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Journal of Visualization
Erschienen in: Journal of Visualization 4/2016

01.11.2016 | Regular Paper

From numerics to combinatorics: a survey of topological methods for vector field visualization

verfasst von: Wentao Wang, Wenke Wang, Sikun Li

Erschienen in: Journal of Visualization | Ausgabe 4/2016

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Abstract

Topological methods are important tools for data analysis, and recently receiving more and more attention in vector field visualization. In this paper, we give an introductory description to some important topological methods in vector field visualization. Besides traditional methods of vector field topology, space-time method and finite-time Lyapunov exponent, we also include in this survey Hodge decomposition, combinatorial vector field topology, Morse decomposition, and robustness, etc. In addition to familiar numerical techniques, more and more combinatorial tools emerge in vector field visualization. The numerical methods often rely on error-prone interpolations and interpolations, while combinatorial techniques produce robust but coarse features. In this survey, we clarify the relevant concepts and hope to guide future topological research in vector field visualization.

Graphical abstract

Vorheriger Artikel On-the-fly simplification of large iso-surfaces with per-cube vertex modifiability detection
Nächster Artikel E-Grid: a visual analytics system for evaluation structures

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Fußnoten
1
Actually \(H_i\simeq T_i\times \underbrace{\mathbb {Z}\times \cdots \times \mathbb {Z}}_{\beta _i}\), where \(T_i\) is the torsion subgroup and usually assumed trivial.
 
Literatur
Abraham R, Marsden JE, Ratiu T (2002) Manifolds, tensor analysis, and applications, 3rd edn. Springer, New YorkMATH
Alexander S (2003) Combinatorial optimization: polyhedra and efficiency, vol 24. Springer, New York
Ali S, Shah M (2007) A Lagrangian particle dynamics approach for crowd flow segmentation and stability analysis. In: IEEE conference on computer vision and pattern recognition (CVPR’07). IEEE, pp 1–6
Asimov D (1993) Notes on the topology of vector fields and flows. Tech. rep., Technical report, NASA Ames Research Center. RNR-93-003 (1993)
Batra R, Hesselink L (1999) Feature comparisons of 3-d vector fields using earth mover’s distance. In: Proceedings of the conference on visualization’99: celebrating ten years. IEEE Computer Society Press, pp 105–114
Bhatia H, Norgard G, Pascucci V, Bremer PT (2013) The Helmholtz-Hodge decomposition: a survey. IEEE Trans Vis Comput Graph 19(8):1386–1404CrossRef
Bhatia H, Gyulassy A, Wang H, Bremer PT, Pascucci V (2014a) Robust detection of singularities in vector fields. In: Topological methods in data analysis and visualization III. Springer, New York, pp 3–18
Bhatia H, Pascucci V, Bremer PT (2014b) The natural Helmholtz-Hodge decomposition for open-boundary flow analysis. IEEE Trans Vis Comput Graph 20(11):1566–1578. doi:10.​1109/​TVCG.​2014.​2312012
Bhatia H, Pascucci V, Kirby RM, Bremer PT (2014c) Extracting features from time-dependent vector fields using internal reference frames. In: Computer graphics forum, vol 33. Wiley Online Library, pp 21–30
Caraballo T, Jara JC, Langa JA, Liu Z (2013) Morse decomposition of attractors for non-autonomous dynamical systems. Adv Nonlinear Stud 13(2):309–329MathSciNetMATHCrossRef
Chen CM, Shen HW (2013) Graph-based seed scheduling for out-of-core FTLE and pathline computation. In: 2013 IEEE symposium on large-scale data analysis and visualization (LDAV), pp 15–23. doi:10.​1109/​LDAV.​2013.​6675154
Chen G, Mischaikow K, Laramee RS, Pilarczyk P, Zhang E (2007) Vector field editing and periodic orbit extraction using morse decomposition. IEEE Trans Vis Comput Graph 13(4):769–785CrossRef
Chen G, Mischaikow K, Laramee RS, Zhang E (2008) Efficient morse decompositions of vector fields. IEEE Trans Vis Comput Graph 14(4):848–862CrossRef
Chen X, Xie C, Wei Z (2009) Feature detection and visualization of ocean flow field sources, sinks based on vector field decomposition. In: International conference on image analysis and signal processing (IASP’09), pp 260–264. doi:10.​1109/​IASP.​2009.​5054628
Chen G, Deng Q, Szymczak A, Laramee RS, Zhang E (2012) Morse set classification and hierarchical refinement using conley index. IEEE Trans Vis Comput Graph 18(5):767–782CrossRef
Chorin AJ, Marsden JE (2000) A mathematical introduction to fluid mechanics, 3rd edn. Springer, New YorkMATH
Conley C (1978) Isolated invariant sets and the Morse index, vol 38. American Mathematical Soc., Providence
De Leeuw W, Van Liere R (1999a) Collapsing flow topology using area metrics. In: Proceedings of the conference on visualization’99: celebrating ten years, pp 349–354. IEEE Computer Society Press
De Leeuw W, Van Liere R (1999b) Visualization of global flow structures using multiple levels of topology. In: Data visualization 99. Springer, New York, , pp 45–52
de Leeuw W, van Liere R (2000) Multi-level topology for flow visualization. Comput Graph 24(3):325–331CrossRef
Edelsbrunner H, Mcke EP (1990) Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans Graph (TOG) 9(1):66–104MATHCrossRef
Edelsbrunner H, Morozov D, Patel A (2011) Quantifying transversality by measuring the robustness of intersections. Found Comput Math 11(3):345–361MathSciNetMATHCrossRef
Firby PA, Gardiner CF (2001) Surface topology, 3rd edn. Elsevier, Amsterdam
Fisher M, Schrder P, Desbrun M, Hoppe H (2007) Design of tangent vector fields. In: ACM transactions on graphics (TOG), vol 26. ACM, p 56
Garth C, Tricoche X, Scheuermann G (2004) Tracking of vector field singularities in unstructured 3d time-dependent datasets. In: Proceedings of the conference on visualization’04. IEEE Computer Society, pp 329–336
Globus A, Levit C, Lasinski T (1991) A tool for visualizing the topology of three-dimensional vector fields. In: Proceedings of the 2nd conference on visualization’91 (VIS’91). IEEE Computer Society Press, Los Alamitos, pp 33–40
Green MA, Rowley CW, Haller G (2007) Detection of lagrangian coherent structures in three-dimensional turbulence. J Fluid Mech 572:111–120MathSciNetMATHCrossRef
Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, vol 42. Springer, New York
Guo Q, Mandal MK, Li MY (2005) Efficient Hodge-Helmholtz decomposition of motion fields. Pattern Recognit Lett 26(4):493–501CrossRef
Haller G (2001) Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Phys D Nonlinear Phenom 149(4):248–277MathSciNetMATHCrossRef
Helman J, Hesselink L (1989) Representation and display of vector field topology in fluid flow data sets. Computer 22(8):27–36CrossRef
Helman JL, Hesselink L (1990) Surface representations of two- and three-dimensional fluid flow topology. In: Proceedings of the 1st conference on visualization’90 (VIS’90). IEEE Computer Society Press, Los Alamitos, pp 6–13
Henle M (1979) A combinatorial introduction to topology. Courier Corporation
Hougardy S, Drake D (2004) Approximation algorithms for the weighted matching problem. Tech. Rep. 28, Oberwolfach
Jiang M, Machiraju R, Thompson D (2002) A novel approach to vortex core region detection. In: Data visualization 2002. Proc. VisSym02, vol 15, p 16
Jiang M, Machiraju R, Thompson D (2005) Detection and visualization of vortices. In: The visualization handbook, p 295
Johnson C (2004) Top scientific visualization research problems. Comput Graph Appl IEEE 24(4):13–17CrossRef
Kalies W, Ban H (2006) A computational approach to Conley’s decomposition theorem. J Comput Nonlinear Dyn 1(4):312–319CrossRef
Kalies WD, Mischaikow K, VanderVorst RCAM (2005) An algorithmic approach to chain recurrence. Found Comput Math 5(4):409–449MathSciNetMATHCrossRef
Kasten J, Reininghaus J, Reich W, Scheuermann G (2014) Toward the extraction of saddle periodic orbits. In: Bremer PT, Hotz I, Pascucci V, Peikert R (eds) Topological methods in data analysis and visualization III, mathematics and visualization. Springer International Publishing, New York, pp 55–69
Klein T, Ertl T (2007) Scale-space tracking of critical points in 3d vector fields. In: Topology-based methods in visualization. Springer, New York, pp 35–49
Kuhn A, Rssl C, Weinkauf T, Theisel H (2012) A benchmark for evaluating FTLE computations. In: Pacific visualization symposium (PacificVis), 2012 IEEE. IEEE, pp 121–128
Laramee RS, Hauser H, Doleisch H, Vrolijk B, Post FH, Weiskopf D (2004) The state of the art in flow visualization: dense and texture-based techniques. In: Computer graphics forum, vol 23. Wiley Online Library, pp 203–221
Laramee RS, Hauser H, Zhao L, Post FH (2007) Topology-based flow visualization, the state of the art. In: Hauser H, Hagen H, Theisel H (eds) Topology-based methods in visualization, mathematics and visualization. Springer, Berlin, pp 1–19CrossRef
Lavin Y, Batra R, Hesselink L (1998) Feature comparisons of vector fields using earth mover’s distance. In: Visualization’98. Proceedings. IEEE, pp 103–109
Li WC, Vallet B, Ray N, Lvy B (2006) Representing higher-order singularities in vector fields on piecewise linear surfaces. IEEE Trans Vis Comput Graph 12(5):1315–1322CrossRef
Mann S, Rockwood A (2002) Computing singularities of 3d vector fields with geometric algebra. In: Proceedings of the conference on visualization’02. IEEE Computer Society, pp 283–290
Mase GT, Mase GE (2010) Continuum mechanics for engineers. CRC Press, Boca Raton
Milnor JW (1963) Morse theory, vol 51. Princeton University Press, Princeton
Mischaikow K (1999) The Conley index theory: a brief introduction. Banach Center Publ 47(1):9–19MathSciNetMATH
Peikert R, Sadlo F (2007) Topology-guided visualization of constrained vector fields. In: Topology-based methods in visualization. Springer, New York, pp 21–33
Peikert R, Sadlo F (2009) Topologically relevant stream surfaces for flow visualization. In: Proceedings of the 25th spring conference on computer graphics (SCCG’09). ACM, New York, pp 35–42. doi:10.​1145/​1980462.​1980472
Pobitzer A, Peikert R, Fuchs R, Schindler B, Kuhn A, Theisel H, Matkovic K, Hauser H (2010) On the way towards topology-based visualization of unsteady flow-the state of the art. H. und E. Reinhard (Hrsg.), Eurographics
Polthier K, Preu E (2000) Variational approach to vector field decomposition. In: de Leeuw WC, van Liere R (eds) Data visualization 2000, Eurographics. Springer, New York, pp 147–155CrossRef
Polthier K, Preuss E (2003) Identifying vector field singularities using a discrete hodge decomposition. In: Visualization and mathematics III. Springer, New York, pp 113–134
Post FH, Vrolijk B, Hauser H, Laramee RS, Doleisch H (2003) The state of the art in flow visualisation: feature extraction and tracking. In: Computer graphics forum, vol 22. Wiley Online Library, New York, pp 775–792
Reich W, Schneider D, Heine C, Wiebel A, Chen G, Scheuermann G (2011) Combinatorial vector field topology in 3 dimensions. In: 4th workshop on topology-based methods in data analysis and visualization (TopoInVis2011)
Reininghaus J, Hotz I (2011) Combinatorial 2d vector field topology extraction and simplification. In: Topological methods in data analysis and visualization. Springer, New York, pp 103–114
Reininghaus J, Kasten J, Weinkauf T, Hotz I (2011) Combinatorial feature flow fields: tracking critical points in discrete scalar fields. Tech. rep., Technical Report 11–02, Zuse Institute Berlin
Rosen KH (2000) Handbook of discrete and combinatorial mathematics. CRC Press, Boca Raton
Sadlo F, Peikert R (2007) Efficient visualization of Lagrangian coherent structures by filtered AMR ridge extraction. IEEE Trans Vis Comput Graph 13(6):1456–1463CrossRef
Sadlo F, Peikert R (2009) Visualizing Lagrangian coherent structures and comparison to vector field topology. In: Topology-based methods in visualization II. Springer, New York, pp 15–29
Sadlo F, Rigazzi A, Peikert R (2011) Time-dependent visualization of Lagrangian coherent structures by grid advection. In: Topological methods in data analysis and visualization. Springer, New York, pp 151–165
Salzbrunn T, Jnicke H, Wischgoll T, Scheuermann G (2008) The state of the art in flow visualization: partition-based techniques. In: SimVis, pp 75–92
Scheuermann G, Kruger H, Menzel M, Rockwood AP (1998) Visualizing nonlinear vector field topology. IEEE Trans Vis Comput Graph 4(2):109–116CrossRef
Shadden SC (2006) A dynamical systems approach to unsteady systems. Ph.D. thesis, California Institute of Technology, Pasadena
Shadden SC, Dabiri JO, Marsden JE (2006) Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys Fluids (1994-present) 18(4):047,105
Sipeki L, Szymczak A (2013) Simplification of Morse decompositions using Morse set mergers. Topo-In-Vis 2013
Skraba P, Wang B (2014) Interpreting feature tracking through the lens of robustness. In: Topological methods in data analysis and visualization III. Springer, New York, pp 19–37
Skraba P, Wang B, Chen G, Rosen P (2014) 2D vector field simplification based on robustness. In: 2014 IEEE pacific visualization symposium (PacificVis), pp 49–56. doi:10.​1109/​PacificVis.​2014.​17
Spanier EH (1982) Algebraic topology. McGraw-Hill, Maidenheach, 1966. Springer, New York
Stter T, Weinkauf T, Seidel HP, Theisel H (2012) Implicit integral surfaces. In: Vision, modeling, and visualization
Szymczak A (2011) Stable morse decompositions for piecewise constant vector fields on surfaces. Comput Graph Forum 30(3):851–860CrossRef
Szymczak A (2013) Hierarchy of stable morse decompositions. IEEE Trans Vis Comput Graph 19(5):799–810CrossRef
Szymczak A, Brunhart-Lupo N (2012) Nearly recurrent components in 3d piecewise constant vector fields. Comput Graph Forum 31(3pt3):1115–1124
Theisel H, Seidel HP (2003) Feature flow fields. In: Proceedings of the symposium on data visualisation 2003. Eurographics Association, pp 141–148
Theisel H, Weinkauf T, Hege HC, Seidel HP (2003) Saddle connectors-an approach to visualizing the topological skeleton of complex 3d vector fields. In: Visualization, 2003 (VIS’03). IEEE, pp 225–232
Theisel H, Weinkauf T, Hege HC, Seidel HP (2004) Grid-independent detection of closed stream lines in 2d vector fields. In: VMV, pp 421–428
Thomas W, Scheuermann G (2002) Locating closed streamlines in 3d vector fields. Methods 16:19
Tong Y, Lombeyda S, Hirani AN, Desbrun M (2003) Discrete multiscale vector field decomposition. In: ACM transactions on graphics (TOG), vol 22. ACM, pp 445–452
Tricoche X (2002) Vector and tensor field topology simplification, tracking, and visualization. Ph.D. thesis, University of Kaiserslautern, Kaiserslautern
Tricoche X, Scheuermann G, Hagen H (2000a) Higher order singularities in piecewise linear vector fields. In: The mathematics of surfaces IX. Springer, New York, pp 99–113
Tricoche X, Scheuermann G, Hagen H (2000b) A topology simplification method for 2d vector fields. In: Visualization 2000. Proceedings. IEEE, pp 359–366
Tricoche X, Scheuermann G, Hagen H (2001a) Continuous topology simplification of planar vector fields. In: Proceedings of the conference on visualization’01. IEEE Computer Society, pp 159–166
Tricoche X, Scheuermann G, Hagen H (2001b) Topology-based visualization of time-dependent 2d vector fields. In: Proceedings of the 3rd joint Eurographics-IEEE TCVG conference on visualization. Eurographics Association, pp 117–126
Tricoche X, Scheuermann G, Hagen H, Clauss S (2001c) Vector and tensor field topology simplification on irregular grids. In: Data visualization 2001 (joint Eurographics-IEEE TCVG symposium on visualization proceedings). Springer, New York, pp 101–116
Tricoche X, Wischgoll T, Scheuermann G, Hagen H (2002) Topology tracking for the visualization of time-dependent two-dimensional flows. Comput Graph 26(2):249–257CrossRef
Tricoche X, Garth C, Sanderson A, Joy KI (2012) Visualizing invariant manifolds in area-preserving maps. In: Topological methods in data analysis and visualization II. Springer, New York, pp 109–124
Trotts I, Kenwright D, Haimes R (2000) Critical points at infinity: a missing link in vector field topology. In: Proc. NSF/DoE Lake Tahoe workshop on hierarchical approximation and geometrical methods for scientific visualization, vol 1. Citeseer
Wang B, Rosen P, Skraba P, Bhatia H, Pascucci V (2013) Visualizing robustness of critical points for 2d time-varying vector fields. Comput Graph Forum 32(3pt2):221–230
Weinkauf T, Theisel H (2010) Streak lines as tangent curves of a derived vector field. IEEE Trans Vis Comput Graph 16(6):1225–1234CrossRef
Weinkauf T, Theisel H, Hege HC, Seidel HP (2004) Boundary switch connectors for topological visualization of complex 3d vector fields. In: Proceedings of the sixth joint Eurographics-IEEE TCVG conference on visualization. Eurographics Association, pp 183–192
Weinkauf T, Theisel H, Shi K, Hege HC, Seidel HP (2005) Extracting higher order critical points and topological simplification of 3d vector fields. In: Visualization, 2005 (VIS’05). IEEE, pp 559–566
Weinkauf T, Theisel H, Van Gelder A, Pang A (2011) Stable feature flow fields. IEEE Trans Vis Comput Graph 17(6):770–780CrossRef
Weinkauf T, Hege HC, Theisel H (2012) Advected tangent curves: a general scheme for characteristic curves of flow fields. Comput Graph Forum 31:825–834CrossRef
Wiebel A (2003) Feature detection in vector fields using the Helmholtz-Hodge decomposition. Ph.D. thesis, University of Kaiserslautern, Kaiserslautern
Wiebel A (2008) Localized flow, particle tracing, and topological separation analysis for flow visualization. Ph.D. thesis, Shaker Verlag, Bremen
Wischgoll T, Scheuermann G, Hagen H (2001) Tracking closed streamlines in time dependent planar flows. In: VMV. Citeseer, pp 447–454
Zhang E, Mischaikow K, Turk G (2006) Vector field design on surfaces. ACM Trans Graph (TOG) 25(4):1294–1326CrossRef
Zomorodian A (2012) Topological data analysis. In: Proceedings of symposia in applied mathematics
Metadaten
Titel
From numerics to combinatorics: a survey of topological methods for vector field visualization
verfasst von
Wentao Wang
Wenke Wang
Sikun Li
Publikationsdatum
01.11.2016
Verlag
Springer Berlin Heidelberg
Erschienen in
Journal of Visualization / Ausgabe 4/2016
Print ISSN: 1343-8875
Elektronische ISSN: 1875-8975
DOI
https://doi.org/10.1007/s12650-016-0348-8

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