Issue No.06 - June (2012 vol.18)
pp: 938-951
Andrzej Szymczak , Colorado School of Mines, Golden
Eugene Zhang , Oregon State University, Corvallis
In this paper, we introduce a new approach to computing a Morse decomposition of a vector field on a triangulated manifold surface. The basic idea is to convert the input vector field to a piecewise constant (PC) vector field, whose trajectories can be computed using simple geometric rules. To overcome the intrinsic difficulty in PC vector fields (in particular, discontinuity along mesh edges), we borrow results from the theory of differential inclusions. The input vector field and its PC variant have similar Morse decompositions. We introduce a robust and efficient algorithm to compute Morse decompositions of a PC vector field. Our approach provides subtriangle precision for Morse sets. In addition, we describe a Morse set classification framework which we use to color code the Morse sets in order to enhance the visualization. We demonstrate the benefits of our approach with three well-known simulation data sets, for which our method has produced Morse decompositions that are similar to or finer than those obtained using existing techniques, and is over an order of magnitude faster.
Morse decomposition, vector field topology.
Andrzej Szymczak, Eugene Zhang, "Robust Morse Decompositions of Piecewise Constant Vector Fields", IEEE Transactions on Visualization & Computer Graphics, vol.18, no. 6, pp. 938-951, June 2012, doi:10.1109/TVCG.2011.88
[1] E. Boczko, W.D. Kalies, and K. Mischaikow, “Polygonal Approximation of Flows,” Topology and Its Applications, vol. 154, no. 13, pp. 2501-2520, 2007.
[2] P.-T. Bremer, H. Edelsbrunner, B. Hamann, and V. Pascucci, “A Multi-Resolution Data Structure for 2-Dimensional Morse Functions,” Proc. IEEE 14th Visualization (VIS '03), pp. 139-146, 2003.
[3] J. Chai, Y. Zhao, W. Guo, and Z. Tang, “A Texture Method for Visualization of Electromagnetic Vector Field,” Proc. Sixth Int'l Conf. Electrical Machines and Systems (ICEMS), vol. 2, pp. 805-808, 2003.
[4] G. Chen, Q. Deng, A. Szymczak, R.S. Laramee, and E. Zhang, “Morse Set Classification and Hierarchical Refinement Using Conley Index,” IEEE Trans. Visualization and Computer Graphics, vol. 18, no. 5, pp. 767-782, May 2012.
[5] G. Chen, K. Mischaikow, R.S. 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.
[6] G. Chen, K. Mischaikow, R.S. Laramee, and E. Zhang, “Efficient Morse Decompositions of Vector Fields,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 4, pp. 848-862, July/Aug. 2008.
[7] C. Conley, Isolated Invariant Sets and Morse Index. American Mathematical Soc., 1978.
[8] E. Early, “On the Euler Characteristic,” The MIT Undergraduate J. Math., vol. 1, pp. 37-48, 1999.
[9] H. Edelsbrunner, J. Harer, and A. Zomorodian, “Hierarchical Morse Complexes for Piecewise Linear 2-Manifolds,” Proc. 17th Ann. Symp. Computational Geometry (SCG '01), pp. 70-79, 2001.
[10] R. Forman, “Combinatorial Vector Fields and Dynamical Systems,” Mathematische Zeitschrift, vol. 228, pp. 629-681, 1998.
[11] L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings: Volume 4 of Topological Fixed Point Theory and Its Applications, second ed. Springer, 2006.
[12] J.L. 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.
[13] J.L. Helman and L. Hesselink, “Visualizing Vector Field Topology in Fluid Flows,” IEEE Computer Graphics and Applications, vol. 11, no. 3, pp. 36-46, May 1991.
[14] T. Kaczynski and M. Mrozek, “Conley Index for Discrete Multivalued Dynamical Systems,” Topology and Its Applications, vol. 65, pp. 83-96, 1997.
[15] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems: Encyclopedia of Mathematics and Its Applications. Cambridge Univ. Press, 1995.
[16] R.S. Laramee, C. Garth, H. Doleisch, J. Schneider, H. Hauser, and H. Hagen, “Visual Analysis and Exploration of Fluid Flow in a Cooling Jacket,” Proc. IEEE Visualization, pp. 623-630, 2005.
[17] R.S. Laramee, H. Hauser, H. Doleisch, B. Vrolijk, F.H. Post, and D. Weiskopf, “The State of the Art in Flow Visualization: Dense and Texture-Based Techniques,” Computer Graphics Forum, vol. 23, no. 2, pp. 203-221, 2004.
[18] R.S. Laramee, H. Hauser, L. Zhao, and F.H. Post, “Topology-Based Flow Visualization, the State of the Art,” Proc. Topology-Based Methods in Visualization (TopoInVis '05), pp. 1-19, 2007.
[19] R.S. Laramee, J.J. van Wijk, B. Jobard, and H. Hauser, “ISA and IBFVS: Image Space-Based Visualization of Flow on Surfaces,” IEEE Trans. Visualization and Computer Graphics, vol. 10, no. 6, pp. 637-648, Nov./Dec. 2004.
[20] R.S. Laramee, D. Weiskopf, J. Schneider, and H. Hauser, “Investigating Swirl and Tumble Flow with a Comparison of Visualization Techniques,” Proc. IEEE Visualization, pp. 51-58, 2004.
[21] P. Lindstrom and V. Pascucci, “Visualization of Large Terrains Made Easy,” Proc. IEEE Visualization, pp. 363-371, 2001.
[22] E.N. Lorenz, “Deterministic Non-Periodic Flow,” J. Atmospheric Science, vol. 20, pp. 130-141, 1963.
[23] R.P. McGehee and T. Wiandt, “Conley Decomposition for Closed Relations,” J. Difference Equations and Applications, vol. 12, no. 1, pp. 1-47, 2006.
[24] T. McLoughlin, R.S. Laramee, R. Peikert, F.H. Post, and M. Chen, “Over Two Decades of Integration-Based, Geometric Flow Visualization,” Proc. Eurographics, pp. 73-92, 2009.
[25] R. Mehran, B.E. Moore, and M. Shah, “A Streakline Representation of Flow in Crowded Scenes,” Proc. 11th European Conf. Computer Vision (ECCV '10), pp. 439-452, 2010.
[26] K. Mischaikow and M. Mrozek, “Chaos in the Lorenz Equations: A Computer-Assisted Proof,” Bull. Am. Math. Soc., vol. 32, pp. 66-72, 1995.
[27] K. Mischaikow, M. Mrozek, and A. Szymczak, “Chaos in the Lorenz Equations: A Computer Assisted Proof. Part III: The Classical Parameter Values,” J. Differential Equations, vol. 169, no. 1, pp. 17-56, 2001.
[28] M. Mrozek, “Index Pairs and the Fixed Point Index for Semidynamical Systems with Discrete Time,” Fundamental Mathematicae, vol. 133, pp. 177-192, 1989.
[29] M. Mrozek, “A Cohomological Index of Conley Type for Multivalued Admissible Flows,” J. Differential Equations, vol. 84, pp. 15-51, 1990.
[30] M. Otto, T. Germer, H.-C. Hege, and H. Theisel, “Uncertain 2D Vector Field Topology,” Computer Graphics Forum, vol. 29, no. 2, pp. 347-356, 2010.
[31] R. Panton, Incompressible Flow. John Wiley & Sons, 1984.
[32] J. Reininghaus, C. Lowen, and I. Hotz, “Fast Combinatorial Vector Field Topology,” IEEE Trans. Visualization and Computer Graphics, vol. pp, no. 99, p. 1, 2010.
[33] E.H. Spanier, Algebraic Topology. Springer, 1966.
[34] A. Szymczak, “Stable Morse Decompositions for Piecewise Constant Vector Fields on Surfaces,” Computer Graphics Forum, to appear, 2011.
[35] R. Tarjan, “Depth-First Search and Linear Graph Algorithms,” SIAM J. Computing, vol. 1, no. 2, pp. 146-160, 1972.
[36] H. Theisel and T. Weinkauf, “Grid-Independent Detection of Closed Stream Lines in 2D Vector Fields,” Proc. Conf. Vision, Modeling and Visualization (VMV '04), pp. 421-428, 2004.
[37] X. Tricoche, C. Garth, and G. Scheuermann, “Fast and Robust Extraction of Separation Line Features,” Proc. Scientific Visualization: The Visual Extraction of Knowledge from Data, pp. 249-263, 2006.
[38] X. Tricoche, G. Scheuermann, and H. Hagen, “Higher Order Singularities in Piecewise Linear Vector Fields,” Proc. IMA Conf. Math. Surfaces, pp. 99-113, 2000.
[39] X. Tricoche, G. Scheuermann, and H. Hagen, “A Topology Simplification Method for 2D Vector Fields,” Proc. Visualization, pp. 359-366, 2000.
[40] T. Wischgoll and G. Scheuermann, “Detection and Visualization of Planar Closed Streamline,” IEEE Trans. Visualization and Computer Graphics, vol. 7, no. 2, pp. 165-172, Apr.-June 2001.
[41] E. Zhang, K. Mischaikow, and G. Turk, “Vector Field Design on Surfaces,” ACM Trans. Graphics, vol. 25, no. 4, pp. 1294-1326, 2006.