Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories

Lyailya Karimova, Alexey Terekhov, Nikolai Makarenko, Andrey Rybintsev, Duc Pham
2020 Cogent Engineering  
The purpose of this article is the development and application of discrete differential geometry methods for digital image analysis within the framework of Topological Data Analysis (TDA). The proposed approach consists of two stages. First of all, topological invariants, Betti numbers, are extracted from the digital image using TDA algorithms. They contain information about the appearance and disappearance of topological properties: the connected components and holes when filtering the image
more » ... ltering the image along with the height of the photometric topography. The interval of heights measuring the lifetime of a property is called the persistence of the property. The most common information about Betti's persistent numbers is presented in the form of a cloud of points on the birth-death diagram, the so-called persistence diagram (PD). The vectorization of PD with the help of a diffuse kernel makes it possible to estimate its pdf. At the second stage, we use the representation of the received pdf on the Riemannian sphere. Here, the Fischer-Rao metric reduces to the Hilbert scalar product of semi-density on the tangent bundle of a sphere. This approach allows you to analyze images of complex, multicomponent natural systems that do not have clear spectral boundaries of the transition between texture classes. Space images of natural landscapes were used as digital images. We ABOUT THE AUTHOR A member of the Board of the executive committee of Russian Neural Networks Society. Research interests: computer modeling of nonlinear systems, computational topology, fractal geometry, deterministic chaos, mathematical morphology, Artificial Neural Networks, analysis of remote sensing data. He has over 100 publications in various sources. Additional information is available at the end of the article © 2020 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. demonstrate this technique to describe the morphological dynamics of wetlands located in arid zones and characterized by extremely high temporal variability. Subjects: Computer Theory; Mathematical Modeling; Theory of Numbers Keywords: TDA; Betti numbers; persistence; fisher-rao information metric; tangent bundle оf Riemannian sphere; remote sensing; long-term dynamic Karimova et al., Cogent Engineering (2020), 7: 1808340 Citation information Cite this article as: Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories, Lyailya Karimova, Alexey Terekhov, Nikolai Makarenko & Andrey Rybintsev, Cogent Engineering (2020), 7: 1808340.
doi:10.1080/23311916.2020.1808340 fatcat:tacgr3sz6ray7ax7mxzl55urwu