Saddle connectors - an approach to visualizing the topological skeleton of complex 3D vector fields

One of the reasons that topological methods have a limited popularity for the visualization of complex 3D flow fields is the fact that such topological structures contain a number of separating stream surfaces. Since these stream surfaces tend to hide each other as well as other topological features, for complex 3D topologies the visualizations become cluttered and hardly interpretable. This paper proposes to use particular stream lines called saddle connectors instead of separating stream surfaces and to depict single surfaces only on user demand. We discuss properties and computational issues of saddle connectors and apply these methods to complex flow data. We show that the use of saddle connectors makes topological skeletons available as a valuable visualization tool even for topologically complex 3D flow data.

[1]  D. Asimov.  Notes on the Topology of Vector Fields and Flows , 2003 .

[2]  Helwig Löffelmann,et al.  Visualizing Dynamical Systems near Critical Points , 1998 .

[3]  Gerik Scheuermann,et al.  Visualizing Nonlinear Vector Field Topology , 1998, IEEE Trans. Vis. Comput. Graph..

[4]  Suresh K. Lodha,et al.  Topology preserving compression of 2D vector fields , 2000, Proceedings Visualization 2000. VIS 2000 (Cat. No.00CH37145).

[5]  Hans Hagen,et al.  A topology simplification method for 2D vector fields , 2000 .

[6]  T. Steinke,et al.  Visualization of Vector Fields in Quantum Chemistry , 1996 .

[7]  Lambertus Hesselink,et al.  Feature comparisons of 3-D vector fields using earth mover's distance , 1999, Proceedings Visualization '99 (Cat. No.99CB37067).

[8]  Robert van Liere,et al.  Visualization of Global Flow Structures Using Multiple Levels of Topology , 1999, VisSym.

[9]  Herbert Edelsbrunner,et al.  Hierarchical morse complexes for piecewise linear 2-manifolds , 2001, SCG '01.

[10]  M. S. Chong,et al.  A general classification of three-dimensional flow fields , 1990 .

[11]  Robin N. Strickland,et al.  Vector Field Analysis and Synthesis Using Three-Dimensional Phase Portraits , 1997, CVGIP Graph. Model. Image Process..

[12]  Valerio Pascucci,et al.  Visualization of scalar topology for structural enhancement , 1998, Proceedings Visualization '98 (Cat. No.98CB36276).

[13]  W. D. Leeuw,et al.  Visualization of Global Flow Structures Using Multiple Levels of Topology , 1999 .

[14]  Jeff P. Hultquist,et al.  Constructing stream surfaces in steady 3D vector fields , 1992, Proceedings Visualization '92.

[15]  Gerik Scheuermann,et al.  Detection and Visualization of Closed Streamlines in Planar Flows , 2001, IEEE Trans. Vis. Comput. Graph..

[16]  Lambertus Hesselink,et al.  Visualizing vector field topology in fluid flows , 1991, IEEE Computer Graphics and Applications.

[17]  Hans Hagen,et al.  Topology-Based Visualization of Time-Dependent 2D Vector Fields , 2001, VisSym.

[18]  Rüdiger Westermann,et al.  Topology-Preserving Smoothing of Vector Fields , 2001, IEEE Trans. Vis. Comput. Graph..

[19]  Holger Theisel.  Designing 2D Vector Fields of Arbitrary Topology , 2002, Comput. Graph. Forum.

[20]  Hans-Peter Seidel,et al.  Feature Flow Fields , 2003, VisSym.

[21]  Allen Van Gelder.  Stream Surface Generation for Fluid Flow Solutions on Curvilinear Grids , 2001, VisSym.

[22]  Hans Hagen,et al.  Topology tracking for the visualization of time-dependent two-dimensional flows , 2002, Comput. Graph..

[23]  Helwig Hauser,et al.  THOROUGH INSIGHTS BY ENHANCED VISUALIZATION OF FLOW TOPOLOGY , 2000 .

[24]  Hans Hagen,et al.  A tetrahedra-based stream surface algorithm , 2001, Proceedings Visualization, 2001. VIS '01..

[25]  B. R. Noack,et al.  On the transition of the cylinder wake , 1995 .

[26]  Lambertus Hesselink,et al.  Feature comparisons of vector fields using Earth mover's distance , 1998, Proceedings Visualization '98 (Cat. No.98CB36276).

[27]  Lambertus Hesselink,et al.  Representation and display of vector field topology in fluid flow data sets , 1989, Computer.

[28]  Jarke J. van Wijk.  Implicit Stream Surfaces , 1993, IEEE Visualization.

[29]  Al Globus,et al.  A tool for visualizing the topology of three-dimensional vector fields , 1991, Proceeding Visualization '91.

[30]  Chandrajit L. Bajaj,et al.  Topology preserving data simplification with error bounds , 1998, Comput. Graph..

[31]  Detlev Stalling,et al.  Fast texture based algorithms for vector field visualization , 1999 .

[32]  HesselinkLambertus,et al.  Representation and Display of Vector Field Topology in Fluid Flow Data Sets , 1989 .

[33]  Robert Haimes,et al.  Critical Points at Infinity: a missing link in vector field topology , 2000 .

[34]  Holger Theisel,et al.  Vector Field Metrics Based on Distance Measures of First Order Critical Points , 2002, WSCG.

[35]  Hans Hagen,et al.  Continuous topology simplification of planar vector fields , 2001, Proceedings Visualization, 2001. VIS '01..

[36]  Robert van Liere,et al.  Collapsing flow topology using area metrics , 1999, Proceedings Visualization '99 (Cat. No.99CB37067).