Homology and Robustness of Level and Interlevel Sets

Given a function f : X → 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. After characterizing these groups, we show how to read their ranks from the same extended persistence diagram. The special case X = R has ramifications in the fields of medical imaging and scientific visualization.