Homology and Robustness of Level and Interlevel Sets

Paul Bendich, Herbert Edelsbrunner, Dmitriy Morozov, Amit Patel.
Homology, Homotopy and Applications, vol. 15, pages 51-72, 2013.
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DOI: 10.4310/HHA.2013.v15.n1.a3
arXiv: 1102.3389
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, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case X = R3 has ramifications in the fields of medical imaging and scientific visualization.
Pyramid diamonds
Gunnar Carlsson, Vin de Silva, and Dmitriy Morozov. Zigzag persistent homology and real-valued functions. Proceedings of the Annual Symposium on Computational Geometry, pages 247–256, 2009.
Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. Quantifying transversality by measuring the robustness of intersections. Manuscript, Deptartment of Computer Science, Duke University, Durham, North Carolina, 2009.