Spatiotemporal Barcodes for Image Sequence Analysis

  • Rocio Gonzalez-DiazEmail author
  • Maria-Jose Jimenez
  • Belen Medrano
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9448)


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.


Persistent homology Barcodes Spatiotemporal data Digital image sequence analysis 



We want to thank the valuable suggestions and comments made by the reviewers to improve the final version of this paper.


  1. 1.
    Adams, H., Carlsson, G.: Evasion paths in mobile sensor networks. I. J. Rob. Res. 34(1), 90–104 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Carlsson, G.E., de Silva, V.: Zigzag persistence. Found. Comput. Math. 10(4), 367–405 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: FOCS 2000, pp. 454–463. IEEE Computer Society (2000)Google Scholar
  4. 4.
    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)zbMATHCrossRefGoogle Scholar
  5. 5.
    Edelsbrunner, H., Harer, J.: Computational Topology - An Introduction. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  6. 6.
    Gamble, J., Chintakunta, H., Krim, H.: Coordinate-free quantification of coverage in dynamic sensor networks. Sign. Proces. 114, 1–18 (2015)CrossRefGoogle Scholar
  7. 7.
    Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45, 61–75 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    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) CrossRefGoogle Scholar
  9. 9.
    Gonzalez-Diaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Appl. Math. 147(2–3), 245–263 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  11. 11.
    Munkres, J.: Elements of Algebraic Topology. Addison-Wesley Co., Reading (1984) zbMATHGoogle Scholar
  12. 12.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Rocio Gonzalez-Diaz
    • 1
    Email author
  • Maria-Jose Jimenez
    • 1
  • Belen Medrano
    • 1
  1. 1.Department of Applied Mathematics (I)University of SevilleSevilleSpain

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