On the Applicability of Topological Methods for Complex Flow Data

In this paper we study the applicability of topological methods for creating expressive, feature revealing visualizations of 3D vector fields. 3D vector fields can become very complex by having a high number of critical points and separatrices. Moreover, they may have a very sparse topology due to a small number of critical points or their total absence. We show that classical topological methods based on the extraction of separation surfaces are poorly suited for creating expressive visualizations of topologically complex fields. We show this fact by pointing out that the number of sectors of different flow behavior grows quadratically with the number of critical points - contrary to 2D vector fields. Although this limits the applicability of topological methods to a certain degree, we demonstrate the extensibility of this limit by using further simplifying methods like saddle connectors. For 3D vector fields with a very sparse topology, topological visualizations may fail to reveal the features inherent to the field. We show how to overcome this problem for a certain class of flow fields by removing the ambient part of the flow.