The extended persistent homology transform of manifolds with boundary

被引：0
作者：
Turner K. [1 ]
Robins V. [1 ]
Morgan J. [2 ]
机构：
[1] Australian National University, Canberra, ACT
[2] University of Sydney, Sydney, NSW
来源：
Journal of Applied and Computational Topology | 2024年 / 8卷 / 7期
基金：
澳大利亚研究理事会;
关键词：
Extended persistent homology; Morse theory of manifolds with boundary; Primary; 55N31; Secondary; 62R40; Statistical shape analysis;
D O I：
10.1007/s41468-024-00175-8
中图分类号：
学科分类号：
摘要：
A shortcoming of persistent homology is that when two domains have different numbers of components or holes the persistence diagrams of any filtration will have an infinite distance between them. We address this issue by revisiting the theory of extended persistence, initially developed by Cohen-Steiner, Edelsbrunner and Harer in 2009 to quantify the support of the essential homology classes for Morse functions on manifolds. We simplify the mathematical treatment of extended persistence by formulating it as a persistence module derived from a sequence of relative homology groups for pairs of spaces. Then, for n-manifolds with boundary embedded in Rn, we use Morse theory to show that the extended persistent homology of a height function over M can be deduced from the extended persistent homology of the same height function restricted to ∂M. As an application, we describe the extended persistent homology transform (XPHT); a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape with respect to that direction. We define a distance between two shapes by integrating over the sphere the distance between the respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different numbers of essential classes. We study the application of the XPHT to binary images; outlining an algorithm for efficient calculation of the XPHT exploiting relationships between the PHT of the boundary curves to the extended persistence of the foreground. © The Author(s) 2024.
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页码：2111 / 2154
页数：43
相关论文
共 27 条
  • [1] Amezquita E.J., Quigley M.Y., Ophelders T., Landis J.B., Koenig D., Munch E., Chitwood D.H., Measuring hidden phenotype: quantifying the shape of barley seeds using the Euler characteristic transform, In Silico Plants, 4, 1, (2022)
  • [2] Bauer U., Lesnick M., Induced matchings of barcodes and the algebraic stability of persistence, In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, pp. 355-364, (2014)
  • [3] Bauer U., Botnan M.B., Fluhr B., (2007)
  • [4] Nicholas B., Vanessa R., Katharine T., Planar symmetry detection and quantification using the extended persistent homology transform, Proceedings of IEEE Vis 2023 Topoinvis Workshop, pp. 1-9, (2023)
  • [5] Braess D., Morse-theorie für berandete mannigfaltigkeiten, Math. Ann, 208, 2, pp. 133-148, (1974)
  • [6] Carlsson G., De Silva V., Kalisnik S., Morozov D., Parametrized homology via zigzag persistence, Algebraic Geom. Topol, 19, 2, pp. 657-700, (2019)
  • [7] Chazal F., Crawley-Boevey W., De Silva, V.: The Observable Structure of Persistence Modules (2014). Arxiv, (1405)
  • [8] Cohen-Steiner D., Edelsbrunner H., Harer J., Erratum to: Extending persistence using Poincaré and Lefschetz duality, Found. Comput. Math, 9, 1, pp. 133-134, (2009)
  • [9] Cohen-Steiner D., Edelsbrunner H., Harer J., Extending persistence using Poincaré and Lefschetz duality, Found. Comput. Math, 9, 1, pp. 79-103, (2009)
  • [10] Crawford L., Monod A., Chen A.X., Mukherjee S., Rabadan R., Predicting clinical outcomes in glioblastoma: an application of topological and functional data analysis, J. Am. Stat. Assoc, 115, 531, pp. 1139-1150, (2020)
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