How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms
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- by Justin Curry, Sayan Mukherjee and Katharine Turner HTML | PDF
- Trans. Amer. Math. Soc. Ser. B 9 (2022), 1006-1043
Abstract:
In this paper we consider two topological transforms that are popular in applied topology: the Persistent Homology Transform and the Euler Characteristic Transform. Both of these transforms are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. Both transforms take a shape, viewed as a tame subset $M$ of $\mathbb { R}^d$, and associates to each direction $v\in S^{d-1}$ a shape summary obtained by scanning $M$ in the direction $v$. These shape summaries are either persistence diagrams or piecewise constant integer-valued functions called Euler curves. By using an inversion theorem of Schapira, we show that both transforms are injective on the space of shapes, i.e. each shape has a unique transform. Moreover, we prove that these transforms determine continuous maps from the sphere to the space of persistence diagrams, equipped with any Wasserstein $p$-distance, or the space of Euler curves, equipped with certain $L^p$ norms. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we prove additional uniqueness results in terms of distributions on the space of Euler curves. Finally, our main result proves that any shape in a certain uncountable space of PL embedded shapes with plausible geometric bounds can be uniquely determined using only finitely many directions.References
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Additional Information
- Justin Curry
- Affiliation: Department of Mathematics and Statistics, University at Albany SUNY, Albany, New York 12222
- MR Author ID: 983660
- ORCID: 0000-0003-2504-8388
- Email: jmcurry@albany.edu
- Sayan Mukherjee
- Affiliation: Departments of Statistical Science, Mathematics, Computer Science, Biostatistics & Bioinformatics, Duke University, Durham, North Carolina 27708; Center for Scalable Data Analytics and Artificial Intelligence at the Universität Leipzig, Germany; and the Max Planck Institute for Mathematics in the Natural Sciences, Leipzig, Germany
- MR Author ID: 683641
- ORCID: 0000-0002-6715-3920
- Email: sayan.mukherjee@mis.mpg.de
- Katharine Turner
- Affiliation: Mathematical Sciences Institute, Australian National University, Canberra, Australia
- MR Author ID: 1046734
- ORCID: 0000-0002-6679-7441
- Email: katharine.turner@anu.edu.au
- Received by editor(s): June 7, 2018
- Received by editor(s) in revised form: October 10, 2019, September 24, 2021, and April 27, 2022
- Published electronically: October 31, 2022
- Additional Notes: The first author was supported by Kathryn Hess and EPFL for his travel to collaborate with the third author on this project. The third author received an Australian Research Council Discovery Early Career Award (project number DE200100056) funded by the Australian Government. The first author was also supported by NSF CCF-1850052 and NASA Contract 80GRC020C0016 for his research. The second author was partially funded by the Human Frontier Science Program, as well as NSF DMS 17-13012, NSF ABI 16-61386, and NSF DMS 16-13261.
- © Copyright 2022 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 9 (2022), 1006-1043
- MSC (2020): Primary 46M20, 52C45; Secondary 60B05
- DOI: https://doi.org/10.1090/btran/122
- MathSciNet review: 4505675