||During the last decade, Scientific Visualization has grown into an active area of research focusing on a variety of different applications. Among the data classes considered in visualization,flow data play an outstanding role. Flow data, obtained both from simulation and measurement processes, usually comes as 2D or 3Dvector fields. Currently, a multitude of different visualization techniques for flow data is available. Among them, topological methods have become a standard tool, because they promise to visualize even complex flow structures by only a limited number of graphical primitives. After their introduction as visualization tools by Helman/Hesslink, a considerable amount of research has been done in the field. The main idea of topological methods is to segment the flow field into regions of different flow behavior.
This is done by extracting critical points and separatrices starting from the saddle points. These separatrices are certain stream lines for 2D vector fields and stream surfaces in the 3D case. Although topological methods are well-established for flow visualization, there is still a number of challenges and open problems to be solved. The research of our group focuses on two main directions: the treatment of the topology of time-dependent flow fields, and the application of topological methods for further problems, features and data classes.
For time-dependent vector fields, two kinds of characteristic curves exist: stream lines and path lines. Since topological methods aim in the segmentation into areas of different flow behavior, two kinds of topologies can be distinguished for time-dependent vector fields: a stream line oriented topology, and a path line oriented topology. For a stream line oriented topology, topological feature of steady vector fields have to be tracked over time. Doing so, certain bifurcations may occur and have to be extracted. While local bifurcations (like Hopf bifurcations and fold bifurcations) are well-known for visualization purposes, our group is working on methods to extracting global bifurcations like saddle connections, closed stream lines and cyclic fold bifurcations. Although most topological methods focus on a stream line oriented topology, there is a demand for segmenting and understanding the behavior of path lines. The group is working on path line oriented approaches for 2D and 3D time-dependent vector fields.
Topological methods can be used not only for visualization purposes but also in two other ways. First, they can also be used to construct, compress, compare and simplify vector fields. Among them, our group works on topology based simplification techniques for 3D vector fields. Second, topological concepts can be applied to other data classes (like volume data or tensor data) and other features of vector fields. In particular, we are working on applying topological methods to extract, segment and classify vortex core lines. Also, topological methods can be applied to vector fields which do not come from a flow simulation environment. In particular, we are working on extracting characteristic 2D and 3D vector fields from surfaces and apply a topological segmentation of them for shape and surface analysis.