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Homology and Robustness of Level and Interlevel Sets

By Paul Bendich, Herbert Edelsbrunner, Dmitriy Morozov and Amit Patel

Abstract

Given a function $f: \Xspace \to \Rspace$ on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of $f$. In addition, we quantify the robustness of the homology classes under perturbations of $f$ using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case $\Xspace = \Rspace^3$ has ramifications in the fields of medical imaging and scientific visualization

Topics: Computer Science - Computational Geometry, Mathematics - Algebraic Topology
Year: 2011
OAI identifier: oai:arXiv.org:1102.3389
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