max planck institut
informatik
informatik
Investigators: Holger Theisel and Kuangyu Shi
The research done in this group mainly focused on the application of topological methods for vector field visualization. Nowadays topological methods are standard tools to visualizing vector fields because they offer to represent 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 vector 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. Here we present results of the research about topological methods of 3D vector fields (section 0.1.1) and of 2D time-dependent vector fields (section 0.1.2). Section 0.1.3 presents further applications of topological methods.
Topological methods for 3D vector fields are still less commonly applied than for 2D vector fields. The main reason for this is the fact that the visualization tends to be cluttered even for rather simple flows: separation surfaces tend to hide each other as well as other topological features. One solution for this is the consideration of saddle connectors [TWHS03]: instead of visualizing the separation surfaces themselves, only their intersection curves are extracted and visualized. These intersection curves build a network of stream lines which gives a good overview of the topological behavior of the vector field. Figure 0.1 illustrates this for a data set describing the electrostatic field of a benzene molecule.
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[WTHS04a] extends the concept of saddle connectors to boundary switch connectors by additionally considering the separation surfaces starting from boundary switch curves. Also for them, their intersection curves (i.e., a number of stream lines) are extracted and visualized instead of the surfaces themselves. Figure 0.2 illustrates this at a data set describing the flow behind a circular cylinder.
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For 2D 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 2D 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, [TWHS04b] presents methods to extracting global bifurcations like saddle connections and cyclic fold bifurcations. In addition, [TWHS05] and [TWHS04a] propose a method to detect and track closed stream lines which does not depend on an underlying grid structure and which does not have to explicitly solve the correspondence problem between adjacent time steps. Figure 0.3 illustrates an example.
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To obtain a path line oriented topology, [TWHS04b] proposes a local analysis of the convering/diverging behavior of the path lines in every point of the flow. This way, the flow is segmented into regions of attracting, repelling, and saddle-like behavior of the path lines.
Topological methods for vector fields can be used for more than just visualization. [WTHS04b] presents a topology based construction approach for 3D vector fields. The idea is to construct the 3D topological skeleton consisting of general critical points and saddle connectors: general (higher order) critical points are constructed by specifying areas of different inflow/outflow behavior as closed polygons on a sphere, while the saddle connectors are constructed as cubic B-spline curves. Then a piecewise linear vector field of exactly the specified topology is automatically created. To do so, an appropriate tetrahedrization of the domain is introduced around the critical points and the saddle connectors. Then the remaining space is filled by a constrained Delaunay tetrahedrization. Figure 0.4 shows an example of this process.
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As part of this approach, [WTHS04b] also introduces an iconic visualization of 3D higher order critical points.
In addition to the construction approach presented above, we also used topological methods to compress [TRS03b] and simplify [TRS03a] vector fields, and to define a topology-based distance metric for vector fields [TRS03c].
