Incremental-Decremental Algorithm for Computing AT-Models and Persistent Homology

  • Rocio Gonzalez-Diaz
  • Adrian Ion
  • Maria Jose Jimenez
  • Regina Poyatos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6854)


In this paper, we establish a correspondence between the incremental algorithm for computing AT-models [8,9] and the one for computing persistent homology [6,14,15]. We also present a decremental algorithm for computing AT-models that allows to extend the persistence computation to a wider setting. Finally, we show how to combine incremental and decremental techniques for persistent homology computation.


Persistent homology AT-model for computing homology cell complex 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B.: Size functions for comparing 3D models. Pattern Recogn. 41(9), 2855–2873 (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Carlsson, G., de Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: SoCG 2009, pp. 247–256. ACM, New York (2009)Google Scholar
  3. 3.
    Chazal, F., Cohen-Steiner, D., Guibas, L., Mémoli, F., Oudot, S.: Gromov-Hausdorff Stable Signatures for Shapes using Persistence. Comput. Graph. Forum 28(5), 1393–1403 (2009)CrossRefGoogle Scholar
  4. 4.
    Chazal, F., Guibas, L., Oudot, S., Skraba, P.: Analysis of scalar fields over point cloud data. In: Proc. of SODA 2009, pp. 1021–2030 (2009)Google Scholar
  5. 5.
    Cerri, A., Ferri, M., Giorgi, D.: Retrieval of trademark images by means of size functions. Graph. Models 68(5), 451–471 (2006)CrossRefGoogle Scholar
  6. 6.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: FOCS 2000, pp. 454–463. IEEE Computer Society, Los Alamitos (2000)Google Scholar
  7. 7.
    Ferri, M., Stanganelli, I.: Size functions for the morphological analysis of melanocytic lesions. Journal of Biomedical Imaging 5, 1–5 (2010)Google Scholar
  8. 8.
    Gonzalez-Diaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Applied Math. 147(2-3), 245–263 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gonzalez-Diaz, R., Medrano, B., Real, P., Sanchez-Pelaez, A.: Simplicial perturbation techniques and effective homology. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194, pp. 166–177. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Gonzalez-Diaz, R., Jimenez, M., Medrano, B., Real, P.: A tool for integer homology computation: Lambda-AT model. Image and Vision Computing, 1–9 (2008)Google Scholar
  11. 11.
    Gonzalez-Diaz, R., Jimenez, M., Medrano, B., Molina, H., Real, P.: Integral operators for computing homology generators at any dimension. In: Ruiz-Shulcloper, J., Kropatsch, W.G. (eds.) CIARP 2008. LNCS, vol. 5197, pp. 356–363. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  13. 13.
    Munkres, J.: Elements of Algebraic Topology. Addison-Wesley Co., Reading (1984)zbMATHGoogle Scholar
  14. 14.
    Zomorodian, A.: Topology for Computing. Cambridge Monographs on Applied and Computational Mathematics (16) (2005)Google Scholar
  15. 15.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete and Computational Geometry 33(2), 249–274 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Rocio Gonzalez-Diaz
    • 1
  • Adrian Ion
    • 2
    • 3
  • Maria Jose Jimenez
    • 1
  • Regina Poyatos
    • 1
  1. 1.Applied Math Dep. (I), School of Computer EngineeringUniversity of SevilleSpain
  2. 2.Pattern Recognition and Image Processing GroupVienna University of TechnologyAustria
  3. 3.Institute of Science and TechnologyAustria

Personalised recommendations