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Vietoris–Rips Filtration
In topological data analysis, the Vietoris–Rips filtration (sometimes shortened to "Rips filtration") is the collection of nested Vietoris–Rips complexes on a metric space created by taking the sequence of Vietoris–Rips complexes over an increasing scale parameter. Often, the Vietoris–Rips filtration is used to create a discrete, simplicial model on point cloud data embedded in an ambient metric space. The Vietoris–Rips filtration is a multiscale extension of the Vietoris–Rips complex that enables researchers to detect and track the persistence of topological features, over a range of parameters, by way of computing the persistent homology of the entire filtration. It is named after Leopold Vietoris and Eliyahu Rips. Definition The Vietoris–Rips filtration is the nested collection of Vietoris–Rips complexes indexed by an increasing scale parameter. The Vietoris–Rips complex is a classical construction in mathematics that dates back to a 1927 paper of Leopold ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Example Of A Vietoris-Rips Filtration On Point Cloud Data
Example may refer to: * ''exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, and example.edu: second-level domain names reserved for use in documentation as examples * HMS ''Example'' (P165), an Archer-class patrol and training vessel of the Royal Navy Arts * ''The Example'', a 1634 play by James Shirley * ''The Example'' (comics), a 2009 graphic novel by Tom Taylor and Colin Wilson * Example (musician), the British dance musician Elliot John Gleave (born 1982) * ''Example'' (album), a 1995 album by American rock band For Squirrels See also * Exemplar (other), a prototype or model which others can use to understand a topic better * Exemplum, medieval collections of short stories to be told in sermons * Eixample The Eixample (, ) is a district of Barcelona between the old city (Ciutat Vella) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Map
A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of a simplex always span a simplex. Simplicial maps can be used to approximate continuous functions between topological spaces that can be triangulated; this is formalized by the simplicial approximation theorem. A simplicial isomorphism is a bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ... simplicial map such that both it and its inverse are simplicial. Definitions A simplicial map is defined in slightly different ways in different contexts. Abstract simplicial complexes Let K and L be two Abstract simplicial complex, abstract simplicial complexes (ASC). A simplicial map of K into L is a function from the vertices of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
