Topological Persistence and Simplification

By Herbert Edelsbrunner, David Letscher, Afra Zomorodian. Proceedings 41st Annual Symposium on Foundations of Computer Science, 2000

2,775 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/cb672955f60bc279474efeed238a4756f0059a4b

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Citations in the corpus

Large‐eddy simulations over Germany using ICON: a comprehensive evaluation (2017, methodology): “The extraction corresponds to a (maximum) watershed algorithm with a persistence-based hierarchical merging step (Edelsbrunner et al., 2002; Kuhn et al., 2015).”
Persistence stability for geometric complexes (2012, background): “One way round this issue is to use topological persistence [11,18], which encodes the homology of the entire nested family Rips(X) = (Rips(X, α))α∈R in a single invariant, the persistence diagram.”
Stability of Persistence Diagrams (2005, methodology): “Here is the connection: the persistence diagram of this function is the set of points whose coordinates are the pairs of indices computed by the persistence algorithm [11], together with points at infinity that correspond to indices left unpaired by the algorithm, together with the diagonal.”
A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology (2019, methodology): “Below we give an overview of PH as it applies to our method, but for more thorough reviews we direct the reader to [11], [24], [25].”
Homological illusions of persistence and stability (2008, methodology): “This thesis comes out of the study of the recently introduced [39] theory of persistent homology; see [37] for a survey.”
Lipschitz Functions Have Lp-Stable Persistence (2010, methodology): “This agrees with the original definition except for classes that do not die for which [7] set the persistence to infinity.”
Barcodes: The persistent topology of data (2007, methodology): “Persistence, as introduced by Edelsbrunner, Letscher, and Zomo-rodian [12] and reﬁned by Carlsson and Zomorodian [22], is a rigorous response to this problem.”
Computing Persistent Homology (2004, background): “A polynomial f ( t ) with coefﬁcients in R is a formal sum P ∞ i =0 a i t i , where a i ∈ R and t is the indeterminate.”
Theory of interleavings on categories with a flow (2017, neutral): “Some commonly used topological signatures include persistence diagrams [13], persistence modules [5, 24], Reeb graphs [9, 21], and Mapper [22].”
Tracing the cosmic web (2017, methodology): “The formalism uses three concepts in computational topology: Persistent Homology and Topological Simplification (Edelsbrunner et al. 2002; Gyulassy 2008; Edelsbrunner & Harer 2010).”
Multi‐scale Feature Extraction on Point‐Sampled Surfaces (2003, methodology): “To improve the robustness and reliability of the classification stage, we apply this operator at multiple scales, which allows us to measure the persistence of a feature [4].”
Morse Theory for Filtrations and Efficient Computation of Persistent Homology (2013, background): “Three fundamental properties account for the importance of persistent homology: (i) being based on algebraic topology, it provides a well understood codification of potentially complicated and/or high dimensional geometric information, (ii) the information it provides is stable with respect to pertu”
The persistent cosmic web and its filamentary structure I: Theory and implementation (2010, background): “The theory was originally described in the context of simplicial homology (for functions defined over a simplicial complex, see Appendix B) and was very nicely exposed in Edelsbrunner et al. (2002).”
Toward unsupervised, multi-object discovery in large-scale image collections (2020, methodology): “Next, we find robust local maxima in the previous saliency map using persistence, a measure used in topological data analysis [5,8,9,25,41] to find critical points of a function (see Section 4.”
Vines and vineyards by updating persistence in linear time (2006, methodology): “Recall that the algorithm in [10] can be interpreted as computing the reduced matrix R = DV , where V is invertible and upper-triangular.”
The observable structure of persistence modules (2014, background): “Topological persistence [7, 15] may be introduced with the observation that a nested sequence of topological spaces X0 ⊆ X1 ⊆ · · · ⊆ Xn”