Abstract
Approaches for computing geometrically accurate Morse-Smale complexes with discrete Morse theory utilize a probabilistic approach to mitigate the negative impact of restricting flow to the edges of a mesh. The most accurate technique (Gyulassy et al., IEEE Trans Vis Comput Graph 18:2014–2022, 2012) builds a discrete gradient field by computing the probability that a cell belongs to any ascending manifold. The limiting factor to the scalability and parallelism in this approach is the reliance on processing cells in order of increasing function values, in practice, involving a global sort of the data. In this paper, we generalize the approach to operate on gradient vector fields, replacing the sorted ordering with a dependency graph. We present a technique to convert a continuous gradient field to a dependency graph with weights on edges, and adapt the accurate Morse-Smale complex algorithm to this setting. We demonstrate the utility of our technique by computing accurate segmentations for both gradient vector fields obtained by sampling scalar functions as well as rotation-free vector fields.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bremer, P.-T., Weber, G., Pascucci, V., Day, M., Bell, J.: Analyzing and tracking burning structures in lean premixed hydrogen flames. IEEE Trans. Vis. Comput. Graph. 16(2), 248–260 (2010)
Cayley, A.: On contour and slope lines. Lond. Edinb. Dublin Philos. Mag. J. Sci. XVIII, 264–268 (1859)
Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics. Springer, Berlin (1993)
Chen, G., Mischaikow, K., Laramee, R.S., Zhang, E.: Efficient Morse decompositions of vector fields. IEEE Trans. Vis. Comput. Grap. 14(4), 848–862 (2008)
Edelsbrunner, H., Mücke, E.P.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph. 9, 66–104 (1990)
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, FOCS ’00, pp. 454–463. IEEE Computer Society, Washington (2000)
Edelsbrunner, H., Harer, J., Zomorodian, A.: Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds. Discrete Comput. Geom. 30, 87–107 (2003)
Forman, R.: A user’s guide to discrete Morse theory. In: Séminare Lotharinen de Combinatore, vol. 48 (2002)
Gyulassy, A., Duchaineau, M., Natarajan, V., Pascucci, V., Bringa, E., Higginbotham, A., Hamann, B.: Topologically clean distance fields. IEEE Trans. Comput. Graph. Vis. 13(6), 1432–1439 (2007)
Gyulassy, A., Bremer, P.-T., Pascucci, V., Hamann, B.: A practical approach to Morse-Smale complex computation: scalability and generality. IEEE Trans. Vis. Comput. Graph. 14(6), 1619–1626 (2008)
Gyulassy, A., Bremer, P.T., Pascucci, V.: Computing Morse-Smale complexes with accurate geometry. IEEE Trans. Vis. Comput. Graph. 18(12), 2014–2022 (2012)
Helman, J., Hesselink, L.: Representation and display of vector field topology in fluid flow data sets. IEEE Comput. 22(8), 27–36 (1989)
Kasten, J., Reininghaus, J., Hotz, I., Hege, H.-C.: Two-dimensional time-dependent vortex regions based on the acceleration magnitude. IEEE Trans. Vis. Comput. Graph. 17(12), 2080–2087 (2011)
King, H., Knudson, K., Neza, M.: Generating discrete Morse functions from point data. Exp. Math. 14(4), 435–444 (2005)
Laney, D., Bremer, P.-T., Mascarenhas, A., Miller, P., Pascucci, V.: Understanding the structure of the turbulent mixing layer in hydrodynamic instabilities. IEEE Trans. Vis. Comput. Graph. 12(5), 1052–1060 (2006)
Levine, J.A., Jadhav, S., Bhatia, H., Pascucci, V., Bremer, P.-T.: A quantized boundary representation of 2D flow. Comput. Graph. Forum (EuroVis Proc.) 31(3pt1), 945–954 (2012)
Lewiner, T.: Constructing discrete Morse functions. Master’s thesis, Department of Mathematics, PUC-Rio (2002)
Maxwell, J.C.: On hills and dales. Lond. Edinb. Dublin Philos. Mag. J. Sci. XL, 421–427 (1870)
Polthier, K., Preuß, E.: Identifying vector fields singularities using a discrete Hodge decomposition. In: Hege, H.C., Polthier, K. (eds.) Mathematical Visualization III, pp. 112–134. Springer Verlag (2003)
Reininghaus, J., Hotz, I.: Combinatorial 2d vector field topology extraction and simplification. In: Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications. Mathematics and Visualization, pp. 103–114. Springer, Berlin (2011)
Reininghaus, J., Hotz, I.: Computational discrete Morse theory for divergence-free 2d vector fields. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds.) Topological Methods in Data Analysis and Visualization II. Mathematics and Visualization, pp. 3–14. Springer, Berlin/Heidelberg (2012)
Reininghaus, J., Lowen, C., Hotz, I.: Fast combinatorial vector field topology. IEEE Trans. Vis. Comput. Graph. 17, 1433–1443 (2011)
Reininghaus, J., Gunther, D., Hotz, I., Weinkauf, T., Seidel, H.-P.: Combinatorial gradient fields for 2d images with empirically convergent separatrices, arxiv, 1(1208.6523) (2012)
Robins, V., Wood, P., Sheppard, A.: Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1646–1658 (2011)
Shivashankar, N., Natarajan, V.: Parallel computation of 3d Morse-Smale complexes. Comput. Graph. Forum 31(3pt1), 965–974 (2012)
Shivashankar, N., Natarajan, V.: Parallel computation of 3D Morse-Smale complexes. Comp. Graph. Forum 31(3pt1), 965–974 (2012). doi:10.1111/j.1467-8659.2012.03089.x
Sousbie, T.: The persistent cosmic web and its filamentary structure - i. Theory and implementation. Mon. Not. R. Astron. Soc. 414(1), 350–383 (2011)
Sousbie, T., Colombi, S., Pichon, C.: The fully connected n-dimensional skeleton: probing the evolution of the cosmic web. Mon. Not. R. Astron. Soc. 393(2), 457–477 (2009)
Szymczak, A., Zhang, E.: Robust Morse decompositions of piecewise constant vector fields. IEEE Trans. Vis. Comput. Grap. 18(6), 938–951 (2012)
Tong, Y., Lombeyda, S., Hirani, A.N., Desbrun, M.: Discrete multiscale vector field decomposition. ACM Trans. Graph. 22(3), 445–452 (2003)
Wiebel, A.: Feature detection in vector fields using the Helmholtz-Hodge decomposition. Diplomarbeit, University of Kaiserslautern (2004)
Acknowledgements
This work is supported in part by NSF OCI-0906379, NSF OCI-0904631, DOE/NEUP 120341, DOE/MAPD DESC000192, DOE/LLNL B597476, DOE/Codesign P01180734, and DOE/SciDAC DESC0007446.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gyulassy, A., Bhatia, H., Bremer, PT., Pascucci, V. (2015). Computing Accurate Morse-Smale Complexes from Gradient Vector Fields. In: Bennett, J., Vivodtzev, F., Pascucci, V. (eds) Topological and Statistical Methods for Complex Data. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44900-4_12
Download citation
DOI: https://doi.org/10.1007/978-3-662-44900-4_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44899-1
Online ISBN: 978-3-662-44900-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)