Using Membrane Computing for Obtaining Homology Groups of Binary 2D Digital Images

  • Hepzibah A. Christinal
  • Daniel Díaz-Pernil
  • Pedro Real Jurado
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5852)


Membrane Computing is a new paradigm inspired from cellular communication. Until now, P systems have been used in research areas like modeling chemical process, several ecosystems, etc. In this paper, we apply P systems to Computational Topology within the context of the Digital Image. We work with a variant of P systems called tissue-like P systems to calculate in a general maximally parallel manner the homology groups of 2D images. In fact, homology computation for binary pixel-based 2D digital images can be reduced to connected component labeling of white and black regions. Finally, we use a software called Tissue Simulator to show with some examples how these systems work.


computational topology homology groups membrane computing P systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adleman, L.M.: Molecular computations of solutions to combinatorial problems. Science 226, 1021–1024 (1994)CrossRefGoogle Scholar
  2. 2.
    Borrego–Ropero, R., Díaz–Pernil, D., Pérez–Jiménez, M.J.: Tissue Simulator: A Graphical Tool for Tissue P Systems. In: Proc. International Workshop, Automata for Cellular and Molecular Computing, MTA SZTAKI, Budapest, pp. 23–34 (2007)Google Scholar
  3. 3.
    Ceterchi, R., Mutyam, M., Păun, G., Subramanian, K.G.: Array-rewriting P systems. Natural Computing 2, 229–249 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chao, J., Nakayama, J.: Cubical Singular Simples Model for 3D Objects and Fast Computation of Homology Groups. In: Proc. ICPR 1996, pp. 190–194. IEEE, Los Alamitos (1996)Google Scholar
  5. 5.
    Ciobanu, G., Păun, G., Pérez–Jiménez, M.J. (eds.): Applications of Membrane Computing. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Dersanambika, K.S., Krithivasan, K., Subramanian, K.G.: P Systems Generating Hexagonal Picture Languages. In: Martín-Vide, C., Mauri, G., Păun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2003. LNCS, vol. 2933, pp. 168–180. Springer, Heidelberg (2004)Google Scholar
  7. 7.
    Díaz–Pernil, D.: Sistemas P de Tejido: Formalización y Eficiencia Computacional. PhD Thesis, University of Seville (2008)Google Scholar
  8. 8.
    Díaz–Pernil, D., Gutiérrez, M.A., Pérez–Jiménez, M.J., Riscos–Núñez, A.: A uniform family of tissue P systems with cell division solving 3-COL in a linear time. Theoretical Computer Science 404(1–2), 76–87 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Díaz–Pernil, D., Pérez–Jiménez, M.J., Romero, A.: Computational efficiency of cellular division in tissue-like P systems. Romanian Journal of Information Science and Technology 11(3), 229–241 (2008)Google Scholar
  10. 10.
    González Díaz, R., Real, P.: Computation of Cohomology Operations of Finite Simplicial Complexes. Homology Homotopy and Applications 2, 83–93 (2003)Google Scholar
  11. 11.
    Gonzalez-Diaz, R., Real, P.: Towards Digital Cohomology. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 92–101. Springer, Heidelberg (2003)Google Scholar
  12. 12.
    Gonzalez-Diaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Applied Mathematics 147, 245–263 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    Gonzalez-Diaz, R., Medrano, B., Real, P., Sánchez, J.: Reusing Integer Homology Information of Binary Digital Images. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 199–210. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Gonzalez-Diaz, R., Medrano, B., Real, P., Sánchez-Peláez, J.: Simplicial Perturbation Technique 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
  16. 16.
    Gonzalez-Diaz, R., Jiménez, M.J., Medrano, B., Real, P.: Extending AT-Models for Integer Homology Computation. In: Escolano, F., Vento, M. (eds.) GbRPR 2007. LNCS, vol. 4538, pp. 330–339. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Gonzalez-Diaz, R., Jiménez, M.J., Medrano, B., Real, P.: A Graph-with-Loop Structure for a Topological Representation of 3D Objects. In: Kropatsch, W.G., Kampel, M., Hanbury, A. (eds.) CAIP 2007. LNCS, vol. 4673, pp. 506–513. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Gonzalez-Diaz, R., Jimenez, M.J., Medrano, B., Molina-Abril, H., Real, R.: 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
  19. 19.
    Gonzalez-Diaz, R., Jimenez, M.J., Medrano, B., Real, P.: Chain homotopies for object topological representations. Discrete Applied Mathematics 157, 490–499 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gutiérrez-Naranjo, M.A., Pérez-Jiménez, M.J.: Riscos-Núñez: On the degree of parallelism in membrane systems. Theoretical Computer Science 372, 183–195 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hatcher, A.: Algebraic Topology. Cambridge Univ. Press, Cambridge (2001)Google Scholar
  22. 22.
    Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)Google Scholar
  23. 23.
    Loos, R., Nagy, B.: Parallelism in DNA and Membrane Computing. In: CiE, Local Proceedings, pp. 283–287 (2007)Google Scholar
  24. 24.
    McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics 5, 115–133 (1943)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Martín–Vide, C., Pazos, J., Păun, G., Rodríguez Patón, A.: A New Class of Symbolic Abstract Neural Nets: Tissue P Systems. In: Ibarra, O.H., Zhang, L. (eds.) COCOON 2002. LNCS, vol. 2387, pp. 290–299. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  26. 26.
    Martín–Vide, C., Pazos, J., Păun, G., Rodríguez Patón, A.: Tissue P systems. Theoretical Computer Science 296, 295–326 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Molina-Abril, H., Real, P.: Cell AT-models for digital volumes. In: Torsello, A., Escolano, F., Brun, L. (eds.) GbRPR 2009. LNCS, vol. 5534, pp. 314–323. Springer, Heidelberg (2009)Google Scholar
  28. 28.
    Molina-Abril, H., Real, P.: Advanced Homological information on 3D Digital volumes. In: da Vitoria Lobo, N., Kasparis, T., Roli, F., Kwok, J.T., Georgiopoulos, M., Anagnostopoulos, G.C., Loog, M. (eds.) S+SSPR 2008. LNCS, vol. 5342, pp. 361–371. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  29. 29.
    Păun, G.: Computing with membranes. Journal of Computer and System Sciences 61(1), 108–143 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Păun, G.: Membrane Computing. An Introduction. Springer, Berlin (2002)zbMATHGoogle Scholar
  31. 31.
    Păun, A., Păun, G.: The power of communication: P systems with symport/antiport. New Generation Computing 20(3), 295–305 (2002)zbMATHCrossRefGoogle Scholar
  32. 32.
    Păun, G., Pérez–Jiménez, M.J.: Recent computing models inspired from biology: DNA and membrane computing. Theoria 18(46), 72–84 (2003)Google Scholar
  33. 33.
    Păun, G., Pérez–Jiménez, M.J., Riscos–Núñez, A.: Tissue P System with cell division. In: Second Brainstorming Week on Membrane Computing, Sevilla, Report RGNC 01/2004, pp. 380–386 (2004)Google Scholar
  34. 34.
    Real, P.: An Algorithm Computing Homotopy Groups. Mathematics and Computers in Simulation 42(4-6), 461–465 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Real, P.: Homological Perturbation Theory and Associativity. Homology Homotopy and Applications, pp. 51–88 (2000)Google Scholar
  36. 36.
    Real, P.: Connectivity forests for homological analysis of digital volumes. In: Cabestany, J., et al. (eds.) IWANN 2009, Part I. LNCS, vol. 5517, pp. 415–423. Springer, Heidelberg (2009)Google Scholar
  37. 37.
    Real, P., Molina-Abril, H., Kropatsch, W.: Homological tree-based strategies for image analysis. In: Computer Analysis and Image Patterns, CAIP (2009)Google Scholar
  38. 38.
    Sergeraert, F.: The computability problem in algebraic topology. Advances in Mathematics 104, 1–29 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    The Tissue Simulator Website:
  40. 40.
    The P Systems Website:

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Hepzibah A. Christinal
    • 1
    • 2
  • Daniel Díaz-Pernil
    • 1
  • Pedro Real Jurado
    • 1
  1. 1.Research Group on Computational Topology and Applied MathematicsUniversity of SevillaSevillaSpain
  2. 2.Karunya UniversityCoimbatore, TamilnaduIndia

Personalised recommendations