Homology and Robustness of Level and Interlevel Sets

TL;DR

This paper links the homology of level and interlevel sets of a function to extended persistence diagrams and introduces a way to measure the robustness of these homology classes under perturbations, with applications in medical imaging.

Contribution

It establishes a method to read homology ranks and robustness measures of level sets directly from extended persistence diagrams, advancing topological data analysis techniques.

Findings

Homology ranks of level sets can be derived from extended persistence diagrams.

Robustness of homology classes under perturbations can be quantified using well groups.

Applications demonstrated in medical imaging and visualization for $ ext{R}^3$.

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.