Abstract
We introduce a new method that identifies and tracks features in arbitrary dimensions using the merge tree—a structure for identifying topological features based on thresholding in scalar fields. This method analyzes the evolution of features of the function by tracking changes in the merge tree and relates features by matching subtrees between consecutive time steps. Using the time-varying merge tree, we present a structural visualization of the changing function that illustrates both features and their temporal evolution. We demonstrate the utility of our approach by applying it to temporal cluster analysis of high-dimensional point clouds.
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
Bache, K., Lichman, M.: UCI machine learning repository. http://archive.ics.uci.edu/ml (2013)
Bhaniramka, P., Wenger, R., Crawfis, R.: Isosurface construction in any dimension using convex hulls. IEEE Trans. Vis. Comput. Graph. 10(2), 130–141 (2004)
Bremer, P., Bringa, E., Duchaineau, M., Gyulassy, A., Laney, D., Mascarenhas, A., Pascucci, V.: Topological feature extraction and tracking. J. Phys. Conf. Ser. 78(1), 012007 (2007)
Bremer, P.T., Weber, G.H., Pascucci, V., Day, M., Bell, J.B.: Analyzing and tracking burning structures in lean premixed hydrogen flames. IEEE Trans. Vis. Comput. Graph. 16(2), 248–260 (2010)
Bremer, P.T., Weber, G.H., Tierny, J., Pascucci, V., Day, M., Bell, J.: Interactive exploration and analysis of large-scale simulations using topology-based data segmentation. IEEE Trans. Vis. Comput. Graph. 17(9), 1307–1324 (2011)
Carr, H., Snoeyink, J., Axen, U.: Computing contour trees in all dimensions. Comput. Geom. 24(2), 75–94 (2003)
Carr, H., Snoeyink, J., van de Panne, M.: Simplifying flexible isosurfaces using local geometric measures. In: Proceedings of IEEE Visualization, pp. 497–504. IEEE Computer Society, Los Alamitos (2004)
Chen, F., Obermaier, H., Hagen, H., Hamann, B., Tierny, J., Pascucci, V.: Topology analysis of time-dependent multi-fluid data using the Reeb graph. Comput. Aided Geom. Des. 30(6), 557–566 (2013)
Cohen-Steiner, D., Edelsbrunner, H., Morozov, D.: Vines and vineyards by updating persistence in linear time. In: Proceedings of Symposium on Computational Geometry, pp. 119–126. Association for Computing Machinery, New York (2006)
Edelsbrunner, H., Harer, J.: Jacobi sets of multiple Morse functions. In: Cucker, F., DeVore, R., Olver, P., Suli, E. (eds.) Foundations of Computational Mathematics. Minneapolis, pp. 37–57. Cambridge University Press, Cambridge (2004)
Edelsbrunner, H., Mücke, E.P.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph. 9(1), 66–104 (1990)
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discret. Comput. Geom. 28(4), 511–533 (2002)
Edelsbrunner, H., Harer, J., Mascarenhas, A., Pascucci, V., Snoeyink, J.: Time-varying Reeb graphs for continuous space-time data. Comput. Geom. 41(3), 149–166 (2008)
Fukunaga, K.: Introduction to Statistical Pattern Recognition, 2nd edn. Academic Press Professional, Inc., San Diego, CA (1990)
Harvey, W., Wang, Y.: Topological landscape ensembles for visualization of scalar-valued functions. Comput. Graph. Forum 29(3), 993–1002 (2010)
Ji, G., Shen, H.W.: Feature tracking using Earth Mover’s distance and global optimization. In: Proceedings of Pacific Graphics. Poster (2006)
Keller, P., Bertram, M.: Modeling and visualization of time-varying topology transitions guided by hyper Reeb graph structures. In: Proceedings of IASTED International Conference on Computer Graphics and Imaging, pp. 15–25 (2007)
Mascarenhas, A., Snoeyink, J.: Implementing time-varying contour trees. In: Proceedings of Symposium on Computational Geometry, pp. 370–371. Association for Computing Machinery, New York (2005)
Mascarenhas, A., Snoeyink, J.: Isocontour based visualization of time-varying scalar fields. In: Möller, T., Hamann, B., Russell, R.D. (eds.) Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration, pp. 41–68. Springer, Berlin (2009)
Oesterling, P., Heine, C., Jänicke, H., Scheuermann, G., Heyer, G.: Visualization of high-dimensional point clouds using their density distribution’s topology. IEEE Trans. Vis. Comput. Graph. 17(11), 1547–1559 (2011)
Oesterling, P., Heine, C., Weber, G.H., Scheuermann, G.: Visualizing nD point clouds as topological landscape profiles to guide local data analysis. IEEE Trans. Vis. Comput. Graph. 19(3), 514–526 (2013)
Reinders, F., Post, F.H., Spoelder, H.J.: Visualization of time-dependent data with feature tracking and event detection. Vis. Comput. 17(1), 55–71 (2001)
Salton, G., Wong, A., Yang, C.S.: A vector space model for automatic indexing. Commun. ACM 18(11), 613–620 (1975)
Sohn, B.S., Bajaj, C.: Time-varying contour topology. IEEE Trans. Vis. Comput. Graph. 12(1), 14–25 (2006)
Szymczak, A.: Subdomain aware contour trees and contour evolution in time-dependent scalar fields. In: Shape Modeling and Applications, pp. 136–144. IEEE Computer Society, Los Alamitos (2005)
Turkay, C., Parulek, J., Reuter, N., Hauser, H.: Interactive visual analysis of temporal cluster structures. Comput. Graph. Forum 30(3), 711–720 (2011)
Weber, G.H., Bremer, P.T., Pascucci, V.: Topological landscapes: a terrain metaphor for scientific data. IEEE Trans. Vis. Comput. Graph. 13(6), 1416–1423 (2007)
Weber, G.H., Bremer, P.T., Day, M., Bell, J., Pascucci, V.: Feature tracking using Reeb graphs. In: Pascucci, V., Tricoche, X., Hagen, H., Tierny, J. (eds.) Topological Methods in Data Analysis and Visualization, pp. 241–253. Springer, New York (2011)
Widanagamaachchi, W., Christensen, C., Pascucci, V., Bremer, P.T.: Interactive exploration of large-scale time-varying data using dynamic tracking graphs. In: IEEE Symposium on Large Data Analysis and Visualization, pp. 9–17 (2012)
Acknowledgements
The authors thank anonymous reviewers for valuable comments and assistance in revising the paper. This work was supported by a grant from the German Research Foundation (DFG) within the strategic research initiative on Scalable Visual Analytics (SPP 1335). This work was supported by the Director, Office of Advanced Scientific Computing Research, Office of Science, of the U.S. DOE under Contract No. DE-AC02-05CH11231 (Lawrence Berkeley National Laboratory) through the grant “Topology-based Visualization and Analysis of High-dimensional Data and Time-varying Data at the Extreme Scale”, program manager Lucy Nowell.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Oesterling, P., Heine, C., Weber, G.H., Morozov, D., Scheuermann, G. (2017). Computing and Visualizing Time-Varying Merge Trees for High-Dimensional Data. In: Carr, H., Garth, C., Weinkauf, T. (eds) Topological Methods in Data Analysis and Visualization IV. TopoInVis 2015. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-44684-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-44684-4_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-44682-0
Online ISBN: 978-3-319-44684-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)