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Issue No.12 - Dec. (2011 vol.17)
pp: 1842-1851
C. Correa , Center for Appl. Sci. Comput. (CASC), Lawrence Livermore Nat. Lab., Livermore, CA, USA
P. Lindstrom , Center for Appl. Sci. Comput. (CASC), Lawrence Livermore Nat. Lab., Livermore, CA, USA
Peer-Timo Bremer , Center for Appl. Sci. Comput. (CASC), Lawrence Livermore Nat. Lab., Livermore, CA, USA
ABSTRACT
We present topological spines-a new visual representation that preserves the topological and geometric structure of a scalar field. This representation encodes the spatial relationships of the extrema of a scalar field together with the local volume and nesting structure of the surrounding contours. Unlike other topological representations, such as contour trees, our approach preserves the local geometric structure of the scalar field, including structural cycles that are useful for exposing symmetries in the data. To obtain this representation, we describe a novel mechanism based on the extraction of extremum graphs-sparse subsets of the Morse-Smale complex that retain the important structural information without the clutter and occlusion problems that arise from visualizing the entire complex directly. Extremum graphs form a natural multiresolution structure that allows the user to suppress noise and enhance topological features via the specification of a persistence range. Applications of our approach include the visualization of 3D scalar fields without occlusion artifacts, and the exploratory analysis of high-dimensional functions.
INDEX TERMS
Topology, Data visualization, Approximation methods, Manifolds, Morse-Smale complex., Scalar field topology, topological spine, extremum graph
CITATION
C. Correa, P. Lindstrom, Peer-Timo Bremer, "Topological Spines: A Structure-preserving Visual Representation of Scalar Fields", IEEE Transactions on Visualization & Computer Graphics, vol.17, no. 12, pp. 1842-1851, Dec. 2011, doi:10.1109/TVCG.2011.244
REFERENCES
[1] D. H. Ackley, A connectionist machine for genetic hillclimbing. Kluwer, Boston, 1987.
[2] R. F. W. Bader, Atoms in Molecules: A Quantum Theory. Oxford University Press, 1994.
[3] K. Beketayev, G. Weber, M. Haranczyk, M. Hlawitschka, P.-T. Bremer, and B. Hamann, Topology-based visualization of transformation pathways in complex chemical systems. In Eurographics Symposium on Visualization, 2011. To appear.
[4] P.-T. Bremer, V. Pascucci, and B. Hamann, Maximizing adaptivity in hierarchical topological models. In Shape Modeling and Applications, pages 298–307, 2005.
[5] P.-T. Bremer, G. Weber, J. Tierny, V. Pascucci, M. Day, and J. B. Bell, Interactive exploration and analysis of large scale simulations using topology-based data segmentation. IEEE Transactions on Visualization and Computer Graphics, 2010. To appear.
[6] H. Carr, J. Snoeyink, and U. Axen, Computing contour trees in all dimensions. Computational Geometry, 24 (2): 75 – 94, 2003.
[7] H. Carr, J. Snoeyink, and M. van de Panne, Flexible isosurfaces: Simplifying and displaying scalar topology using the contour tree. Computational Geometry, 43 (1):42–58, 2010.
[8] F. Chazal, L. Guibas, S. Oudot, and P. Skraba, Analysis of scalar fields over point cloud data. In ACM-SIAM Symposium on Discrete Algorithms, pages 1021–1030, 2009.
[9] H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, Morse-Smale complexes for piecewise linear 3-manifolds. In ACM Symposium on Computational geometry, pages 361–370, 2003.
[10] H. Edelsbrunner, D. Morozov, and V. Pascucci, Persistence-sensitive simplification functions on 2-manifolds. In Proceedings of the twenty-second annual symposium on Computational geometry, SCG '06, pages 127– 134, New York, NY, USA, 2006. ACM.
[11] E. Espinosa, M. Souhassou, H. Lachekar, and C. Lecomte, Topological analysis of the electron density in hydrogen bonds. Acta Crystallograph-ica Section B-structural Science, 55: 563–572, 1999.
[12] I. Fujishiro, Y. Takeshima, T. Azuma, and S. Takahashi, Volume data mining using 3D field topology analysis. IEEE Computer Graphics & Applications, 20 (5): 46–51, 2000.
[13] S. Gerber, P.-T. Bremer, V. Pascucci, and R. T. Whitaker, Visual exploration of high dimensional scalar functions. IEEE Transactions on Visualization and Computer Graphics, 16 (6): 1271–1280, 2010.
[14] A. Gyulassy, P.-T. Bremer, V. Pascucci, and B. Hamann, A practical approach to Morse-Smale complex computation: Scalability and generality. IEEE Transactions on Visualization and Computer Graphics, 14 (6): 1619–1626, 2008.
[15] A. Gyulassy, V. Natarajan, V. Pascucci, P.-T. Bremer, and B. Hamann, Topology-based simplification for feature extraction from 3D scalar fields. IEEE Transactions on Computer Graphics and Visualization, 12 (4): 474–484, 2006.
[16] A. Gyulassy, V. Natarajan, V. Pascucci, and B. Hamann, , Efficient computation of Morse-Smale complexes for three-dimensional scalar functions. IEEE Transactions on Visualization and Computer Graphics, 13 (6): 1440–1447, 2007.
[17] W. Harvey and Y. Wang, Topological landscape ensembles for visualization of scalar-valued functions. Computer Graphics Forum, 29 (3): 993–1002, 2010.
[18] J. L. Helman and L Hesselink, Visualizing vector field topology in fluid flows. IEEE Computer Graphics & Applications, 11 (3):36–46, 1991.
[19] C. K. Johnson, M. N. Burnett, and W. D. Dunbar, Crystallographic Topology and its Applications, pages 1–25. 1996.
[20] D. Laney, P.-T. Bremer, A. Mascarenhas, P. Miller, and V. Pascucci, Understanding the structure of the turbulent mixing layer in hydrodynamic instabilities. IEEE Transactions on Visualization and Computer Graphics, 12 (5): 1052–1060, 2006.
[21] L. Leherte, J. Glasgow, K. Baxter, E. Steeg, and S. Fortier, Analysis of Three-Dimensional Protein Images. Journal of Artificial Intelligence Research, 7: 125–159, 1997.
[22] P. Oesterling, C. Heine, H. Janicke, G. Scheuermann, and G. Heyer, Visualization of high dimensional point clouds using their density distribution's topology. IEEE Transactions on Visualization and Computer Graphics, 2011. To appear.
[23] V. Pascucci, K. Cole-MacLaughlin, and G. Scorzelli, Multi-resolution computation and presentation of contour trees. Technical Report UCRL-PROC-208680, Lawrence Livermore National Laboratory, 2005.
[24] V. Pascucci and K. Cole-McLaughlin, Parallel computation of the topology of level sets. Algorithmica, 38: 249–268, 2003.
[25] V. Pascucci, G. Scorzelli, P.-T. Bremer, and A. Mascarenhas, Robust on-line computation of Reeb graphs: Simplicity and speed. ACM Transactions on Graphics, 26 (3):58. 1–58.9, 2007.
[26] J. Sahner, B. Weber, S. Prohaska, and H. Lamecker, Extraction of feature lines on surface meshes based on discrete Morse theory. Computer Graphics Forum, 27 (3): 735–742, 2008.
[27] H. Schwefel, Numerical optimization of computer models. Wiley, 1981.
[28] Y. Shinagawa and T. Kunii, Constructing a Reeb graph automatically from cross sections. IEEE Computer Graphics & Applications, 11 (5): 44– 51, 1991.
[29] S. Takahashi, Y. Takeshima, and I. Fujishiro, Topological volume skele-tonization and its application to transfer function design. Graphical Models, 66 (1): 24–49, 2004.
[30] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel, Saddle connectors — An approach to visualizing the topological skeleton of complex 3D vector fields. In IEEE Visualization, pages 225–232, 2003.
[31] J. Tierny, A. Gyulassy, E. Simon, and V. Pascucci, Loop surgery for volumetric meshes: Reeb graphs reduced to contour trees. IEEE Transactions on Visualization and Computer Graphics, 15 (6): 1177–1184, 2009.
[32] I. G. Tollis, G. Di Battista, P. Eades, and R. Tamassia, Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, July 1998.
[33] M. J. van Kreveld, R. van Oostrum, C. L. Bajaj, V. Pascucci, and D. Schikore, Contour trees and small seed sets for isosurface traversal. In ACM Symposium on Computational Geometry, pages 212–220, 1997.
[34] G. Weber, P.-T. Bremer, and V. Pascucci, Topological landscapes: A terrain metaphor for scientific data. IEEE Transactions on Visualization and Computer Graphics, 13 (6): 1077– 2626, 2007.
[35] G. Weber, G. Scheuermann, H. Hagen, and B. Hamann, Exploring scalar fields using critical isovalues. In IEEE Visualization, pages 171–178, 2002.
[36] T. Weinkauf and D. Günther, Separatrix persistence: Extraction of salient edges on surfaces using topological methods. Computer Graphics Forum, 28 (5): 1519 –1528, 2009.
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