# Combinatorial 2D Vector Field Topology Extraction and Simplification

## Summary

This paper investigates a combinatorial approach to vector field topology. The theoretical basis is given by Robin Forman’s work on a combinatorial Morse theory for dynamical systems defined on general simplicial complexes. We formulateForman’s theory in a graph theoretic setting and provide a simple algorithm for the construction and topological simplification of combinatorial vector fields on 2D manifolds. Given a combinatorial vector field we are able to extract its topological skeleton including allperiodic orbits. Due to the solid theoretical foundation we know that the resulting structure is always topologically consistent. We explore the applicability and limitations of this combinatorial approach with several examples and determine its robustness with respect to noise.

## Keywords

Periodic Orbit Simplicial Graph Morse Theory Singular Homology Morse Decomposition## Preview

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## Notes

### Acknowledgements

We would like to thank David Günther and Tino Weinkauf for many fruitful discussions on this topic and Christian Löwen for his great implementational efforts. This work was funded by the DFG Emmy-Noether research programm. All visualizations in this paper have been created using AMIRA - a system for advanced visual data analysis (see http://amira.zib.de/).

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