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Topology Preserving Top-Down Compression of 2D Vector Fields Using Bintree and Triangular Quadtrees
October-December 2003 (vol. 9 no. 4)
pp. 433-442

Abstract—We present a hierarchical top-down refinement algorithm for compressing 2D vector fields that preserves topology. Our approach is to reconstruct the data set using adaptive refinement that considers topology. The algorithms start with little data and subdivide regions that are most likely to reconstruct the original topology of the given data set. We use two different refinement techniques. The first technique uses bintree subdivision and linear interpolation. The second algorithm is driven by triangular quadtree subdivision with Coons patch quadratic interpolation. We employ local error metrics to measure the quality of compression and as a global metric we compute Earth Mover's Distance (EMD) to measure the deviation from the original topology. Experiments with both analytic and simulated data sets are presented. Results indicate that one can obtain significant compression with low errors without losing topological information. Advantages and disadvantages of different topology preserving compression algorithms are also discussed in the paper.

[1] C.L. Bajaj, V. Pascucci, and D.R. Schikore, Visualization of Scalar Topology for Structural Enhancement Proc. Visualization '98, pp. 51-58, Oct. 1999.
[2] C.L. Bajaj and D.R. Schikore, Topology Preserving Data Simplification with Error Bounds J. Computers and Graphics, vol. 22, no. 1, pp. 3-12, 1998.
[3] R.K. Batra and L. Hesselink, Feature Comparisons of 3D Vector Fields Proc. Visualization '99, pp. 105-114, Oct. 1999.
[4] S. Chen, D. Holm, L. Margolin, and R. Zhang, Direct Numerical Simulations of the Navier-Stokes Alpha Model Predictability: Quantifying Uncertainty in Models of Complex Phenomena, S. Chen, L. Margolin, and D. Sharp, eds., pp. 66-83, Amsterdam: Elsevier, 1999.
[5] M.S. Chong, A.E. Perry, and B.J. Cantwell, A General Classification of Three-Dimensional Flow Fields Physicd of Fluids A, vol. 2, no. 5, pp. 765-777, 1980.
[6] S.A. Coons, Surfaces for Computer-Aided Design of Space Forms Technical Report MIT/LCS/TR-41, Massachusetts Inst. of Tech nology, June 1967.
[7] U. Dallmann, Topological Structures of Three-Dimensional Vortex Flow Separation Technical Report AIAA-83-1935, AIAA 16th Fluid and Plasma Dynamics Conf., 1983.
[8] W. de Leeuw and R. van Liere, Visualization of Global Flow Structures Using Multiple Levels of Topology Eurographics, pp. 45-52, 1999.
[9] W.C. de Leeuw and R. van Liere, Collapsing Flow Topology Using Area Metrics Proc. Visualization '99, pp. 349-354, Oct. 1999.
[10] M.A. Duchaineau, M. Wolinsky, D.E. Sigeti, M.C. Miller, C. Aldrich, and M.B. Mineev-Weinstein, “ROAMing Terrain: Real-Time Optimally Adapting Meshes,” Proc. IEEE Visualization '97, pp. 81-88, Nov. 1997.
[11] A. Friedrich, K. Polthier, and M. Schmies, Interpolation of Triangle Hierarchies Proc. IEEE Visualization '98, pp. 391-396, 1998.
[12] A. Globus, C. Levit, and T. Lasinski, "A Tool for Visualizing the Topology of Three-Dimensional Vector Fields," Proc. IEEE Visualization '91, pp. 33-40, 1991.
[13] K. Hanson, A Framework for Assessing Uncertainties in Simulation Predictions Predictability: Quantifying Uncertainty in Models of Complex Phenomena, S. Chen, L. Margolin, and D. Sharp, eds., pp. 179-188, Amsterdam: Elsevier, 1999.
[14] B. Heckel, G. Weber, B. Hamann, and K.I. Joy, “Construction of Vector Field Hierarchies,” Proc. IEEE Visualization '99, pp. 19-25, 1999.
[15] J.L. Helman and L. Hesselink, "Visualization of Vector Field Topology in Fluid Flows," IEEE Computer Graphics and Applications, vol. 11, no. 3, pp. 36-46, 1991.
[16] D.N. Kenwright, Automatic Detection of Open and Closed Separation and Attachment Lines Proc. IEEE Visualization 1998, pp. 151-158, 1998.
[17] Y. Lavin, R. Batra, and L. Hesselink, Feature Comparison of Vector Fields Using Earth Mover's Distance Proc. IEEE Visualization 1998, pp. 103-110, 1998.
[18] P. Lindstrom et al., "Real-Time, Continuous Level of Detail Rendering of Height Fields," Proc. Siggraph 96, ACM Press, New York, 1996, pp. 109-118.
[19] S. Lodha, J.C. Renteria, and K.M. Roskin, Topology Preserving Compression of 2D Vector Fields Proc. IEEE Visualization '00, pp. 343-350, Oct. 2000.
[20] G.M. Nielson, D.J. Holliday, and T. Roxborough, Cracking the Cracking Prpblem with Coons Patches Proc. IEEE Visualization '99, pp. 285-289, Nov. 1999.
[21] G.M. Nielson, I.H. Jung, and J. Song, Wavelets over Curvilinear Grids Proc. Visualization '98, pp. 313-317, Oct. 1998.
[22] M. Rivara, Algorithms for Refining Triangular Grids Suitable for Adaptive and Multigrid Techniques Int'l J. Numerical Methods in Eng., 1984.
[23] Y. Rubner and C. Tomasi, The Earth Mover's Distance as a Metric for Image Retrieval Technical Report STAN-CS-TN-98-86, Dept. of Computer Science, Stanford Univ., Sept. 1998.
[24] G. Scheuermann, H. Hagen, H. Kruger, M. Menzel, and A. Rockwood, Visualization of Higher-Order Singularities in Vector Field Proc. Visualization '97, pp. 67-74, Oct. 1997.
[25] G. Scheuermann, X. Tricoche, and H. Hagen, C1-Interpolation for Vector Field Topology Visualization Proc. Visualization '99, pp. 271-278, Oct. 1999.
[26] A.C. Telea and J.J. van Wijk, “Simplified Representation of Vector Fields,” Proc. IEEE Visualization '99, pp. 35-42, 1999.
[27] X. Tricoche, G. Scheuermann, and H. Hagen, A Topology Simplification Method for 2D Vector Fields Proc. Visualization '00, pp. 359-366, Oct. 2000.
[28] X. Tricoche, G. Scheuermann, and H. Hagen, Continuous Topology Simplification of Planar Vector Fields Proc. Visualization '01, pp. 159-166, Oct. 2001.
[29] X. Tricoche, G. Scheuermann, H. Hagen, and S. Clauss, Vector and Tensor Field Topology Simplification on Irregular Grids Eurographics, pp. 107-116, 2001.
[30] R. Verstappen and A. Veldman, Spectro-Consistent Discretization of Navier-Stokes: A Challenge to RANS and LES J. Eng. Math., vol. 34, pp. 163-179, 1998.
[31] J. Wilhelms and A. Van Gelder, “Multi-Dimensional Trees for Controlled Volume Rendering and Compression,” Proc. 1994 Symp. Volume Visualization, pp. 27-34, Oct. 1994.

Index Terms:
Vector fields, compression, topology, top-down, bintree, quadtree, hierarchical algorithm.
Suresh K. Lodha, Nikolai M. Faaland, Jose C. Renteria, "Topology Preserving Top-Down Compression of 2D Vector Fields Using Bintree and Triangular Quadtrees," IEEE Transactions on Visualization and Computer Graphics, vol. 9, no. 4, pp. 433-442, Oct.-Dec. 2003, doi:10.1109/TVCG.2003.1260738
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