Homology, Homotopy and Applications

Homology and robustness of level and interlevel sets

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.

Article information

Source
Homology Homotopy Appl., Volume 15, Number 1 (2013), 51-72.

Dates
First available in Project Euclid: 8 November 2013

https://projecteuclid.org/euclid.hha/1383943667

Mathematical Reviews number (MathSciNet)
MR3031814

Zentralblatt MATH identifier
1266.55004

Subjects
Primary: 55 68

Citation

Bendich, Paul; Edelsbrunner, Herbert; Morozov, Dmitriy; Patel, Amit. Homology and robustness of level and interlevel sets. Homology Homotopy Appl. 15 (2013), no. 1, 51--72. https://projecteuclid.org/euclid.hha/1383943667