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Computational Topology: An Introduction
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TLDR
In this article, the authors present an introduction to the field of computational topology, combining concepts from topology and algorithms, and the main approach is the discovery of topology through algorithms.Abstract:
Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.read more
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Journal ArticleDOI
A roadmap for the computation of persistent homology
Nina Otter,Nina Otter,Mason A. Porter,Mason A. Porter,Ulrike Tillmann,Ulrike Tillmann,Peter Grindrod,Heather A. Harrington +7 more
TL;DR: A friendly introduction to PH is given, the pipeline for the computation of PH is navigated with an eye towards applications, and a range of synthetic and real-world data sets are used to evaluate currently available open-source implementations for the computations of PH.
Journal ArticleDOI
Homological scaffolds of brain functional networks
Giovanni Petri,Paul Expert,Federico Turkheimer,Robin L. Carhart-Harris,David J. Nutt,Peter J. Hellyer,Francesco Vaccarino,Francesco Vaccarino +7 more
TL;DR: The results show that the homological structure of the brain's functional patterns undergoes a dramatic change post-psilocybin, characterized by the appearance of many transient structures of low stability and of a small number of persistent ones that are not observed in the case of placebo.
Journal ArticleDOI
Geometric frustration of icosahedron in metallic glasses
Akihiko Hirata,L. J. Kang,Takeshi Fujita,B. Klumov,Kaname Matsue,Motoko Kotani,A.R. Yavari,A.R. Yavari,Mingwei Chen,Mingwei Chen +9 more
TL;DR: Experimental observation of local icosahedral order in metallic glasses by means of angstrom-beam electron diffraction of single icosahedra found to be distorted with partially face-centered cubic symmetry, presenting compelling evidence on geometric frustration of local Icosahedron order in Metallic glasses.
Proceedings ArticleDOI
A stable multi-scale kernel for topological machine learning
TL;DR: In this paper, a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data, is proposed for 3D shape classification/retrieval and texture recognition.
Journal Article
Persistence images: a stable vector representation of persistent homology
Henry Adams,Tegan Emerson,Michael Kirby,Rachel Neville,Chris Peterson,Patrick D. Shipman,Sofya Chepushtanova,Eric Hanson,Francis C. Motta,Lori Ziegelmeier +9 more
TL;DR: In this article, a persistence diagram (PD) is converted to a finite-dimensional vector representation which is called a persistence image (PI) and proved the stability of this transformation with respect to small perturbations in the inputs.