Nonlinear differential equations and dynamical systems
Nonlinear differential equations and dynamical systems
Visualization of higher order singularities in vector fields
VIS '97 Proceedings of the 8th conference on Visualization '97
VIS '97 Proceedings of the 8th conference on Visualization '97
An interesting class of polynomial vector fields
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
Visualizing Vector Field Topology in Fluid Flows
IEEE Computer Graphics and Applications
Surface representations of two- and three-dimensional fluid flow topology
VIS '90 Proceedings of the 1st conference on Visualization '90
Feature comparisons of vector fields using earth mover's distance
Proceedings of the conference on Visualization '98
Locating closed streamlines in 3D vector fields
VISSYM '02 Proceedings of the symposium on Data Visualisation 2002
Detection and Visualization of Closed Streamlines in Planar Flows
IEEE Transactions on Visualization and Computer Graphics
Scientific Visualization - Methods and Applications
Informatics - 10 Years Back. 10 Years Ahead.
Combining Topological Simplification and Topology Preserving Compression for 2D Vector Fields
PG '03 Proceedings of the 11th Pacific Conference on Computer Graphics and Applications
Scientific Visualization: methods and applications
SCCG '03 Proceedings of the 19th spring conference on Computer graphics
Stream Line and Path Line Oriented Topology for 2D Time-Dependent Vector Fields
VIS '04 Proceedings of the conference on Visualization '04
Topological Methods for 2D Time-Dependent Vector Fields Based on Stream Lines and Path Lines
IEEE Transactions on Visualization and Computer Graphics
Clifford Fourier Transform on Vector Fields
IEEE Transactions on Visualization and Computer Graphics
Texture analysis using directional local extrema
Machine Graphics & Vision International Journal
Saddle Connectors - An Approach to Visualizing the Topological Skeleton of Complex 3D Vector Fields
Proceedings of the 14th IEEE Visualization 2003 (VIS'03)
Vector field design on surfaces
ACM Transactions on Graphics (TOG)
Representing Higher-Order Singularities in Vector Fields on Piecewise Linear Surfaces
IEEE Transactions on Visualization and Computer Graphics
Nested OpenMP for efficient computation of 3D critical points in multi-block CFD datasets
Proceedings of the 2006 ACM/IEEE conference on Supercomputing
Vector Field Editing and Periodic Orbit Extraction Using Morse Decomposition
IEEE Transactions on Visualization and Computer Graphics
Computation of Localized Flow for Steady and Unsteady Vector Fields and Its Applications
IEEE Transactions on Visualization and Computer Graphics
Rotational symmetry field design on surfaces
ACM SIGGRAPH 2007 papers
Parallel hierarchical visualization of large time-varying 3D vector fields
Proceedings of the 2007 ACM/IEEE conference on Supercomputing
Measuring the Similarity of Vector Fields Using Global Distributions
SSPR & SPR '08 Proceedings of the 2008 Joint IAPR International Workshop on Structural, Syntactic, and Statistical Pattern Recognition
Localized flow analysis of 2D and 3D vector fields
EUROVIS'05 Proceedings of the Seventh Joint Eurographics / IEEE VGTC conference on Visualization
Galilean invariant extraction and iconic representation of vortex core lines
EUROVIS'05 Proceedings of the Seventh Joint Eurographics / IEEE VGTC conference on Visualization
Boundary switch connectors for topological visualization of complex 3D vector fields
VISSYM'04 Proceedings of the Sixth Joint Eurographics - IEEE TCVG conference on Visualization
Path line oriented topology for periodic 2D time-dependent vector fields
EUROVIS'06 Proceedings of the Eighth Joint Eurographics / IEEE VGTC conference on Visualization
Singular points analysis in fingerprints based on topological structure and orientation field
ICB'07 Proceedings of the 2007 international conference on Advances in Biometrics
Hierarchical vortex regions in swirling flow
EuroVis'09 Proceedings of the 11th Eurographics / IEEE - VGTC conference on Visualization
Visualizing robustness of critical points for 2D time-varying vector fields
EuroVis '13 Proceedings of the 15th Eurographics Conference on Visualization
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We present our results on the visualization of nonlinear vector field topology. The underlying mathematics is done in Clifford algebra, a system describing geometry by extending the usual vector space by a multiplication of vectors. We started with the observation that all known algorithms for vector field topology are based on piecewise linear or bilinear approximation, and that these methods destroy the local topology if nonlinear behavior is present. Our algorithm looks for such situations, chooses an appropriate polynomial approximation in these areas, and, finally, visualizes the topology. This overcomes the problem, and the algorithm is still very fast because we are using linear approximation outside these small but important areas. The paper contains a detailed description of the algorithm and a basic introduction to Clifford algebra.