Homology and Robustness of Level and Interlevel Sets
  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. 
Paul Bendich and Herbert Edelsbrunner and Dmitriy Morozov and Amit   Patel
cs.CG math.AT