Abstract
Taking as input a time-varying sequence of two-dimensional (2D) binary images, we develop an algorithm for computing a spatiotemporal 0–barcode encoding lifetime of connected components on the image sequence over time. This information may not coincide with the one provided by the 0–barcode encoding the 0–persistent homology, since the latter does not respect the principle that it is not possible to move backwards in time. A cell complex K is computed from the given sequence, being the cells of K classified as spatial or temporal depending on whether they connect two consecutive frames or not. A spatiotemporal path is defined as a sequence of edges of K forming a path such that two edges of the path cannot connect the same two consecutive frames. In our algorithm, for each vertex \(v\in K\), a spatiotemporal path from v to the “oldest” spatiotemporally-connected vertex is computed and the corresponding spatiotemporal 0–bar is added to the spatiotemporal 0–barcode.
Author partially supported by IMUS, Junta de Andalucia under grant FQM-369, Spanish Ministry under grant MTM2012-32706 and ESF ACAT program.
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References
Adams, H., Carlsson, G.: Evasion paths in mobile sensor networks. I. J. Rob. Res. 34(1), 90–104 (2015)
Carlsson, G.E., de Silva, V.: Zigzag persistence. Found. Comput. Math. 10(4), 367–405 (2010)
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: FOCS 2000, pp. 454–463. IEEE Computer Society (2000)
de Silva, V., Ghrist, R.: Coordinate-free coverage in sensor networks with controlled boundaries via homology. I. J. Rob. Res. 25(12), 1205–1222 (2006)
Edelsbrunner, H., Harer, J.: Computational Topology - An Introduction. American Mathematical Society, Providence (2010)
Gamble, J., Chintakunta, H., Krim, H.: Coordinate-free quantification of coverage in dynamic sensor networks. Sign. Proces. 114, 1–18 (2015)
Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45, 61–75 (2008)
Gonzalez-Diaz, R., Ion, A., Jimenez, M.J., Poyatos, R.: Incremental-decremental algorithm for computing AT-models and persistent homology. In: Real, P., Diaz-Pernil, D., Molina-Abril, H., Berciano, A., Kropatsch, W. (eds.) CAIP 2011, Part I. LNCS, vol. 6854, pp. 286–293. Springer, Heidelberg (2011)
Gonzalez-Diaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Appl. Math. 147(2–3), 245–263 (2005)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Munkres, J.: Elements of Algebraic Topology. Addison-Wesley Co., Reading (1984)
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)
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We want to thank the valuable suggestions and comments made by the reviewers to improve the final version of this paper.
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Gonzalez-Diaz, R., Jimenez, MJ., Medrano, B. (2015). Spatiotemporal Barcodes for Image Sequence Analysis. In: Barneva, R., Bhattacharya, B., Brimkov, V. (eds) Combinatorial Image Analysis. IWCIA 2015. Lecture Notes in Computer Science(), vol 9448. Springer, Cham. https://doi.org/10.1007/978-3-319-26145-4_5
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DOI: https://doi.org/10.1007/978-3-319-26145-4_5
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