Skip to main content
Springer Nature Link
Log in

Connectivity Forests for Homological Analysis of Digital Volumes

  • Conference paper
Bio-Inspired Systems: Computational and Ambient Intelligence (IWANN 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5517))

Included in the following conference series:

Abstract

In this paper, we provide a graph-based representation of the homology (information related to the different “holes” the object has) of a binary digital volume. We analyze the digital volume AT-model representation [8] from this point of view and the cellular version of the AT-model [5] is precisely described here as three forests (connectivity forests), from which, for instance, we can straightforwardly determine representative curves of “tunnels” and “holes”, classify cycles in the complex, computing higher (co)homology operations,... Depending of the order in which we gradually construct these trees, tools so important in Computer Vision and Digital Image Processing as Reeb graphs and topological skeletons appear as results of pruning these graphs.

This work has been partially supported by “Computational Topology and Applied Mathematics” PAICYT research project FQM-296, “Andalusian research project PO6-TIC-02268 and Spanish MEC project MTM2006-03722.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Damiand, G., Peltier, S., Fuchs, L., Lienhardt, P.: Topological Map: An Efficient Tool to Compute Incrementally Topological Features on 3D Images. In: Reulke, R., Eckardt, U., Flach, B., Knauer, U., Polthier, K. (eds.) IWCIA 2006. LNCS, vol. 4040, pp. 1–15. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  2. Delfinado, C.J.A., Edelsbrunner, H.: An Incremental Algorithm for Betti Numbers of Simplicial Complexes on the 3–Sphere. Comput. Aided Geom. Design 12, 771–784 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Forman, R.: A Discrete Morse Theory for Cell Complexes. In: Yau, S.T. (ed.) Geometry, Topology & Physics for Raoul Bott. International Press (1995)

    Google Scholar 

  4. Gonzalez-Diaz, R., Medrano, B., Real, P., Sanchez-Pelaez, J.: Algebraic Topological Analysis of Time-sequence of Digital Images. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 208–219. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  5. Molina-Abril, H., Real, P.: Advanced homological information of digital volumen via cell complexes. In: da Vitora Lobo, N., et al. (eds.) SSPR&SPR 2008. LNCS, vol. 5342, pp. 361–371. Springer, Heidelberg (2008)

    Google Scholar 

  6. Gonzalez-Diaz, R., Jimenez, M.J., Medrano, B., Molina-Abril, 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)

    Chapter  Google Scholar 

  7. Gonzalez-Diaz, R., Jiménez, M.J., Medrano, B., Real, P.: Chain Homotopies for Object Topological Representations. Discrete and Applied Mathematics 157, 490–499 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gonzalez-Diaz, R., Real, P.: On the Cohomology of 3D Digital Images. Discrete Applied Math. 147, 245–263 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hatcher, A.: Algebraic Topology. Cambridge University Press, Canbridge (2001)

    MATH  Google Scholar 

  10. Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley Co., Reading (1984)

    MATH  Google Scholar 

  11. Peltier, S., Alayrangues, S., Fuchs, L., Lachaud, J.O.: Computation of homology groups and generators. Computers & Graphics 30(1), 62–69 (2006)

    Article  MATH  Google Scholar 

  12. Peltier, S., Ion, A., Haxhimusa, Y., Kropatsch, W.G., Damiand, G.: Computing Homology Group Generators of Images Using Irregular Graph Pyramids. In: Escolano, F., Vento, M. (eds.) GbRPR 2007. LNCS, vol. 4538, pp. 283–294. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  13. Suuriniemi, S., Tarhasaari, T., Kettunen, L.: Generalization of the spanning-tree technique. IEEE Transactions on Magnetics 38(2), 525–528 (2002)

    Article  Google Scholar 

  14. Taubin, G., Rossignag, J.: Geometric compression through topological surgery. ACM Transactions on Graphics 17(2), 84–115 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dpto. Matematica Aplicada I, E.T.S.I. Informatica, Universidad de Sevilla, Avda. Reina Mercedes, s/n, 41012, Sevilla, Spain

    Pedro Real

Authors
  1. Pedro Real
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Ingeniería Electrónica, Universitat Politècnica de Catalunya (UPC). E.T.S.I. de Telecomunicación., , , ,, Campus Norte, Edificio C4, C/ Jordi Girona, 1-3, E08034, Barcelona, Spain

    Joan Cabestany

  2. Grupo ISIS, Dpto. Tecnología Electrónica ETSI Telecomunicación, Universidad de Málaga, Campus de Teatinos, 29071, Málaga, Spain

    Francisco Sandoval

  3. Department of Computer Architecture and Computer Technology, University of Granada, Spain

    Alberto Prieto

  4. Department of Informatics, University of Salamanca, Salamanca, Spain

    Juan M. Corchado

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Real, P. (2009). Connectivity Forests for Homological Analysis of Digital Volumes. In: Cabestany, J., Sandoval, F., Prieto, A., Corchado, J.M. (eds) Bio-Inspired Systems: Computational and Ambient Intelligence. IWANN 2009. Lecture Notes in Computer Science, vol 5517. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02478-8_52

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02478-8_52

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02477-1

  • Online ISBN: 978-3-642-02478-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics