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From numerics to combinatorics: a survey of topological methods for vector field visualization

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Abstract

Topological methods are important tools for data analysis, and recently receiving more and more attention in vector field visualization. In this paper, we give an introductory description to some important topological methods in vector field visualization. Besides traditional methods of vector field topology, space-time method and finite-time Lyapunov exponent, we also include in this survey Hodge decomposition, combinatorial vector field topology, Morse decomposition, and robustness, etc. In addition to familiar numerical techniques, more and more combinatorial tools emerge in vector field visualization. The numerical methods often rely on error-prone interpolations and interpolations, while combinatorial techniques produce robust but coarse features. In this survey, we clarify the relevant concepts and hope to guide future topological research in vector field visualization.

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Notes

  1. Actually \(H_i\simeq T_i\times \underbrace{\mathbb {Z}\times \cdots \times \mathbb {Z}}_{\beta _i}\), where \(T_i\) is the torsion subgroup and usually assumed trivial.

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Acknowledgments

This work is supported by Chinese 973 Program (2015CB755604), the National Science Foundation of China (61202335, 61170157). We would like to thank Dr. He Ou-Yang for his guidance and John Lucynski for his generous advices.

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  1. College of Computer, National University of Defense Technology, Changsha, 410073, China

    Wentao Wang & Sikun Li

  2. State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, 410073, China

    Wentao Wang & Sikun Li

  3. Institute of Ocean Science and Engineering, National University of Defense Technology, Changsha, 410073, China

    Wenke Wang

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Wang, W., Wang, W. & Li, S. From numerics to combinatorics: a survey of topological methods for vector field visualization. J Vis 19, 727–752 (2016). https://doi.org/10.1007/s12650-016-0348-8

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