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Given a function $f: \Xspace \to \Rspace$ on a topological space, we considerthe preimages of intervals and their homology groups and show how to read theranks of these groups from the extended persistence diagram of $f$. Inaddition, we quantify the robustness of the homology classes underperturbations of $f$ using well groups, and we show how to read the ranks ofthese groups from the same extended persistence diagram. The special case$\Xspace = \Rspace^3$ has ramifications in the fields of medical imaging andscientific visualization.

Homology and robustness of level and interlevel sets