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Issue No.11 - November (2011 vol.17)
pp: 1599-1611
Christian Heine , Universität Leipzig, Leipzig
Dominic Schneider , Universität Leipzig, Leipzig
Hamish Carr , University of Leeds, Leeds
Gerik Scheuermann , Universität Leipzig, Leipzig
The contour tree compactly describes scalar field topology. From the viewpoint of graph drawing, it is a tree with attributes at vertices and optionally on edges. Standard tree drawing algorithms emphasize structural properties of the tree and neglect the attributes. Applying known techniques to convey this information proves hard and sometimes even impossible. We present several adaptions of popular graph drawing approaches to the problem of contour tree drawing and evaluate them. We identify five esthetic criteria for drawing contour trees and present a novel algorithm for drawing contour trees in the plane that satisfies four of these criteria. Our implementation is fast and effective for contour tree sizes usually used in interactive systems (around 100 branches) and also produces readable pictures for larger trees, as is shown for an 800 branch example.
Contour tree, graph layout.
Christian Heine, Dominic Schneider, Hamish Carr, Gerik Scheuermann, "Drawing Contour Trees in the Plane", IEEE Transactions on Visualization & Computer Graphics, vol.17, no. 11, pp. 1599-1611, November 2011, doi:10.1109/TVCG.2010.270
[1] G. Reeb, “Sur le Points Singuliers D'une Forme de Pfaff Complètement Intégrable ou D'une Fonction Numérique,” Comptes Rendus de l'Acadèmie des Sciences de Paris, vol. 222, pp. 847-849, 1946.
[2] C.L. Bajaj, V. Pascucci, and D.R. Schikore, “The Contour Spectrum,” Proc. Conf. Visualization '97, pp. 167-173, 1997.
[3] C. Ware, Information Visualization: Perception for Design, second ed. Morgan Kaufman, 2004.
[4] R.L. Boyell and H. Ruston, “Hybrid Techniques for Real-Time Radar Simulation,” Proc. 1963 Fall Joint Computer Conf., pp. 445-458, 1963.
[5] J.W. Milnor, Morse Theory. Princeton Univ. Press, 1963.
[6] H. Carr and J. Snoeyink, “Path Seeds and Flexible Isosurfaces Using Topology for Exploratory Visualization,” VISSYM '03: Proc. Symp. Data Visualisation, pp. 49-58, 2003.
[7] Y.-J. Chiang and X. Lu, “Progressive Simplification of Tetrahedral Meshes Preserving All Isosurface Topologies,” Computer Graphics Forum, vol. 22, no. 3, pp. 493-504, 2003.
[8] H. Carr, J. Snoeyink, and M. van de Panne, “Simplifying Flexible Isosurfaces Using Local Geometric Measures,” Proc. IEEE Visualization, pp. 497-504, 2004.
[9] V. Pascucci, M. Cole-McLaughlin, and G. Scorzelli, “Multi-Resolution Representation of Topology,” Proc. IASTED Visualization, Imaging, and Image Processing, pp. 452-290, 2004.
[10] S. Takahashi, G.M. Nielson, Y. Takeshima, and I. Fujishiro, “Topological Volume Skeletonization Using Adaptive Tetrahedralization,” Proc. Geometric Modelling and Processing, 2004.
[11] X. Zhang, C.L. Bajaj, and N. Baker, “Fast Matching of Volumetric Functions Using Multi-Resolution Dual Contour Trees,” technical report, Texas Inst. for Computational and Applied Math., 2004.
[12] D. Schneider, A. Wiebel, H. Carr, M. Hlawitschka, and G. Scheuermann, “Interactive Comparison of Scalar Fields Based on Largest Contours with Applications to Flow Visualization,” Proc. IEEE Visualization, pp. 1475-1482, 2008.
[13] Y. Takeshima, S. Takahashi, I. Fujishiro, and G.M. Nielson, “Introducing Topological Attributes for Objective-Based Visualization of Simulated Datasets,” Proc. Int'l Workshop Vol. Graphics, pp. 137-236, 2005.
[14] G. Weber, S. Dillard, H. Carr, V. Pascucci, and B. Hamann, “Topology-Controlled Volume Rendering,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 2, pp. 330-341, Mar./Apr. 2007.
[15] Y. Shinagawa, T.L. Kunii, and Y.L. Kergosien, “Surface Coding Based on Morse Theory,” IEEE Computer Graphics and Applications, vol. 11, no. 5, pp. 66-78, Sept. 1991.
[16] E.R. Gansner, E. Koutsofios, S.C. North, and K.-P. Vo, “A Technique for Drawing Directed Graphs,” IEEE Trans. Software Eng., vol. 19, no. 3, pp. 214-230, Mar. 1993.
[17] G. Weber, P.-T. Bremer, and V. Pascucci, “Topological Landscapes: A Terrain Metaphor for Scientific Data,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1416-1423, Nov./Dec. 2007.
[18] H. Doraiswamy and V. Natarajan, “Efficient Output-Sensitive Construction of Reeb Graphs,” Proc. Int'l Symp. Algorithms and Computation (ISAAC), S.-H. Hong, H. Nagamochi, and T. Fukunaga, eds., pp. 556-567, 2008.
[19] H. Doraiswamy and V. Natarajan, “Efficient Algorithms for Computing Reeb Graphs,” Computational Geometry, vol. 42, nos. 6/7, pp. 606-616, 2009.
[20] D. Auber, “Tulip - A Huge Graph Visualization Framework,” Graph Drawing Software, M. Junger and P. Mutzel, eds., pp. 105-126, Springer-Verlag, 2004.
[21] S. Takahashi, I. Fujishiro, and M. Okada, “Applying Manifold Learning to Plotting Approximate Contour Trees,” IEEE Trans. Visualization and Computer Graphics, vol. 15, no. 6, pp. 1185-1192, Nov./Dec. 2009.
[22] J.B. Tenenbaum, V.d. Silva, and J.C. Langford, “A Global Geometric Framework for Nonlinear Dimensionality Reduction.”
[23] G.D. Battista, P. Eades, R. Tamassia, and I. Tollis, Graph Drawing. Prentice Hall, 1998.
[24] I. Herman, G. Melançon, and M.S. Marshall, “Graph Visualization and Navigation in Information Visualization: A Survey,” IEEE Trans. Visualization and Computer Graphics, vol. 6, no. 1, pp. 24-43, Jan.-Mar. 2000.
[25] E. Reingold and J. Tilford, “Tidier Drawings of Trees,” IEEE Trans. Software Eng., vol. SE-7, no. 2, pp. 223-228, Mar. 1981.
[26] P. Eades, “Drawing Free Trees,” Bull. of the Inst. of Combinatorics and Its Applications, vol. 5, pp. 10-36, 1992.
[27] J. Hartigan, Clustering Algorithms. Wiley, 1975.
[28] C.S. Parr, B. Lee, D. Campbell, and B.B. Bederson, “Visualizations for Taxonomic and Phylogenetic Trees,” Bioinformatics, vol. 20, pp. 2997-3004, 2004.
[29] K. Sugiyama, S. Tagawa, and M. Toda, “Methods for Visual Understanding of Hierarchical Systems,” IEEE Trans. Systems, Man, and Cybernetics, vol. 11, no. 2, pp. 109-125, 1981.
[30] A. Frick, “Upper Bounds on the Number of Hidden Nodes in Sugiyama's Algorithm,” Graph Drawing, S.C. North, ed., pp. 169-183, Springer, 1996.
[31] M. Eiglsperger, M. Siebenhaller, and M. Kaufmann, “An Efficient Implementation of Sugiyama's Algorithm for Layered Graph Drawing,” Graph Drawing, J. Pach, ed., pp. 155-166, Springer, 2004.
[32] T. Kamada and S. Kawai, “An Algorithm for Drawing General Undirected Graphs,” Information Processing Letters, vol. 31, no. 1, pp. 7-15, 1989.
[33] E.R. Gansner, Y. Koren, and S.C. North, “Graph Drawing by Stress Majorization,” Graph Drawing, J. Pach, ed., pp. 239-250, Springer, 2004.
[34] Y. Koren and D. Harel, “Axis-by-Axis Stress Minimization,” Graph Drawing, G. Liotta, ed., pp. 450-459, Springer, 2003.
[35] D. Harel and Y. Koren, “Graph Drawing by High-Dimensional Embedding,” J. Graph Algorithms and Applications, vol. 8, no. 2, pp. 195-214, 2004.
[36] J. Tierny, A. Gyulassy, E. Simon, and V. Pascucci, “Loop Surgery for Volumetric Meshes: Reeb Graphs Reduced to Contour Trees,” IEEE Trans. Visualization and Computer Graphics, vol. 15, no. 6, pp. 1177-1184, Nov./Dec. 2009.
[37] R. Wiese, M. Eiglsperger, and M. Kaufmann, “Yfiles: Visualization and Automatic Layout of Graphs,” Graph Drawing, pp. 453-454, Springer, 2001.
[38] E.R. Gansner and S.C. North, “An Open Graph Visualization System and Its Applications to Software Engineering,” Software—Practice and Experience, vol. 30, pp. 1203-1233, 1999.
[39] M. Galassi, GNU Scientific Library Reference Manual, third ed.
[40] H.C. Purchase, “Which Aesthetic Has the Greatest Effect on Human Understanding?,” Graph Drawing, G.D. Battista, ed., pp. 248-261, Springer, 1997.
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