Gromov‐Hausdorff Stable Signatures for Shapes using Persistence
Abstract
We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on the persistence diagrams of suitable filtrations built on top of these spaces. We prove the stability of our signatures under Gromov‐Hausdorff perturbations of the spaces. We also extend these results to metric spaces equipped with measures. Our signatures are well‐suited for the study of unstructured point cloud data, which we illustrate through an application in shape classification.
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For any non-negative integer k, the function β_k assigning to each finite metric space its k-dimensional homology barcode is distance non-increasing.
“Let β k : F → B be the function which assigns to each finite metric space its k -dimensional homology barcode. Then β k is distance non-increasing.”
From Topological pattern recognition for point cloud data (2014) · cites this paper as Chazal et al. (2009)
Cites
Topological Persistence and Simplification
Edelsbrunner, Letscher, & Zomorodian (2002) · Discrete & Computational Geometry · doi:10.1007/s00454-002-2885-2 · Cited by 1,779 · see citing works
Zomorodian, A., & Carlsson, G. (2005) · Discrete & Computational Geometry · doi:10.1007/s00454-004-1146-y · Cited by 1,591 · see citing works
Stability of Persistence Diagrams
Cohen-Steiner, D., Edelsbrunner, H., & Harer, J. (2007) · Discrete & Computational Geometry · doi:10.1007/s00454-006-1276-5 · Cited by 1,280 · see citing works
Self-citationPERSISTENCE BARCODES FOR SHAPES
CARLSSON, G., ZOMORODIAN, A., COLLINS, A., & GUIBAS, L. J. (2005) · International Journal of Shape Modeling · doi:10.1142/s0218654305000761 · Cited by 287 · see citing works
Self-citationBronstein, A. M., Bronstein, M. M., Kimmel, R., Mahmoudi, M., & Sapiro, G. (2010) · International Journal of Computer Vision · doi:10.1007/s11263-009-0301-6 · Cited by 267 · see citing works
Stability of persistence diagrams
Cohen-Steiner, D., Edelsbrunner, H., & Harer, J. (2005) · Proceedings of the twenty-first annual symposium on Computational geometry · doi:10.1145/1064092.1064133 · Cited by 251 · see citing works
Self-citationDeformation transfer for triangle meshes
Sumner, R. W., & Popović, J. (2004) · ACM SIGGRAPH 2004 Papers · doi:10.1145/1186562.1015736 · Cited by 203 · see citing works
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Topological pattern recognition for point cloud data
Carlsson, G. (2014) · Acta Numerica · doi:10.1017/s0962492914000051 · Cited by 230 · see citing works
- Authors
- Chazal, F., Cohen‐Steiner, D., Guibas, L. J., Mémoli, F., & Oudot, S. Y.
- Affiliations
- Institut national de recherche en sciences et technologies du numérique (FR)
- Stanford University (US)
- Venue
- Computer Graphics Forum
- Publisher
- Wiley
- Published
- 2009-07
- Volume
- 28
- Issue
- 5
- Pages
- 1393-1403
- ISSN
- 0167-7055 (ISSN-L)
- 1467-8659
- OpenAlex
- W2108776879
- Cited by
- 225
- References
- 7
- Citations by year
- 2026: 5 · 2025: 10 · 2024: 16 · 2023: 19 · 2022: 15 · 2021: 18 · 2020: 16 · 2019: 19 · 2018: 23 · 2017: 14 · 2016: 12 · 2015: 7 · 2014: 18 · 2013: 12 · 2012: 9
- FWCI
- 6.83
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Computer Graphics Forum (publishedVersion)
- HAL (submittedVersion)
- Adelaide Research & Scholarship (AR&S) (University of Adelaide) (submittedVersion)
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Engineering › Computational Mechanics › 3D Shape Modeling and Analysis
- Keywords
- shape signatures, persistence diagrams, finite metric spaces, filtration, shape classification, point cloud data, metric measure spaces, topological data analysis
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