Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Meduri, Dhruv Sharma, Mohit Natarajan, Vijay |
| author_facet | Meduri, Dhruv Sharma, Mohit Natarajan, Vijay |
| contents | The Jacobi set of a bivariate scalar field is the set of points where the gradients of the two constituent scalar fields align with each other. It captures the regions of topological changes in the bivariate field. The Jacobi set is a bivariate analog of critical points, and may correspond to features of interest. In the specific case of time-varying fields and when one of the scalar fields is time, the Jacobi set corresponds to temporal tracks of critical points, and serves as a feature-tracking graph. The Jacobi set of a bivariate field or a time-varying scalar field is complex, resulting in cluttered visualizations that are difficult to analyze. This paper addresses the problem of Jacobi set simplification. Specifically, we use the time-varying scalar field scenario to introduce a method that computes a reduced Jacobi set. The method is based on a stability measure called robustness that was originally developed for vector fields and helps capture the structural stability of critical points. We also present a mathematical analysis for the method, and describe an implementation for 2D time-varying scalar fields. Applications to both synthetic and real-world datasets demonstrate the effectiveness of the method for tracking features. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03348 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields Meduri, Dhruv Sharma, Mohit Natarajan, Vijay Numerical Analysis Computational Geometry Computer Vision and Pattern Recognition Graphics The Jacobi set of a bivariate scalar field is the set of points where the gradients of the two constituent scalar fields align with each other. It captures the regions of topological changes in the bivariate field. The Jacobi set is a bivariate analog of critical points, and may correspond to features of interest. In the specific case of time-varying fields and when one of the scalar fields is time, the Jacobi set corresponds to temporal tracks of critical points, and serves as a feature-tracking graph. The Jacobi set of a bivariate field or a time-varying scalar field is complex, resulting in cluttered visualizations that are difficult to analyze. This paper addresses the problem of Jacobi set simplification. Specifically, we use the time-varying scalar field scenario to introduce a method that computes a reduced Jacobi set. The method is based on a stability measure called robustness that was originally developed for vector fields and helps capture the structural stability of critical points. We also present a mathematical analysis for the method, and describe an implementation for 2D time-varying scalar fields. Applications to both synthetic and real-world datasets demonstrate the effectiveness of the method for tracking features. |
| title | Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields |
| topic | Numerical Analysis Computational Geometry Computer Vision and Pattern Recognition Graphics |
| url | https://arxiv.org/abs/2407.03348 |