The Community for Technology Leaders
RSS Icon
Issue No.08 - August (2011 vol.17)
pp: 1148-1163
Marcel Hlawatsch , University of Stuttgart, Stuttgart
Filip Sadlo , University of Stuttgart, Stuttgart
Daniel Weiskopf , University of Stuttgart, Stuttgart
This paper presents an acceleration scheme for the numerical computation of sets of trajectories in vector fields or iterated solutions in maps, possibly with simultaneous evaluation of quantities along the curves such as integrals or extrema. It addresses cases with a dense evaluation on the domain, where straightforward approaches are subject to redundant calculations. These are avoided by first calculating short solutions for the whole domain. From these, longer solutions are then constructed in a hierarchical manner until the designated length is achieved. While the computational complexity of the straightforward approach depends linearly on the length of the solutions, the computational cost with the proposed scheme grows only logarithmically with increasing length. Due to independence of subtasks and memory locality, our algorithm is suitable for parallel execution on many-core architectures like GPUs. The trade-offs of the method—lower accuracy and increased memory consumption—are analyzed, including error order as well as numerical error for discrete computation grids. The usefulness and flexibility of the scheme are demonstrated with two example applications: line integral convolution and the computation of the finite-time Lyapunov exponent. Finally, results and performance measurements of our GPU implementation are presented for both synthetic and simulated vector fields from computational fluid dynamics.
Flow visualization, integral curves, hierarchical computation, FTLE, LIC, GPU.
Marcel Hlawatsch, Filip Sadlo, Daniel Weiskopf, "Hierarchical Line Integration", IEEE Transactions on Visualization & Computer Graphics, vol.17, no. 8, pp. 1148-1163, August 2011, doi:10.1109/TVCG.2010.227
[1] S. Brunton and C. Rowley, “A Method for Fast Computation of FTLE Fields,” Proc. 61st Ann. Meeting of the APS Division of Fluid Dynamics, , 2008.
[2] P.J. Burt, “Fast Filter Transform for Image Processing,” Computer Graphics and Image Processing, vol. 16, no. 1, pp. 20-51, 1981.
[3] B. Cabral and L.C. Leedom, “Imaging Vector Fields Using Line Integral Convolution,” Proc. ACM SIGGRAPH '93, pp. 263-270, 1993.
[4] B. Csébfalvi, “An Evaluation of Prefiltered Reconstruction Schemes for Volume Rendering,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 2, pp. 289-301, Mar./Apr. 2008.
[5] G. Erlebacher, B. Jobard, and D. Weiskopf, “Flow Textures: High-Resolution Flow Visualization,” The Visualization Handbook, C. Hansen and C. Johnson, eds., pp. 279-293, Elsevier, 2005.
[6] G. Farin, Curves and Surfaces for CAGD: A Practical Guide. Morgan Kaufmann Publishers, 2002.
[7] C. Garth, F. Gerhardt, X. Tricoche, and H. Hagen, “Efficient Computation and Visualization of Coherent Structures in Fluid Flow Applications,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1464-1471, Nov./Dec. 2007.
[8] D. Goldberg, “What Every Computer Scientist Should Know about Floating-Point Arithmetic,” ACM Computing Surveys, vol. 23, no. 1, pp. 5-48, 1991.
[9] I. Goldhirsch, P.-L. Sulem, and S.A. Orszag, “Stability and Lyapunov Stability of Dynamical Systems: A Differential Approach and a Numerical Method,” Physica D, vol. 27, no. 3, pp. 311-337, 1987.
[10] G. Haller, “Distinguished Material Surfaces and Coherent Structures in Three-Dimensional Fluid Flows,” Physica D, vol. 149, no. 4, pp. 248-277, 2001.
[11] G. Haller, “An Objective Definition of a Vortex,” J. Fluid Mechanics, vol. 525, pp. 1-26, 2005.
[12] P.S. Heckbert, “Filtering by Repeated Integration,” Computer Graphics, vol. 20, no. 4, pp. 315-321, 1986.
[13] H.-C. Hege and D. Stalling, “Fast LIC with Piecewise Polynomial Filter Kernels,” Mathematical Visualization—Algorithms and Applications, H.-C. Hege and K. Polthier, eds., pp. 295-314, Springer, 1998.
[14] W. Heidrich, R. Westermann, H.-P. Seidel, and T. Ertl, “Applications of Pixel Textures in Visualization and Realistic Image Synthesis,” Proc. ACM Symp. Interactive 3D Graphics '99, pp. 127-134, 1999.
[15] 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.
[16] B. Jobard, G. Erlebacher, and M.Y. Hussaini, “Hardware-Accelerated Texture Advection for Unsteady Flow Visualization,” Proc. IEEE Visualization '00, pp. 155-162, 2000.
[17] M. Kraus and M. Strengert, “Pyramid Filters Based on Bilinear Interpolation,” Proc. Int'l Conf. Computer Graphics Theory and Applications (GRAPP '07), pp. 21-28, 2007.
[18] R. Laramee, H. Hauser, H. Doleisch, B. Vrolijk, F. 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.
[19] L. Li and H.-W. Shen, “View-Dependent Multi-Resolutional Flow Texture Advection,” Proc. IS&T/SPIE Visualization and Data Analysis '06, pp. 24-34, 2006.
[20] E.N. Lorenz, “A Study of the Predictability of a 28-Variable Atmospheric Model,” Tellus, vol. 17, pp. 321-333, 1965.
[21] E. Lum, B. Wilson, and K.-L. Ma, “High-Quality Lighting and Efficient Pre-Integration for Volume Rendering,” Proc. Eurographics/IEEE Visualization and Graphics Technical Committee (VGTC) Symp. Visualization, pp. 25-34, 2004.
[22] nVidia “CUDA Programming Guide 2.3.1,” http://developer. cuda/2_3/toolkit/docs NVIDIA_CUDA_Programming_Guide_2.3.pdf , 2009.
[23] J. Ogden, E. Adelson, J. Bergen, and P. Burt, “Pyramid-Based Computer Graphics,” RCA Engineer, vol. 30, no. 5, pp. 4-15, 1985.
[24] R. Peikert and F. Sadlo, “Flow Topology beyond Skeletons: Visualization of Features in Recirculating Flow,” Topology-Based Methods in Visualization II, H.-C. Hege, K. Polthier, and G. Scheuermann, eds., pp. 145-160, Springer, 2009.
[25] F. Sadlo and R. Peikert, “Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1456-1463, Nov. 2007.
[26] F. Sadlo and D. Weiskopf, “Time-Dependent 2-D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures,” Computer Graphics Forum, vol. 29, no. 1, pp. 88-100, 2010.
[27] H.R. Schwarz, Numerische Mathematik. B.G. Teubner, 1997.
[28] C. Sigg and M. Hadwiger, “Fast Third-Order Texture Filtering,” GPU Gems 2, M. Pharr and R. Fernando, eds., pp. 313-329, Addison-Wesley, 2005.
[29] D. Stalling and H.-C. Hege, “Fast and Resolution Independent Line Integral Convolution,” Proc. ACM SIGGRAPH '95, pp. 249-256, 1995.
[30] M. Unser, A. Aldroubi, and M. Eden, “B-Spline Signal Processing: Part I-Theory,” IEEE Trans. Signal Processing, vol. 41, no. 2, pp. 821-833, Feb. 1993.
[31] M. Unser, A. Aldroubi, and M. Eden, “B-Spline Signal Processing: Part II-Efficient Design and Applications,” IEEE Trans. Signal Processing, vol. 41, no. 2, pp. 834-848, Feb. 1993.
[32] J.J. van Wijk, “Implicit Stream Surfaces,” Proc. IEEE Visualization '93, pp. 245-252, 1993.
[33] J.J. van Wijk, “Image Based Flow Visualization,” ACM Trans. Graphics, vol. 21, no. 3, pp. 745-754, 2002.
[34] D. Weiskopf, “Iterative Twofold Line Integral Convolution for Texture-Based Vector Field Visualization,” Math. Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration, T. Möller, B. Hamann, and R. Russell, eds., pp. 191-211, Springer, 2009.
[35] D. Weiskopf, G. Erlebacher, and T. Ertl, “A Texture-Based Framework for Spacetime-Coherent Visualization of Time-Dependent Vector Fields,” Proc. IEEE Visualization '03, pp. 107-114, 2003.
[36] D. Weiskopf, M. Hopf, and T. Ertl, “Hardware-Accelerated Visualization of Time-Varying 2D and 3D Vector Fields by Texture Advection via Programmable Per-Pixel Operations,” Proc. Vision, Modeling, and Visualization Conf. '01, pp. 439-446, 2001.
[37] R. Westermann, C. Johnson, and T. Ertl, “A Level-Set Method for Flow Visualization,” Proc. IEEE Visualization '00, pp. 147-154, 2000.
18 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool