous duality results we can translate the barcodes of images in relative cohomology to those in absolute homology. This forms the basis for an implementation of image persistence computations for inclusions of filtrations of Vietoris-Rips complexes in the framework of the software Ripser.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Computational geometry
  • Persistent homology
  • image persistence
  • barcode computation


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  1. Ulrich Bauer. Ripser: efficient computation of Vietoris-Rips persistence barcodes. J. Appl. Comput. Topol., 5(3):391-423, 2021. URL: https://doi.org/10.1007/s41468-021-00071-5.
  2. Ulrich Bauer and Michael Lesnick. Induced matchings and the algebraic stability of persistence barcodes. J. Comput. Geom., 6(2):162-191, 2015. URL: https://doi.org/10.20382/jocg.v6i2a9.
  3. Ulrich Bauer and Michael Lesnick. Persistence diagrams as diagrams: A categorification of the stability theorem. In Nils A. Baas, Gunnar E. Carlsson, Gereon Quick, Markus Szymik, and Marius Thaule, editors, Topological Data Analysis, pages 67-96, Cham, 2020. Springer. URL: https://doi.org/10.1007/978-3-030-43408-3_3.
  4. Ulrich Bauer and Maximilian Schmahl. Efficient computation of image persistence. Preprint, 2022. URL: https://arxiv.org/abs/2201.04170.
  5. Ulrich Bauer and Maximilian Schmahl. Lifespan functors and natural dualities in persistent homology. Homology Homotopy Appl., 2023. To appear. URL: https://arxiv.org/abs/2012.12881.
  6. Ulrich Bauer and Maximilian Schmahl. Ripser for image persistence, 2023. GitHub. URL: https://github.com/Ripser/ripser/tree/image-persistence-simple.
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