Towards a Certified Computation of Homology Groups for Digital Images

  • Jónathan Heras
  • Maxime Dénès
  • Gadea Mata
  • Anders Mörtberg
  • María Poza
  • Vincent Siles
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7309)


In this paper we report on a project to obtain a verified computation of homology groups of digital images. The methodology is based on programming and executing inside the Coq proof assistant. Though more research is needed to integrate and make efficient more processing tools, we present some examples partially computed in Coq from real biomedical images.


Homology Discrete Morse Theory Proof assistant tools Coq SSReflect Synapses 


  1. 1.
  2. 2.
    Armand, M., Grégoire, B., Spiwack, A., Théry, L.: Extending Coq with Imperative Features and Its Application to SAT Verification. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 83–98. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Cazals, F., Chazal, F., Lewiner, T.: Molecular shape analysis based upon Morse-Smale complex and the Connolly function. In: Proceedings 19th ACM Symposium on Computational Geometry (SCG 2003), pp. 351–360 (2003)Google Scholar
  4. 4.
    Cohen, C., Dénès, M., Mörtberg, A., Siles, V.: Smith Normal form and executable rank for matrices,
  5. 5.
    Coq development team. The Coq Proof Assistant Reference Manual, version 8.3. Technical report (2010)Google Scholar
  6. 6.
    Cuesto, G., et al.: Phosphoinositide-3-Kinase Activation Controls Synaptogenesis and Spinogenesis in Hippocampal Neurons. The Journal of Neuroscience 31(8), 2721–2733 (2011)CrossRefGoogle Scholar
  7. 7.
    Forman, R.: Morse theory for cell complexes. Advances in Mathematics 134, 90–145 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gonthier, G.: Formal proof - The Four-Color Theorem, vol. 55. Notices of the American Mathematical Society (2008)Google Scholar
  9. 9.
    Gonthier, G., Mahboubi, A.: A Small Scale Reflection Extension for the Coq system. Technical report, Microsoft Research INRIA (2009),
  10. 10.
    Grégoire, B., Leroy, X.: A compiled implementation of strong reduction. In: Proceedings of the Seventh ACM SIGPLAN international Conference on Functional Programming, ICFP 2002, pp. 235–246. ACM, New York (2002)CrossRefGoogle Scholar
  11. 11.
    Gyulassy, A., Bremer, P., Hamann, B., Pascucci, V.: A practical approach to Morse-Smale complex computation: Scalability and generality. IEEE Transactions on Visualization and Computer Graphics 14(6), 1619–1626 (2008)CrossRefGoogle Scholar
  12. 12.
    Harker, S., et al.: The Efficiency of a Homology Algorithm based on Discrete Morse Theory and Coreductions. In: Proceedings 3rd International Workshop on Computational Topology in Image Context (CTIC 2010). Image A, vol. 1, pp. 41–47 (2010)Google Scholar
  13. 13.
    Heras, J., Mata, G., Poza, M., Rubio, J.: Homological processing of biomedical digital images: automation and certification. Technical report (2010),
  14. 14.
    Heras, J., Poza, M., Dénès, M., Rideau, L.: Incidence Simplicial Matrices Formalized in Coq/SSReflect. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) Calculemus/ MKM 2011. LNCS (LNAI), vol. 6824, pp. 30–44. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Jerse, G., Kosta, N.M.: Tracking features in image sequences using discrete Morse functions. In: Proceedings 3rd International Workshop on Computational Topology in Image Context (CTIC 2010). Image A, vol. 1, pp. 27–32 (2010)Google Scholar
  16. 16.
    Krebbers, R., Spitters, B.: Computer Certified Efficient Exact Reals in Coq. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) Calculemus/MKM 2011. LNCS (LNAI), vol. 6824, pp. 90–106. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Mata, G.: SynapsCountJ. University of La Rioja (2011),
  18. 18.
    Boespflug, M., Dénès, M., Grégoire, B.: Full Reduction at Full Throttle. In: Jouannaud, J.-P., Shao, Z. (eds.) CPP 2011. LNCS, vol. 7086, pp. 362–377. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  19. 19.
    Molina-Abril, H., Real, P.: A Homological–Based Description of Subdivided nD Objects. In: Real, P., Diaz-Pernil, D., Molina-Abril, H., Berciano, A., Kropatsch, W. (eds.) CAIP 2011, Part I. LNCS, vol. 6854, pp. 42–50. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Mrozek, M., et al.: Homological methods for extraction and analysis of linear features in multidimensional images. Pattern Recognition 45(1), 285–298 (2012)CrossRefGoogle Scholar
  21. 21.
    F.L.R.: node. From Digital Images to Simplicial Complexes: A report. Technical report (2011),
  22. 22.
    Rasband, W.S.: ImageJ: Image Processing and Analysis in Java (2003),
  23. 23.
    Real, P., Molina-Abril, H.: Towards Optimality in Discrete Morse Theory through Chain Homotopies. In: Proceedings 3rd International Workshop on Computational Topology in Image Context (CTIC 2010). Image A, vol. 1, pp. 33–40 (2010)Google Scholar
  24. 24.
    Robins, V., Wood, P., Sheppard, A.: Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(8), 1646–1658 (2011)CrossRefGoogle Scholar
  25. 25.
    Romero, A., Sergeraert, F.: Discrete Vector Fields and Fundamental Algebraic Topology (2010),
  26. 26.
    Selkoe, D.J.: Alzheimer’s disease is a synaptic failure. Science 298(5594), 789–791 (2002)CrossRefGoogle Scholar
  27. 27.
    Ziou, D., Allili, M.: Generating Cubical Complexes from Image Data and Computation of the Euler number. Pattern Recognition 35, 2833–2839 (2002)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jónathan Heras
    • 1
  • Maxime Dénès
    • 2
  • Gadea Mata
    • 1
  • Anders Mörtberg
    • 3
  • María Poza
    • 1
  • Vincent Siles
    • 3
  1. 1.Department of Mathematics and Computer ScienceUniversity of La RiojaSpain
  2. 2.INRIA Sophia AntipolisMéditerranéeFrance
  3. 3.University of GothenburgSweden

Personalised recommendations