Efficient Computation of Topological Integral Transforms

Authors Vadim Lebovici , Steve Oudot, Hugo Passe

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Author Details

Vadim Lebovici
  • Mathematical Institute, University of Oxford, United Kingdom
Steve Oudot
  • Inria Saclay, Palaiseau, France
Hugo Passe
  • Ecole Normale Supérieure de Lyon, France

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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)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Topological data analysis
  • Euler calculus
  • Topological integral transform
  • Euler characteristic transform
  • Hybrid transforms


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  1. Erik J Amézquita. demeter. https://github.com/amezqui3/demeter, 2021.
  2. Erik J Amézquita, Michelle Y Quigley, Tim Ophelders, Jacob B Landis, Daniel Koenig, Elizabeth Munch, and Daniel H Chitwood. Measuring hidden phenotype: Quantifying the shape of barley seeds using the euler characteristic transform. in silico Plants, 4(1), 2022. Google Scholar
  3. Mladen Bestvina and Noel Brady. Morse theory and finiteness properties of groups. Inventiones mathematicae, 129(3):445-470, 1997. Google Scholar
  4. Leo Betthauser. Topological-Reconstruction. https://github.com/lbetthauser/Topological-Reconstruction, 2018.
  5. Lorin Crawford, Anthea Monod, Andrew X. Chen, Sayan Mukherjee, and Raúl Rabadán. Predicting clinical outcomes in glioblastoma: An application of topological and functional data analysis. Journal of the American Statistical Association, 115(531):1139-1150, 2020. Google Scholar
  6. Justin Curry, Robert Ghrist, and Michael Robinson. Euler calculus with applications to signals and sensing. In Proceedings of Symposia in Applied Mathematics, volume 70, pages 75-146, 2012. Google Scholar
  7. Justin Curry, Sayan Mukherjee, and Katharine Turner. How many directions determine a shape and other sufficiency results for two topological transforms. Transactions of the American Mathematical Society, Series B, 9(32):1006-1043, 2022. Google Scholar
  8. Robert Ghrist, Rachel Levanger, and Huy Mai. Persistent homology and euler integral transforms. Journal of Applied and Computational Topology, 2:55-60, 2018. Google Scholar
  9. Robert Ghrist and Michael Robinson. EulerendashBessel and EulerendashFourier transforms. Inverse Problems, 27(12), 2011. Google Scholar
  10. Olympio Hacquard and Vadim Lebovici. Euler characteristic tools for topological data analysis. arXiv preprint:2303.14040, 2023. Google Scholar
  11. Vadim Lebovici. Hybrid transforms of constructible functions. Foundations of Computational Mathematics, pages 1-47, 2022. Google Scholar
  12. Hugo Passe. eucalc. https://github.com/HugoPasse/Eucalc, 2021.
  13. Pierre Schapira. Operations on constructible functions. Journal of Pure and Applied Algebra, 72(1):83-93, 1991. Google Scholar
  14. Pierre Schapira. Tomography of constructible functions. In International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pages 427-435. Springer, 1995. Google Scholar
  15. Wai Shing Tang, Gabriel Monteiro da Silva, Henry Kirveslahti, Erin Skeens, Bibo Feng, Timothy Sudijono, Kevin K. Yang, Sayan Mukherjee, Brenda Rubenstein, and Lorin Crawford. A topological data analytic approach for discovering biophysical signatures in protein dynamics. PLOS Computational Biology, 18(5):1-42, May 2022. Google Scholar
  16. Katharine Turner, Sayan Mukherjee, and Doug M. Boyer. Persistent homology transform for modeling shapes and surfaces. Information and Inference: A Journal of the IMA, 3(4):310-344, 2014. Google Scholar
  17. Oleg Yanovich Viro. Some integral calculus based on euler characteristic. In Topology and geometry—Rohlin seminar, pages 127-138. Springer, 1988. Google Scholar
  18. Günter M Ziegler. Lectures on Polytopes, Updated seventh printing of the first edition, volume 152. Springer, 2006. Google Scholar