Homology, Homotopy and Applications

Volume 15 (2013)

Number 1

Homology and robustness of level and interlevel sets

Pages: 51 – 72

DOI: https://dx.doi.org/10.4310/HHA.2013.v15.n1.a3

Authors

Paul Bendich (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.; IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria)

Herbert Edelsbrunner (Departments of Computer Science and Mathematics, Duke University, Durham, North Carolina, U.S.A.; IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria)

Dmitriy Morozov (Departments of Computer Science and of Mathematics, Stanford University, Stanford, Calif., U.S.A.)

Amit Patel (Department of Computer Science, Duke University, Durham, North Carolina, U.S.A.; IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria)

Abstract

Given a continuous function $f\colon \mathbb{X} \to \mathbb{R}$ 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 $\mathbb{X} = \mathbb{R}^3$ has ramifications in the fields of medical imaging and scientific visualization.

Keywords

zigzag, levelset, homology, perturbation, well group, robustness

2010 Mathematics Subject Classification

55-xx, 68-xx

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Published 1 May 2013