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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2409.05532 |
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| _version_ | 1866912019061407744 |
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| author | Thanh, Son Le Ankele, Michael Weinkauf, Tino |
| author_facet | Thanh, Son Le Ankele, Michael Weinkauf, Tino |
| contents | The Morse-Smale complex is a standard tool in visual data analysis. The classic definition is based on a continuous view of the gradient of a scalar function where its zeros are the critical points. These points are connected via gradient curves and surfaces emanating from saddle points, known as separatrices. In a discrete setting, the Morse-Smale complex is commonly extracted by constructing a combinatorial gradient assuming the steepest descent direction. Previous works have shown that this method results in a geometric embedding of the separatrices that can be fundamentally different from those in the continuous case. To achieve a similar embedding, different approaches for constructing a combinatorial gradient were proposed. In this paper, we show that these approaches generate a different topology, i.e., the connectivity between critical points changes. Additionally, we demonstrate that the steepest descent method can compute topologically and geometrically accurate Morse-Smale complexes when applied to certain types of grids. Based on these observations, we suggest a method to attain both geometric and topological accuracy for the Morse-Smale complex of data sampled on a uniform grid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05532 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Revisiting Accurate Geometry for Morse-Smale Complexes Thanh, Son Le Ankele, Michael Weinkauf, Tino Computational Geometry The Morse-Smale complex is a standard tool in visual data analysis. The classic definition is based on a continuous view of the gradient of a scalar function where its zeros are the critical points. These points are connected via gradient curves and surfaces emanating from saddle points, known as separatrices. In a discrete setting, the Morse-Smale complex is commonly extracted by constructing a combinatorial gradient assuming the steepest descent direction. Previous works have shown that this method results in a geometric embedding of the separatrices that can be fundamentally different from those in the continuous case. To achieve a similar embedding, different approaches for constructing a combinatorial gradient were proposed. In this paper, we show that these approaches generate a different topology, i.e., the connectivity between critical points changes. Additionally, we demonstrate that the steepest descent method can compute topologically and geometrically accurate Morse-Smale complexes when applied to certain types of grids. Based on these observations, we suggest a method to attain both geometric and topological accuracy for the Morse-Smale complex of data sampled on a uniform grid. |
| title | Revisiting Accurate Geometry for Morse-Smale Complexes |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2409.05532 |