Persistent Homology for Kernels, Images, and Cokernels

David Cohen-Steiner, Herbert Edelsbrunner, John Harer, Dmitriy Morozov.
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1011-1020, 2009.
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Motivated by the measurement of local homology and of functions on noisy domains, we extend the notion of persistent homology to sequences of kernels, images, and cokernels of maps induced by inclusions in a filtration of pairs of spaces. Specifically, we note that persistence in this context is well defined, we prove that the persistence diagrams are stable, and we explain how to compute them.
Local homology vineyards
Paul Bendich, David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Dmitriy Morozov. Inferring Local Homology from Sampled Stratified Spaces. Proceedings of the Symposium on Foundations of Computer Science, pages 536-546, Providence, Rhode Island, October 2007.
David Cohen-Steiner, Herbert Edelsbrunner and Dmitriy Morozov. Vines and vineyards by updating persistence in linear time. Proceedings of the 22nd Annual ACM Symposium on Computational Geometry, pages 119-126, New York, NY, USA, 2006.