This Article 
 Bibliographic References 
 Add to: 
Efficient Computation of Combinatorial Feature Flow Fields
Sept. 2012 (vol. 18 no. 9)
pp. 1563-1573
J. Kasten, Konrad-Zuse-Zentrum fuer Informationstechnik, Zuse Inst., Berlin, Germany
J. Reininghaus, Konrad-Zuse-Zentrum fuer Informationstechnik, Zuse Inst., Berlin, Germany
T. Weinkauf, Dept. 4: Comput. Graphics, Max Planck Inst. for Inf., Saarbrucken, Germany
I. Hotz, Konrad-Zuse-Zentrum fuer Informationstechnik, Zuse Inst., Berlin, Germany
We propose a combinatorial algorithm to track critical points of 2D time-dependent scalar fields. Existing tracking algorithms such as Feature Flow Fields apply numerical schemes utilizing derivatives of the data, which makes them prone to noise and involve a large number of computational parameters. In contrast, our method is robust against noise since it does not require derivatives, interpolation, and numerical integration. Furthermore, we propose an importance measure that combines the spatial persistence of a critical point with its temporal evolution. This leads to a time-aware feature hierarchy, which allows us to discriminate important from spurious features. Our method requires only a single, easy-to-tune computational parameter and is naturally formulated in an out-of-core fashion, which enables the analysis of large data sets. We apply our method to synthetic data and data sets from computational fluid dynamics and compare it to the stabilized continuous Feature Flow Field tracking algorithm.

[1] H. Theisel and H.-P. Seidel, "Feature Flow Fields," VISSYM '03: Proc. Symp. Data Visualization, pp. 141-148, 2003.
[2] R. Forman, "Combinatorial Vector Fields and Dynamical Systems," Math. Zeitschrift, vol. 228, no. 4, pp. 629-681, Aug. 1998.
[3] H. Edelsbrunner, J. Harer, and A. Zomorodian, "Hierarchical Morse Complexes for Piecewise Linear 2-Manifolds," Proc. 17th Symp. Computational Geometry, pp. 70-79, 2001.
[4] H. King, K. Knudson, and N. Mramor, "Birth and Death in Discrete Morse Theory," arXiv:0808.0051v1, http://adsabs. , 2008.
[5] A. Yilmaz, O. Javed, and M. Shah, "Object Tracking: A Survey," ACM Computing Surveys, vol. 38, no. 4, p. 13, 2006.
[6] F.H. Post, "The State of the Art in Flow Visualization: Feature Extraction and Tracking," Computer Graphics Forum, vol. 22, no. 4, pp. 775-792, , June 2003.
[7] J. Caban, A. Joshi, and P. Rheingans, "Texture-Based Feature Tracking for Effective Time-Varying Data Visualizations," IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1472-1479, Nov./Dec. 2007.
[8] R. Samtaney, D. Silver, N. Zabusky, and J. Cao, "Visualizing Features and Tracking Their Evolution," Computer, vol. 27, no. 7, pp. 20-27, July 1994.
[9] F. Reinders, F. Post, and H. Spoelder, "Attribute-Based Feature Tracking," Proc. Data Visualization, 1999.
[10] D. Laney, P.T. Bremer, A. Mascarenhas, P. Miller, and V. Pascucci, "Understanding the Structure of the Turbulent Mixing Layer in Hydrodynamic Instabilities," IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 5, pp. 1053-1060, Sept./Oct. 2006.
[11] W. de Leeuw and R. van Liere, "Chromatin Decondensation: A Case Study of Tracking Features in Confocal Data," Proc. IEEE Conf. Visualization (VIS '01), pp. 441-444, 2001.
[12] D. Silver and X. Wang, "Tracking and Visualizing Turbulent 3D Features," IEEE Trans. Visualization and Computer Graphics, vol. 3, no. 2, pp. 129-141, www.caip.rutgers.eduvislab.html, Apr.-June 1997.
[13] G. Ji, "Feature Tracking and Viewing for Time-Varying Data Sets," PhD dissertation, Ohio State Univ., 2006.
[14] F. Reinders, I.A. Sadarjoen, B. Vrolijk, and F.H. Post, "Vortex Tracking and Visualisation in a Flow Past a Tapered Cylinder," Computer Graphics Forum, vol. 21, no. 4, pp. 675-682, 2002.
[15] C. Bajaj, A. Shamir, and B.-S. Sohn, "Progressive Tracking of Isosurfaces in Time-Varying Scalar Fields," Technical Report TR-02-4, Dept. of Computer Sciences and TICAM, Univ. of Texas, CS and TICAM, 2002.
[16] B.-S. Sohn and C. Bajaj, "Time-Varying Contour Topology," IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 1, pp. 14-25, Jan./Feb. 2006.
[17] C. Weigle and D.C. Banks, "Extracting Iso-Valued Features in 4D Scalar Fields," VVS '98: Proc. IEEE Symp. Vol. Visualization, pp. 103-110, 1998.
[18] G. Ji, H.-W. Shen, and R. Wenger, "Volume Tracking Using Higher Dimensional Isosurfacing," VIS '03: Proc. IEEE 14th Visualization, pp. 209-216, 2003.
[19] G.H. Weber, P.-T. Bremer, M.S. Day, J.B. Bell, and V. Pascucci, "Feature Tracking Using Reeb Graphs," Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications, V. Pascucci, X. Tricoche, H. Hagen, and J. Tierny, eds., pp. 241-253, Springer Verlag, 2011.
[20] P.-T. Bremer, G.H. Weber, V. Pascucci, M. Day, and J.B. Bell, "Analyzing and Tracking Burning Structures in Lean Premixed Hydrogen Flames," IEEE Trans. Visualization and Computer Graphics, vol. 16, no. 2, pp. 248-260, Mar./Apr. 2010.
[21] X. Tricoche, T. Wischgoll, G. Scheuermann, and H. Hagen, "Topology Tracking for the Visualization of Time-Dependent 2D Flows," Computer and Graphics, vol. 26, pp. 249-257, 2002.
[22] C. Garth, X. Tricoche, and G. Scheuermann, "Tracking of Vector Field Singularities in Unstructured 3D Time-Dependent Datasets," VIS '04: Proc. Conf. Visualization, pp. 329-336, 2004.
[23] D. Bauer and R. Peikert, "Vortex Tracking in Scale Space," Proc. Symp. Data Visualization, pp. 140-147, , May 2002.
[24] H. Edelsbrunner and J. Harer, "Jacobi Sets of Multiple Morse Functions," Foundations of Computational Math., Minneapolis 2002, F. Cucker, R. DeVore, P. Olver, and E. Sueli, eds., pp. 37-57, Cambridge Univ. Press, 2004.
[25] H. Edelsbrunner, J. Harer, A. Mascarenhas, J. Snoeyink, and V. Pascucci, "Time-Varying Reeb Graphs for Continuous Space-Time Data," Computation Geometry: Theory and Applications, vol. 41, no. 3, pp. 149-166, 2008.
[26] M.K. Chari, "On Discrete Morse Functions and Combinatorial Decompositions," Discrete Math., vol. 217, nos. 1-3, pp. 101-113, 2000.
[27] T. Lewiner, "Geometric Discrete Morse Complexes," PhD dissertation, Dept. of Math., PUC-Rio, advised by Hélio Lopes and Geovan Tavares, ~tomlew phd_thesis_puc_uk.pdf, 2005.
[28] U. Bauer, C. Lange, and M. Wardetzky, "Optimal Topological Simplification of Discrete Functions on Surfaces," arXiv:1001.1269v2, 2010.
[29] J. Reininghaus, D. Günther, I. Hotz, S. Prohaska, and H.-C. Hege, "TADD: A Computational Framework for Data Analysis Using Discrete Morse Theory," Proc. Third Int'l Congress Conf. Math. Software (ICMS '10), 2010.
[30] R. Forman, "Morse Theory for Cell Complexes," Advances in Math., vol. 134, pp. 90-145, 1998.
[31] R. Forman, "A User's Guide to Discrete Morse Theory," Seminaire Lotharingien de Combinatoire, vol. B48c, pp. 1-35, 2002.
[32] A. Gyulassy, P.-T. Bremer, B. Hamann, and V. Pascucci, "A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality," IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 6, pp. 1619-1626, Nov./Dec. 2008.
[33] H. Edelsbrunner, D. Letscher, and A. Zomorodian, "Topological Persistence and Simplification," Discrete and Computational Geometry, vol. 28, pp. 511-533, 2002.
[34] V. Robins, P. Wood, and A. Sheppard, "Discrete Morse Theory for Grayscale Digital Images," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 33, no. 8, pp. 1646-1658, Aug. 2011.
[35] D. Cohen-Steiner, H. Edelsbrunner, and D. Morozov, "Vines and Vineyards by Updating Persistence in Linear Time," Proc. 22nd Ann. Symp. Computational Geometry, pp. 119-126, http://doi.acm. org/10.11451137856.1137877 , 2006.
[36] T. Weinkauf, H. Theisel, A.V. Gelder, and A. Pang, "Stable Feature Flow Fields," IEEE Trans. Visualization and Computer Graphics, vol. 17, no. 6, pp. 770-780, http:/, June 2011.
[37] B.R. Noack, M. Schlegel, B. Ahlborn, G. Mutschke, M. Morzyński, P. Comte, and G. Tadmor, "A Finite-Time Thermodynamics of Unsteady Fluid Flows," J. NonEquilibrium Thermodynamics, vol. 33, no. 2, pp. 103-148, 2008.
[38] J. Kasten, I. Hotz, B. Noack, and H.-C. Hege, "On the Extraction of Long-Living Features in Unsteady Fluid Flows," Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications, V. Pascucci, X. Tricoche, H. Hagen, and J. Tierny, eds., pp. 115-126, Springer, 2010.
[39] E. Caraballo, M. Samimy, and J. DeBonis, "Low Dimensional Modeling of Flow for Closed-Loop Flow Control," Am. Inst. Aeronautics and Astronautics (AIAA Paper), vol. 59, 2003.
[40] R. Fuchs, J. Kemmler, B. Schindler, F. Sadlo, H. Hauser, and R. Peikert, "Toward a Lagrangian Vector Field Topology," Computer Graphics Forum, vol. 29, no. 3, pp. 1163-1172, 2010.
[41] T. Weinkauf, J. Sahner, H. Theisel, and H.-C. Hege, "Cores of Swirling Particle Motion in Unsteady Flows," IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1759-1766, Nov./Dec. 2007.

Index Terms:
numerical analysis,combinatorial mathematics,computational fluid dynamics,flow visualisation,flow visualization,combinatorial feature flow field computation,combinatorial algorithm,critical points tracking,2D time-dependent scalar fields,tracking algorithms,numerical schemes,computational parameters,noise robustness,importance measure,spatial persistence,temporal evolution,time-aware feature hierarchy,computational fluid dynamics,Feature extraction,Algorithm design and analysis,Joining processes,Manifolds,Noise measurement,Jacobian matrices,Noise,graph algorithms.,Flow visualization
J. Kasten, J. Reininghaus, T. Weinkauf, I. Hotz, "Efficient Computation of Combinatorial Feature Flow Fields," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 9, pp. 1563-1573, Sept. 2012, doi:10.1109/TVCG.2011.269
Usage of this product signifies your acceptance of the Terms of Use.