Persistent Homology: Theory and Practice
نویسندگان
چکیده
Persistent homology is a recent grandchild of homology that has found use in science and engineering as well as in mathematics. This paper surveys the method as well as the applications, neglecting completeness in favor of highlighting ideas and directions. 2010 Mathematics Subject Classification. Primary 55N99; Secondary 68W30.
منابع مشابه
The Basic Theory of Persistent Homology
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 ...
متن کاملStabilizing the output of persistent homology computations
We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is generally seen as a procedure which starts with a filtered complex and ends with a persistence diagram. This procedure is stable (at least to certain types of perturbations of the input). This justifies the use of the...
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 کاملComputing homology and persistent homology using iterated Morse decomposition
In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph theoretical operations. We use iterated Morse decomposition, which allows us to sidetrack many problems related to the standard discrete Morse theory. In particu...
متن کاملMorse Theory for Filtrations and Efficient Computation of Persistent Homology
We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012