Formation of features based on computational topology methods

Abstract

The use of traditional methods of algebraic topology to obtain information about the shape of an object is associated with the problem of forming a small amount of information, namely, Betti numbers and Euler characteristics. The central tool for topological data analysis is the persistent homology method, which summarizes the geometric and topological information in the data using persistent diagrams and barcodes. Based on persistent homology methods, topological data can be analyzed to obtain information about the shape of an object. The construction of persistent barcodes and persistent diagrams in computational topology does not allow one to construct a Hilbert space with a scalar product. The possibility of applying the methods of topological data analysis is based on mapping persistent diagrams into a Hilbert space; one of the ways of such mapping is a method of constructing a persistence landscape. It has an advantage of being reversible, so it does not lose any information and has persistence properties. The paper considers mathematical models and functions for representing persistence landscape objects based on the persistent homology method. Methods for converting persistent barcodes and persistent diagrams into persistence landscape functions are considered. Associated with persistence landscape functions is a persistence landscape kernel that forms a mapping into a Hilbert space with a dot product. A formula is proposed for determining a distance between the persistence landscapes, which allows the distance between images of objects to be found. The persistence landscape functions map persistent diagrams into a Hilbert space. Examples of determining the distance between images based on the construction of persistence landscape functions for these images are given. Representations of topological characteristics in various models of computational topology are considered. Results for one-parameter persistence modules are extended onto multi-parameter persistence modules.