Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories
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 extracte...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Taylor & Francis Group
2020-01-01
|
| Series: | Cogent Engineering |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1080/23311916.2020.1808340 |
| _version_ | 1797713172153499648 |
|---|---|
| author | Lyailya Karimova Alexey Terekhov Nikolai Makarenko Andrey Rybintsev |
| author_facet | Lyailya Karimova Alexey Terekhov Nikolai Makarenko Andrey Rybintsev |
| author_sort | Lyailya Karimova |
| collection | DOAJ |
| description | 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 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 demonstrate this technique to describe the morphological dynamics of wetlands located in arid zones and characterized by extremely high temporal variability. |
| first_indexed | 2024-03-12T07:32:13Z |
| format | Article |
| id | doaj.art-c5b61fe85b6a4fd68ce8a1acb5848457 |
| institution | Directory Open Access Journal |
| issn | 2331-1916 |
| language | English |
| last_indexed | 2024-03-12T07:32:13Z |
| publishDate | 2020-01-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | Cogent Engineering |
| spelling | doaj.art-c5b61fe85b6a4fd68ce8a1acb58484572023-09-02T21:43:57ZengTaylor & Francis GroupCogent Engineering2331-19162020-01-017110.1080/23311916.2020.18083401808340Methods of computational topology and discrete Riemannian geometry for the analysis of arid territoriesLyailya Karimova0Alexey Terekhov1Nikolai Makarenko2Andrey Rybintsev3Institute of Information and Computational TechnologiesInstitute of Information and Computational TechnologiesInstitute of Information and Computational TechnologiesThe Central Astronomical Observatory of the Russian Academy of Sciences at PulkovoThe 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 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 demonstrate this technique to describe the morphological dynamics of wetlands located in arid zones and characterized by extremely high temporal variability.http://dx.doi.org/10.1080/23311916.2020.1808340tdabetti numberspersistencefisher-rao information metrictangent bundle оf riemannian sphereremote sensinglong-term dynamic |
| spellingShingle | Lyailya Karimova Alexey Terekhov Nikolai Makarenko Andrey Rybintsev Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories Cogent Engineering tda betti numbers persistence fisher-rao information metric tangent bundle оf riemannian sphere remote sensing long-term dynamic |
| title | Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories |
| title_full | Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories |
| title_fullStr | Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories |
| title_full_unstemmed | Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories |
| title_short | Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories |
| title_sort | methods of computational topology and discrete riemannian geometry for the analysis of arid territories |
| topic | tda betti numbers persistence fisher-rao information metric tangent bundle оf riemannian sphere remote sensing long-term dynamic |
| url | http://dx.doi.org/10.1080/23311916.2020.1808340 |
| work_keys_str_mv | AT lyailyakarimova methodsofcomputationaltopologyanddiscreteriemanniangeometryfortheanalysisofaridterritories AT alexeyterekhov methodsofcomputationaltopologyanddiscreteriemanniangeometryfortheanalysisofaridterritories AT nikolaimakarenko methodsofcomputationaltopologyanddiscreteriemanniangeometryfortheanalysisofaridterritories AT andreyrybintsev methodsofcomputationaltopologyanddiscreteriemanniangeometryfortheanalysisofaridterritories |