Localized Evaluation for Constructing Discrete Vector Fields
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| Titel: | Localized Evaluation for Constructing Discrete Vector Fields |
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| Autoren: | Tanner Finken, Julien Tierny, Joshua A. Levine |
| Weitere Verfasser: | Tierny, Julien |
| Quelle: | IEEE Transactions on Visualization and Computer Graphics |
| Publication Status: | Preprint |
| Verlagsinformationen: | Institute of Electrical and Electronics Engineers (IEEE), 2025. |
| Publikationsjahr: | 2025 |
| Schlagwörter: | topological data analysis, Computational Geometry (cs.CG), FOS: Computer and information sciences, Computer Science - Graphics, Computer Science - Computational Geometry, [INFO] Computer Science [cs], discrete Morse theory, Flow visualization discrete Morse theory topological data analysis, Flow visualization, Graphics (cs.GR) |
| Beschreibung: | Topological abstractions offer a method to summarize the behavior of vector fields but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. We couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows. 11 pages, Accepted at IEEE Vis Conference 2024 |
| Publikationsart: | Article |
| Dateibeschreibung: | application/pdf |
| ISSN: | 2160-9306 1077-2626 |
| DOI: | 10.1109/tvcg.2024.3456355 |
| DOI: | 10.48550/arxiv.2408.04769 |
| Zugangs-URL: | https://pubmed.ncbi.nlm.nih.gov/39250396 http://arxiv.org/abs/2408.04769 https://hal.science/hal-04674219v1 |
| Rights: | IEEE Copyright arXiv Non-Exclusive Distribution |
| Dokumentencode: | edsair.doi.dedup.....005f6df861a765b6de33a9f6b71faadf |
| Datenbank: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | Topological abstractions offer a method to summarize the behavior of vector fields but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. We couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.<br />11 pages, Accepted at IEEE Vis Conference 2024 |
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| ISSN: | 21609306 10772626 |
| DOI: | 10.1109/tvcg.2024.3456355 |