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

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.