The Basic Theory of Persistent Homology
نویسنده
چکیده
Persistent homology has widespread applications in computer vision and image analysis. This paper first motivates the use of persistent homology as a suitable tool to solve the problem of extracting global topological information from a discrete sample of points. The remainder of this paper develops the mathematical theory behind persistent homology. Persistent homology will be developed as an extension to simplicial homology. We then discuss an algebraic interpretation, as well as graphical representations, of persistence.
منابع مشابه
Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology
A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally persistence, is a central tool in topological data analysis, which examines the structure of data through topology. The basic techniques have been extended in several different directions, encoding topological features by so called b...
متن کاملOptimising the topological information of the $A_\infty$-persistence groups
Persistent homology typically studies the evolution of homology groups Hp(X) (with coefficients in a field) along a filtration of topological spaces. A∞-persistence extends this theory by analysing the evolution of subspaces such as V := Ker ∆n|Hp(X) ⊆ Hp(X), where {∆m}m≥1 denotes a structure of A∞-coalgebra on H∗(X). In this paper we illustrate how A∞-persistence can be useful beyond persisten...
متن کاملMultiresolution Topological Simplification
Persistent homology has been advocated as a new strategy for the topological simplification of complex data. However, it is computationally intractable for large data sets. In this work, we introduce multiresolution persistent homology for tackling large datasets. Our basic idea is to match the resolution with the scale of interest so as to create a topological microscopy for the underlying dat...
متن کاملGeneralized Local Homology Modules of Complexes
The theory of local homology modules was initiated by Matlis in 1974. It is a dual version of the theory of local cohomology modules. Mohammadi and Divaani-Aazar (2012) studied the connection between local homology and Gorenstein flat modules by using Gorenstein flat resolutions. In this paper, we introduce generalized local homology modules for complexes and we give several ways for computing ...
متن کاملPersistent Intersection Homology
The theory of intersection homology was developed to study the singularities of a topologically stratified space. This paper incorporates this theory into the already developed framework of persistent homology. We demonstrate that persistent intersection homology gives useful information about the relationship between an embedded stratified space and its singularities. We give, and prove the co...
متن کامل