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Alexander duality for functions: the persistent behavior of land and water and shore

Authors:

Herbert Edelsbrunner,

Michael KerberAuthors Info & Claims

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

Pages 249 - 258

https://doi.org/10.1145/2261250.2261287

Published: 17 June 2012 Publication History

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Abstract

This note contributes to the point calculus of persistent homology by extending Alexander duality from spaces to real-valued functions. Given a perfect Morse function f: Sspaceⁿ⁺¹ -> [0,1] and a decomposition Sspaceⁿ⁺¹ = Uspace ∪ Vspace into two (n+1)-manifolds with common boundary Mspace, we prove elementary relationships between the persistence diagrams of f restricted to Uspace, to Vspace, and to Mspace.

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Turner KRobins VMorgan J(2024)The extended persistent homology transform of manifolds with boundaryJournal of Applied and Computational Topology10.1007/s41468-024-00175-88:7(2111-2154)Online publication date: 6-May-2024
https://doi.org/10.1007/s41468-024-00175-8
Patel ARask T(2024)Poincaré duality for generalized persistence diagrams of (co)filtrationsJournal of Applied and Computational Topology10.1007/s41468-023-00159-08:2(427-442)Online publication date: 19-Feb-2024
https://doi.org/10.1007/s41468-023-00159-0
Garside KGjoka AHenderson RJohnson HMakarenko I(2020)Event history and topological data analysisBiometrika10.1093/biomet/asaa097Online publication date: 16-Nov-2020
https://doi.org/10.1093/biomet/asaa097
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Index Terms

Alexander duality for functions: the persistent behavior of land and water and shore
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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Published In

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

June 2012

436 pages

ISBN:9781450312998

DOI:10.1145/2261250

Program Chairs:
Tamal Dey
The Ohio State University, USA
,
Sue Whitesides
University of Victoria, Canada

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Association for Computing Machinery

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Published: 17 June 2012

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SoCG '12: Symposium on Computational Geometry 2012

June 17 - 20, 2012

North Carolina, Chapel Hill, USA

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Turner KRobins VMorgan J(2024)The extended persistent homology transform of manifolds with boundaryJournal of Applied and Computational Topology10.1007/s41468-024-00175-88:7(2111-2154)Online publication date: 6-May-2024
https://doi.org/10.1007/s41468-024-00175-8
Patel ARask T(2024)Poincaré duality for generalized persistence diagrams of (co)filtrationsJournal of Applied and Computational Topology10.1007/s41468-023-00159-08:2(427-442)Online publication date: 19-Feb-2024
https://doi.org/10.1007/s41468-023-00159-0
Garside KGjoka AHenderson RJohnson HMakarenko I(2020)Event history and topological data analysisBiometrika10.1093/biomet/asaa097Online publication date: 16-Nov-2020
https://doi.org/10.1093/biomet/asaa097
Attali DBauer UDevillers OGlisse MLieutier Ada Fonseca GLewiner TPeñaranda LChan TKlein R(2013)Homological reconstruction and simplification in R3Proceedings of the twenty-ninth annual symposium on Computational geometry10.1145/2462356.2462373(117-126)Online publication date: 17-Jun-2013
https://dl.acm.org/doi/10.1145/2462356.2462373
Edelsbrunner HSymonova O(2012)The Adaptive Topology of a Digital ImageProceedings of the 2012 Ninth International Symposium on Voronoi Diagrams in Science and Engineering10.1109/ISVD.2012.11(41-48)Online publication date: 27-Jun-2012
https://dl.acm.org/doi/10.1109/ISVD.2012.11

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Note: Combinatorial Alexander Duality—A Short and Elementary Proof

A Note on the Homotopy Type of the Alexander Dual

Measuring and computing natural generators for homology groups

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