The Community for Technology Leaders
RSS Icon
Issue No.12 - Dec. (2012 vol.18)
pp: 2140-2148
Wieland Reich , University of Leipzig
Gerik Scheuermann , University of Leipzig
Existing methods for analyzing separation of streamlines are often restricted to a finite time or a local area. In our paper we introduce a new method that complements them by allowing an infinite-time-evaluation of steady planar vector fields. Our algorithm unifies combinatorial and probabilistic methods and introduces the concept of separation in time-discrete Markov-Chains. We compute particle distributions instead of the streamlines of single particles. We encode the flow into a map and then into a transition matrix for each time direction. Finally, we compare the results of our grid-independent algorithm to the popular Finite-Time-Lyapunov-Exponents and discuss the discrepancies.
Vectors, Eigenvalues and eigenfunctions, Markov processes, Transmission line matrix methods, Sparse matrices, Approximation methods, Topology, uncertainty, Vector field topology, flow visualization, feature extraction
Wieland Reich, Gerik Scheuermann, "Analysis of Streamline Separation at Infinity Using Time-Discrete Markov Chains", IEEE Transactions on Visualization & Computer Graphics, vol.18, no. 12, pp. 2140-2148, Dec. 2012, doi:10.1109/TVCG.2012.198
[1] S. L. Brunton and C. W. Rowley., Fast computation of finite-time lyapunov exponent fields for unsteady flows. In Chaos, 20, 2010.
[2] G. Chen, K. Mischakow, R.S. Laramee,, and E. Zhang., Efficient morse decompositions of vector fields. In IEEE Transactions on Visualization and Computer Graphics, 14, pages 848-862, 2008.
[3] M. Dellnitz and O. Junge., On the approximation of complicated dynamical behavior. In SIAM Journal on Numerical Analysis, 36, pages 491-515, 1999.
[4] R. Forman., Morse theory for cell complexes. In Advances in Mathemat-ics, 134, pages 90-145, 1998.
[5] C. Garth, G.-S. u, X. Tricoche., C.-D. Hansen, and H. Hagen., Visualization of coherent structures in transient 2d flows. In Topology-Based Methods in Visualization II, 2009.
[6] C. Garth, A. Wiebel, X. Tricoche., K. Joy, and G. Scheuermann., Lagrangian visualization of flow-embedded surface structures. In Computer Graphics Forum, 27, pages 767-774, 2008.
[7] G. Haller., Distinguished material surfaces and coherent structures in three-dimensional flows. In Physica D, 149, pages 248-277, 2001.
[8] J. Helman and L. Hesselink., Visualizing vector field topology in fluid flows. In IEEE Computer Graphics and Applications, 11, pages 36-46, 1991.
[9] V. Hernandez, J.-E. Roman, and V. Vidal, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems ACM Transactions on Mathematical Software, 31: 351-362, 2005.
[10] M. Hirsch, S. Smale, and R. Devaney., Differential Equations, Dynamical Svstems and An Introduction to Chaos. Elsevier, second edition, 2004.
[11] C. Johnson., Top scientific visualization research problems. In IEEE Computer Graphics and Applications, 24, pages 13-17, 2004.
[12] W. Kalies and H. Ban., A computational approach to conley's decomposition theorem. In J. Computational and Non-Linear Dynamics, 1, pages 312-319, 2006.
[13] J. Kasten, C. Petz, I. Hotz., B. Noack, and H.-C. Hege. In , Proceedings Vision, Modeling and Visualization 2008, pages 265-274.
[14] D. N. Kenwright, C. Henze, and C. Levit., Feature extraction of separation and attachment lines IEEE Trans. Vis. Comput. Graph., 5: 135-144, 1999.
[15] K. M. Mahrous,B. H. J. C. Bennett,, and K. I. Joy., Improving topological segmentation of three-dimensional vector fields. In IEEE TCVG Symposium on Visualization, 2003.
[16] M. Mrozek and P. Zgliczynski., Set arithmetic and the enclosing problem in dynamics. In Annales Polonici Mathematici, pages 237-259, 2000.
[17] M. Otto, T. Germer, H.-C. Hege,, and H. Theisel., Uncertain 2d vector field topology. In Computer Graphics Forum, 29, pages 347-356, 2010.
[18] J. Paixao, M. Lage, F. Petronetto., A. Laier, S. Pesco., G. Tavares, T. Lewiner,, and H. Lopes., Random walks for vector field denoising. In Computer Graphics and Image Processing (SIBGRAPI), 2009 XXII Brazilian Symposium on, pages 112-119, 2009.
[19] A.-T. Pang, C.-M. Wittenbrink, and S.-K. Lodh, Approaches to uncertainty visualization The Visual Computer, 13: 370-390, 1996.
[20] R. Peikert and M. Roth., The parallel vector operator - a vector field visualization primitive. In Proc. IEEE Visualization Conf., pages 263-270, 1999.
[21] F. Post, B. Vrolijk, H. Hauser., R. Laramee, and H. Doleisch., The state of art in flow visualization: Feature extraction and tracking. In Computer Graphics Forum, 22, pages 775-792, 2003.
[22] J. Reininghaus and I. Hotz., Combinatorial 2d vector field topology extraction and simplification. In Topology in Visualization, 2010.
[23] F. Sadlo and R. Peikert., Visualizing lagrangian coherent structures and comparison to vector field topology. In Topology-Based Methods in Visualization II, 2009.
[24] G. Scheuermann, B. Hamann, K. Joy,, and W. Kollmann., Visualizing Local Vector Field Topology. Journal of Electronic Imaging, 9(4): 356-367, 2000.
[25] D. Schneider, J. Fuhrmann, W. Reich,, and G. Scheuermann., A variance based ftle-like method for unsteady uncertain vector fields. In Topological Methods in Data Analysis and Visualization II, pages 255-268, 2012.
[26] D. Stalling and H.-C. Hege., Fast and resolution independent line integral convolution. In ACM SIGGRAPH, pages 249-256, 1995.
[27] D.-W. Stroock., An Introduction to Markov Processes. Springer, 2005.
[28] L.-N. Trefethen and D. Bau., Numerical Linear Algebra. SIAM, 1997.
[29] T. Weinkauf., Extraction of Topological Structures in 2D and 3D Vector Fields. PhD thesis, University Magdeburg, 2008.
[30] D. Weiskopf and B. Erlebacher., Overview of flow visualization. In The Visualization Handbook, pages 261-278, 2005.
[31] T. Wischgoll and G. Scheuermann., Detection and visualization of planar closed streamlines. In IEEE Trans. Visualization and CG, 7, pages 165-172, 2001.
503 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool