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Issue No.10 - October (2011 vol.17)
pp: 1433-1443
Jan Reininghaus , Zuse Institute Berlin, Berlin
Christian Löwen , Zuse Institute Berlin, Berlin
Ingrid Hotz , Zuse Institute Berlin, Berlin
ABSTRACT
This paper introduces a novel approximation algorithm for the fundamental graph problem of combinatorial vector field topology (CVT). CVT is a combinatorial approach based on a sound theoretical basis given by Forman's work on a discrete Morse theory for dynamical systems. A computational framework for this mathematical model of vector field topology has been developed recently. The applicability of this framework is however severely limited by the quadratic complexity of its main computational kernel. In this work, we present an approximation algorithm for CVT with a significantly lower complexity. This new algorithm reduces the runtime by several orders of magnitude and maintains the main advantages of CVT over the continuous approach. Due to the simplicity of our algorithm it can be easily parallelized to improve the runtime further.
INDEX TERMS
Flow visualization, graph algorithms.
CITATION
Jan Reininghaus, Christian Löwen, Ingrid Hotz, "Fast Combinatorial Vector Field Topology", IEEE Transactions on Visualization & Computer Graphics, vol.17, no. 10, pp. 1433-1443, October 2011, doi:10.1109/TVCG.2010.235
REFERENCES
[1] T. Weinkauf, “Extraction of Topological Structures in 2d and 3d Vector Fields,” PhD dissertation, Univ. of Magdeburg, http://www.zib.deweinkauf/, 2008.
[2] X. Tricoche, G. Scheuermann, H. Hagen, and S. Clauss, “Vector, Tensor Field Topology Simplification on Irregular Grids,” Proc. Symp. Data Visualization (VisSym '01), D. Ebert J. M. Favre, and R. Peikert, eds., pp. 107-116, May 2001.
[3] X. Tricoche, G. Scheuermann, and H. Hagen, “Continuous Topology Simplification of Planar Vector Fields,” Proc. IEEE Conf. Visualization 2001 (VIS '01), pp. 159-166, 2001.
[4] T. Weinkauf, H. Theisel, K. Shi, H.-C. Hege, and H.-P. Seidel, “Extracting Higher Order Critical Points and Topological Simplification of 3D Vector Fields,” Proc. IEEE Conf. Visualization, pp. 559-566, Oct. 2005.
[5] T. Klein and T. Ertl, “Scale-Space Tracking of Critical Points in 3d Vector Fields,” Topology-Based Methods in Visualization, H.H. Helwig Hauser and H. Theisel, eds., pp. 35-49, Springer, May 2007.
[6] G. Chen, K. Mischaikow, R. Laramee, P. Pilarczyk, and E. Zhang, “Vector Field Editing and Periodic Orbit Extraction Using Morse Decomposition,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 4, pp. 769-785, July/Aug. 2007.
[7] J. Reininghaus and I. Hotz, “Combinatorial 2d Vector Field Topology Extraction and Simplification,” Proc.Workshop Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications (TopoInVis '09), 2009.
[8] H. Edelsbrunner, D. Letscher, and A. Zomorodian, “Topological Persistence and Simplification,” Discrete and Computational Geometry, vol. 28, pp. 511-533, 2002.
[9] H. Edelsbrunner and J. Harer, “Persistent Homology—A Survey,” Surveys on Discrete and Computational Geometry: Twenty Years Later, J.E. Goodman, J. Pach, and R. Pollack, eds., vol. 458, pp. 257-282, AMS Bookstore, 2008.
[10] J. Helman and L. Hesselink, “Representation and Display of Vector Field Topology in Fluid Flow Data Sets,” Computer, vol. 22, no. 8, pp. 27-36, Aug. 1989.
[11] T. Wischgoll and G. Scheuermann, “Detection and Visualization of Closed Streamlines in Planar Flows,” IEEE Trans. Visualization and Computer Graphics, vol. 7, no. 2 pp. 165-172, Apr.-June 2001.
[12] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel, “Grid-Independent Detection of Closed Stream Lines in 2d Vector Fields,” Proc. Vision, Modeling, and Visualization (VMV) Conf., p. 665, Nov. 2004.
[13] R.S. Laramee, H. Hauser, L. Zhao, and F.H. Post, “Topology-Based Flow Visualization, the State of the Art,” Topology-Based Methods in Visualization, H.H. Helwig Hauser and H. Theisel, eds., pp. 1-19, Springer, May 2007.
[14] H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, “Morse-Smale Complexes for Piecewise Linear 3-Manifolds,” Proc. 19th Ann. Symp. Computational Geometry (SCG '03), pp. 361-370, 2003.
[15] A. Gyulassy, V. Natarajan, V. Pascucci, P.-T. Bremer, and B. Hamann, “A Topological Approach to Simplification of Three-Dimensional Scalar Functions,” IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 4, pp. 474-484, July/Aug. 2006.
[16] A. Gyulassy, “Combinatorial Construction of Morse-Smale Complexes for Data Analysis and Visualization,” PhD dissertation, Univ. of California at Davis, 2008.
[17] T. Lewiner, H. Lopes, and G. Tavares, “Applications of Forman's Discrete Morse Theory to Topology Visualization and Mesh Compression,” IEEE Trans. Visualization and Computer Graphics, vol. 10, no. 5, pp. 499-508, Sept./Oct. 2004.
[18] R. Forman, “A User's Guide to Discrete Morse Theory,” Proc. Int'l Conf. Formal Power Series and Algebraic Combinatorics, citeseer.ist. psu.eduforman01users.html, 2001.
[19] T. Lewiner, “Geometric Discrete Morse Complexes,” PhD dissertation, Dept. of Math., PUC-Rio, http://www.matmidia.mat. puc-rio.br/tomlew phd_thesis_puc_uk.pdf, 2005.
[20] R. Forman, “Morse Theory for Cell Complexes,” Advances in Math., vol. 134, pp. 90-145, 1998.
[21] R. Forman, “Combinatorial Vector Fields and Dynamical Systems,” Mathematische Zeitschrift, vol. 228, pp. 629-681, 1998.
[22] A. Schrijver, Combinatorial Optimization, R. Graham, B. Korte, L. Lovasz, A. Widgerson, and G. Ziegler, eds. Springer, 2003.
[23] S. Hougardy and D. Drake, “Approximation Algorithms for the Weighted Matching Problem,” Oberwolfach report, no. 28, 2004.
[24] A. Hatcher, Algebraic Topology. Cambridge Univ. Press, http://www.math.cornell.edu/~hatcher/ATATpage.html , 2002.
[25] J. Reininghaus, D. Günther, I. Hotz, S. Prohaska, and H.-C. Hege, “A Computational Framework for Data Analysis Using Discrete Morse Theory,” Proc. Third Int'l Congress on Math. Software (ICMS '10), Sept. 2010.
[26] H. Kuhn, “The Hungarian Method for the Assignment Problem,” Naval Research Logistics Quarterly, vol. 2, pp. 83-97, 1955.
[27] P. Harish and P. Narayanan, “Accelerating Large Graph Algorithms on the GPU Using CUDA,” Proc. IEEE Int'l Conf. High Performance Computing, 2007.
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