@inproceedings{3559,
  abstract     = {Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its per- tinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in [10] computes the pairs from an ordering of the simplices in a triangulation and takes worst-case time cubic in the number of simplices. The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering. A side-effect of the algorithm’s anal- ysis is an elementary proof of the stability of persistence diagrams [7] in the special case of piecewise-linear functions. We use the algorithm to compute 1-parameter families of diagrams which we apply to the study of protein folding trajectories.},
  author       = {Cohen Steiner, David and Edelsbrunner, Herbert and Morozov, Dmitriy},
  booktitle    = {Proceedings of the twenty-second annual symposium on Computational geometry},
  isbn         = {9781595933409},
  location     = {Sedona, AZ, United States},
  pages        = {119 -- 126},
  publisher    = {ACM},
  title        = {{Vines and vineyards by updating persistence in linear time}},
  doi          = {10.1145/1137856.1137877},
  year         = {2006},
}

@inproceedings{3560,
  abstract     = {We continue the study of topological persistence [5] by investigat- ing the problem of simplifying a function f in a way that removes topological noise as determined by its persistence diagram [2]. To state our results, we call a function g an ε-simplification of another function f if ∥f − g∥∞ ≤ ε, and the persistence diagrams of g are the same as those of f except all points within L1-distance at most ε from the diagonal have been removed. We prove that for func- tions f on a 2-manifold such ε-simplification exists, and we give an algorithm to construct them in the piecewise linear case.},
  author       = {Edelsbrunner, Herbert and Morozov, Dmitriy and Pascucci, Valerio},
  booktitle    = {Proceedings of the twenty-second annual symposium on Computational geometry},
  isbn         = {9781595933409},
  location     = {Sedona, AZ, United States},
  pages        = {127 -- 134},
  publisher    = {ACM},
  title        = {{Persistence-sensitive simplification of functions on 2-manifolds}},
  doi          = {10.1145/1137856.1137878},
  year         = {2006},
}