Abstract
In this paper we present an efficient framework for computation of persistent homology of cubical data in arbitrary dimensions. An existing algorithm using simplicial complexes is adapted to the setting of cubical complexes. The proposed approach enables efficient application of persistent homology in domains where the data is naturally given in a cubical form. By avoiding triangulation of the data, we significantly reduce the size of the complex. We also present a data-structure designed to compactly store and quickly manipulate cubical complexes. By means of numerical experiments, we show high speed and memory efficiency of our approach. We compare our framework to other available implementations, showing its superiority. Finally, we report performance on selected 3D and 4D data-sets.
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References
Bajaj, C.L., Pascucci, V., Schikore, D.: The contour spectrum. In: Proceedings of IEEE Visualization, pp. 167–174 (1997)
Bendich, P., Edelsbrunner, H., Kerber, M.: Computing robustness and persistence for images. In: Proceedings of IEEE Visualization, vol. 16, pp. 1251–1260 (2010)
Biasotti, S., Cerri, A., Frosini, P., Giorgi, D., Landi, C.: Multidimensional size functions for shape comparison. J. Math. Imaging Vis. 32(2), 161–179 (2008)
Carr, H., Snoeyink, J., van de Panne, M.: Flexible isosurfaces: Simplifying and displaying scalar topology using the contour tree. Comput. Geom. 43(1), 42–58 (2010)
Chen, C., Kerber, M.: An output-sensitive algorithm for persistent homology. In: Proceedings of the 27th annual symposium on Computational geometry 207–215 (2011)
Chen, C., Kerber, M.: Persistent homology computation with a twist. In: 27th European Workshop on Computational Geometry (EuroCG 2011) (2011)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)
Computer Assisted Proofs in Dynamics: CAPD Homology Library, http://capd.ii.uj.edu.pl.
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithms. MIT, MA (2009)
Edelsbrunner, H., Harer, J.: Computational topology, an introduction. American Mathematical Society, RI (2010)
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002)
Freedman D., Chen C.: Algebraic topology for computer vision, Computer Vision. Nova Science (to appear)
Freudenthal, H.: Simplizialzerlegungen von beschränkter Flachheit. Ann. Math. 43(3), 580–582 (1942)
Gyulassy, A., Natarajan, V., Pascucci, V., Hamann, B.: Efficient computation of morse-smale complexes for three-dimensional scalar functions. IEEE Trans. Vis. Comput. Graph. 13(6), 1440–1447 (2007)
Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational homology, vol. 157 of Applied Mathematical Sciences. Springer, Berlin (2004)
Kedenburg, G.: Persistent Cubical Homology. Master’s thesis, University of Hamburg (2010)
Kershner, R.: The number of circles covering a set. Am. J. Math. 61(3), 665–671 (1939)
Milosavljevic, N., Morozov, D., Skraba, P.: Zigzag persistent homology in matrix multiplication time. In: Proceedings of the 27th annual symposium on Computational geometry (2011)
Morozov, D.: Dionysus: A C++ library for computing persistent homology. http://www.mrzv.org/software/dionysus/
Morozov, D.: Persistence algorithm takes cubic time in worst case. BioGeometry News. Dept. Comput. Sci., Duke Univ., Durham, NC (2005)
Mrozek, M., Wanner, T.: Coreduction homology algorithm for inclusions and persistent homology. Comput. Math. Appl. 2812–2833 (2010)
Mrozek, M., Zelawski, M., Gryglewski, A., Han, S., Krajniak, A.: Homological methods for extraction and analysis of linear features in multidimensional images. Pattern Recogn, 45(1), 285–298 (2012). doiL 10.1016/j.patcog.2011.04.020
Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, CA (1984)
Pascucci, V., Scorzelli, G., Bremer, P.-T., Mascarenhas, A.: Robust on-line computation of reeb graphs: simplicity and speed. ACM Trans. Graph. 26(58), 1–8 (2007)
Strömbom, D.: Persistent homology in the cubical setting: theory, implementations and applications. Master’s thesis, Luleå University of Technology (2007)
Zong, C.: What is known about unit cubes. Bull. Am. Math. Soc. 42(2005), 181–211 (2005)
Acknowledgements
This work was supported by the Austrian Science Fund (FWF) grant no. P20134-N13 and the Austrian COMET program. The first author is also supported by the Foundation for Polish Science IPP Programme “Geometry and Topology in Physical Models,” co-financed by the EU European Regional Development Fund, Operational Program Innovative Economy 2007–2013.
The authors would like to thank Prof. Herbert Edelsbrunner and Dr. Michael Kerber for fruitful discussions.
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Wagner, H., Chen, C., Vuçini, E. (2012). Efficient Computation of Persistent Homology for Cubical Data. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds) Topological Methods in Data Analysis and Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23175-9_7
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DOI: https://doi.org/10.1007/978-3-642-23175-9_7
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