Towards Multifield Scalar Topology Based on Pareto Optimality
How can the notion of topological structures for single scalar fields be extended to multifields? In this paper we propose a definition for such structures using the concepts of Pareto optimality and Pareto dominance. Given a set of piecewise‐linear, scalar functions over a common simplical complex...
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| Vydané v: | Computer graphics forum Ročník 32; číslo 3pt3; s. 341 - 350 |
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| Hlavní autori: | , , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Oxford, UK
Blackwell Publishing Ltd
01.06.2013
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | How can the notion of topological structures for single scalar fields be extended to multifields? In this paper we propose a definition for such structures using the concepts of Pareto optimality and Pareto dominance. Given a set of piecewise‐linear, scalar functions over a common simplical complex of any dimension, our method finds regions of “consensus” among single fields’ critical points and their connectivity relations. We show that our concepts are useful to data analysis on real‐world examples originating from fluid‐flow simulations; in two cases where the consensus of multiple scalar vortex predictors is of interest and in another case where one predictor is studied under different simulation parameters. We also compare the properties of our approach with current alternatives. |
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| AbstractList | How can the notion of topological structures for single scalar fields be extended to multifields? In this paper we propose a definition for such structures using the concepts of Pareto optimality and Pareto dominance. Given a set of piecewise‐linear, scalar functions over a common simplical complex of any dimension, our method finds regions of “consensus” among single fields’ critical points and their connectivity relations. We show that our concepts are useful to data analysis on real‐world examples originating from fluid‐flow simulations; in two cases where the consensus of multiple scalar vortex predictors is of interest and in another case where one predictor is studied under different simulation parameters. We also compare the properties of our approach with current alternatives. How can the notion of topological structures for single scalar fields be extended to multifields? In this paper we propose a definition for such structures using the concepts of Pareto optimality and Pareto dominance. Given a set of piecewise-linear, scalar functions over a common simplical complex of any dimension, our method finds regions of "consensus" among single fields' critical points and their connectivity relations. We show that our concepts are useful to data analysis on real-world examples originating from fluid-flow simulations; in two cases where the consensus of multiple scalar vortex predictors is of interest and in another case where one predictor is studied under different simulation parameters. We also compare the properties of our approach with current alternatives [PUBLICATION ABSTRACT]. |
| Author | Huettenberger, L. Garth, C. Scheuermann, G. Carr, H. Heine, C. |
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| References_xml | – reference: Lampe O. D., Correa C., Ma K.-L., Hauser H.: Curve-centric volume reformation for comparative visualization. IEEE Transactions on Visualization and Computer Graphics 15, 6 (2009), 1235-1242. 2. – reference: Edelsbrunner H., Harer J., Mascarenhas A., Pascucci V., Snoeyink J.: Time-varying Reeb graphs for continuous space-time data. Computational Geometry 41, 3 (2008), 149-166. 8. – reference: Bremer P.-T., Bringa E. M., Duchaineau M. A., Gyulassy A. G., Laney D., Mascarenhas A., Pascucci V.: Topological feature extraction and tracking. Journal of Physics: Conference Series 78 (2007). 2. – reference: Kniss J., Kindlmann G., Hansen C. D.: Interactive volume rendering using multi-dimensional transfer functions and direct manipulation widgets. In Proc. of Visualization 2001 (2001), pp. 255-262, 562. 2. – reference: Carr H., Snoeyink J.: Path seeds and flexible isosurfaces using topology for exploratory visualization. In VISSYM '03: Proceedings of the symposium on Data visualisation 2003 (2003), pp. 49-58. 2. – reference: Jänicke H., Böttinger M., Scheuermann G.: Brushing of attribute clouds for the visualization of multivariate data. IEEE Transactions on Visualization and Computer Graphics 14, 6 (2008), 1459-1466. 2. – reference: Jeong J., Hussain F.: On the Identification of a Vortex. Journal of Fluid Mechanics 285 (1995), 69-94. 6. – reference: De Berg M., Cheong O., Van Kreveld M., Overmars M.: Computational Geometry: Algorithms and Applications, 3rd. ed. Springer, 2008. 4. – reference: Sauber N., Theisel H., Seidel H.-P.: Multifieldgraphs: An approach to visualizing correlations in multifield scalar data. IEEE Transactions on Visualization and Computer Graphics 12, 5 (2006), 917-924. 2. – reference: Nagaraj S., Natarajan V., Nanjundiah R. S.: A gradient-based comparison measure for visual analysis of multifield data. Comput. Graph. Forum 30, 3 (2011), 1101-1110. 2. – reference: Weber G. H., Dillard S. E., Carr H., Pascucci V., Hamann B.: Topology-controlled volume rendering. IEEE Transactions on Visualization and Computer Graphics 13, 2 (2007), 330-341. 2. – reference: Nagaraj S., Natarajan V.: Relation-aware isosurface extraction in multifield data. IEEE Transactions on Visualization and Computer Graphics 17, 2 (2011), 182-191. 2. – reference: Fuchs R., Waser J., Gröller M. E.: Visual human+machine learning. IEEE Transactions on Visualization and Computer Graphics 15, 6 (2009), 1327-1334. 2. – reference: Popinet S.: Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 2 (2003), 572-600. 6. – reference: Stadler P. F., Flamm C.: Barrier Trees on Poset-Valued Landscapes. Genetic Programming and Evolvable Machines 4 (2003), 7-20. 2. – reference: Weber G., Bremer P.-T., Day M., Bell J., Pascucci V.: Feature tracking using Reeb graphs. Mathematics and Visualization. Springer, 2011, pp. 241-253. 2. – reference: Banchoff T.: Critical points and curvature for embedded polyhedral surfaces. Am. Math. Monthly 77 (1970), 475-485. 2. – reference: Bremer P.-T., Weber G., Pascucci V., Day M., Bell J.: Analyzing and tracking burning structures in lean pre-mixed hydrogen flames. IEEE Transactions on Visualization and Computer Graphics 16, 2 (2010), 248-260. 2. – reference: Edelsbrunner H., Harer J.: Computational Topology - an Introduction. American Mathematical Society, 2010. 2. – reference: Szymczak A.: Stable Morse Decompositions for Piece-wise Constant Vector Fields on Surfaces. Eurographics / IEEE Symposium on Visualisation 30, 3 (2011), 1-10. 8, 9. – reference: Houh Chen C., Härdle W., Unwin A.: Handbook of Data Visualization. Springer Handbooks Comp. Statistics. Springer, 2008. 2. – reference: Woodring J., Shen H.-W.: Multi-variate, time varying, and comparative visualization with contextual cues. IEEE Transactions on Visualization and Computer Graphics 12 (2006), 909-916. 2. – reference: Schneider D., Wiebel A., Carr H., Hlawitschka M., Scheuermann G.: Interactive comparison of scalar fields based on largest contours with applications to flow visualization. IEEE Transactions on Visualization and Computer Graphics 14, 6 (2008), 1475-1482. 8. – reference: Johnson C., Hansen C.: Visualization Handbook. Academic Press, Inc., Orlando , FL , USA , 2004. 4. – reference: Chiang Y.-J., Lenz T., Lu X., Rote G.: Simple and optimal output-sensitive construction of contour trees using monotone paths. 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| Snippet | How can the notion of topological structures for single scalar fields be extended to multifields? In this paper we propose a definition for such structures... |
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| SubjectTerms | Analysis and systems Computational fluid dynamics Computer graphics Computer simulation Data processing Dominance I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems languages Mathematical analysis Pareto optimality Scalars Studies Topological manifolds Topology Visualization |
| Title | Towards Multifield Scalar Topology Based on Pareto Optimality |
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