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# Homology, Homotopy and Applications

## Volume 15 (2013)

### Number 1

### Homology and robustness of level and interlevel sets

Pages: 51 – 72

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

#### Authors

#### 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