Abstract
In this work, we present concepts for the analysis of the evolution of two-dimensional skeletons. By introducing novel persistence concepts, we are able to reduce typical temporal incoherence, and provide insight in skeleton dynamics. We exemplify our approach by means of a simulation of viscous fingering—a highly dynamic process whose analysis is a hot topic in porous media research.
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References
Berndt, D.J., Clifford, J.: Using dynamic time warping to find patterns in time series. Technical Report WS-94-03, AAAI (1994)
Carstens, C., Horadam, K.: Persistent homology of collaboration networks. Math. Prob. Eng. 2013, 815,035:1–815,035:7 (2013)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Mileyko, Y.: Lipschitz functions have Lp-stable persistence. Found. Comput. Math. 10(2), 127–139 (2010)
Comaniciu, D., Meer, P.: Mean shift: a robust approach toward feature space analysis. IEEE Trans. Pattern Anal. Mach. Intell. 24(5), 603–619 (2002)
Delgado-Friedrichs, O., Robins, V., Sheppard, A.: Skeletonization and partitioning of digital images using discrete Morse theory. IEEE Trans. Pattern Anal. Mach. Intell. 37(3), 654–666 (2015)
Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. AMS, Providence (2010)
Edelsbrunner, H., Letscher, D., Zomorodian, A.J.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002)
Edelsbrunner, H., Morozov, D., Pascucci, V.: Persistence-sensitive simplification of functions on 2-manifolds. In: Proceedings of the 22nd Annual Symposium on Computational Geometry, pp. 127–134. ACM Press, New York (2006)
Farin, G.: Curves and Surfaces for Computer-aided Geometric Design: A Practical Guide, 3rd edn. Elsevier, New York (1993)
Kurlin, V.: A one-dimensional homologically persistent skeleton of an unstructured point cloud in any metric space. Comput. Graph. Forum 34(5), 253–262 (2015)
Lam, L., Lee, S.W., Suen, C.Y.: Thinning methodologies – a comprehensive survey. IEEE Trans. Pattern Anal. Mach. Intell. 14(9), 869–885 (1992)
Lukasczyk, J., Aldrich, G., Steptoe, M., Favelier, G., Gueunet, C., Tierny, J., Maciejewski, R., Hamann, B., Leitte, H.: Viscous fingering: a topological visual analytic approach. In: Physical Modeling for Virtual Manufacturing Systems and Processes. Applied Mechanics and Materials, vol. 869, pp. 9–19 (2017)
Rieck, B., Leitte, H.: Exploring and comparing clusterings of multivariate data sets using persistent homology. Comput. Graph. Forum 35(3), 81–90 (2016)
Rieck, B., Mara, H., Leitte, H.: Multivariate data analysis using persistence-based filtering and topological signatures. IEEE Trans. Vis. Comput. Graph. 18(12), 2382–2391 (2012)
Viscous fingering displacement. https://www.youtube.com/watch?v=NZEB8tQ3eOM
Zhang, T.Y., Suen, C.Y.: A fast parallel algorithm for thinning digital patterns. Commun. ACM 27(3), 236–239 (1984)
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Rieck, B., Sadlo, F., Leitte, H. (2020). Persistence Concepts for 2D Skeleton Evolution Analysis. In: Carr, H., Fujishiro, I., Sadlo, F., Takahashi, S. (eds) Topological Methods in Data Analysis and Visualization V. TopoInVis 2017. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-030-43036-8_9
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