A Quasi-linear Algorithm to Compute the Tree of Shapes of nD Images

  • Thierry Géraud
  • Edwin Carlinet
  • Sébastien Crozet
  • Laurent Najman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7883)


To compute the morphological self-dual representation of images, namely the tree of shapes, the state-of-the-art algorithms do not have a satisfactory time complexity. Furthermore the proposed algorithms are only effective for 2D images and they are far from being simple to implement. That is really penalizing since a self-dual representation of images is a structure that gives rise to many powerful operators and applications, and that could be very useful for 3D images. In this paper we propose a simple-to-write algorithm to compute the tree of shapes; it works for nD images and has a quasi-linear complexity when data quantization is low, typically 12 bits or less. To get that result, this paper introduces a novel representation of images that has some amazing properties of continuity, while remaining discrete.


Parent Function Image Representation Cubical Complex Morphological Tree Canonical Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Modern Birkhäuser Classics, Birkhäuser (2008)Google Scholar
  2. 2.
    Berger, C., Géraud, T., Levillain, R., Widynski, N., Baillard, A., Bertin, E.: Effective component tree computation with application to pattern recognition in astronomical imaging. In: Proceedings of ICIP, vol. 4, pp. 41–44 (2007)Google Scholar
  3. 3.
    Carlinet, E., Géraud, T.: A (fair?) comparison of many max-tree computation algorithms. In: Luengo Hendriks, C.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 73–95. Springer, Heidelberg (2013)Google Scholar
  4. 4.
    Caselles, V., Meinhardt, E., Monasse, P.: Constructing the tree of shapes of an image by fusion of the trees of connected components of upper and lower level sets. Positivity 12(1), 55–73 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Caselles, V., Monasse, P.: Geometric Description of Images as Topographic Maps. Lecture Notes in Mathematics Series, vol. 1984. Springer (2009)Google Scholar
  6. 6.
    Géraud, T.: Ruminations on tarjan’s union-find algorithm and connected operators. In: Proceedings of ISMM. CIVS, vol. 30, pp. 105–116. Springer (2005)Google Scholar
  7. 7.
    Géraud, T., Talbot, H., Van Droogenbroeck, M.: Mathematical Morphology—From Theory to Applications, ch. 12, pp. 323–353. ISTE & Wiley (2010)Google Scholar
  8. 8.
    Henle, M.: A Combinatorial Introduction to Topology. Dover Publications Inc. (1994)Google Scholar
  9. 9.
    Latecki, L., Eckhardt, U., Rosenfeld, A.: Well-composed sets. Computer Vision and Image Understanding 61, 70–83 (1995)CrossRefGoogle Scholar
  10. 10.
    Levillain, R., Géraud, T., Najman, L.: Why and how to design a generic and efficient image processing framework: The case of the Milena library. In: Proceedings of ICIP, pp. 1941–1944 (2010),
  11. 11.
    Levillain, R., Géraud, T., Najman, L.: Writing reusable digital topology algorithms in a generic image processing framework. In: Köthe, U., Montanvert, A., Soille, P. (eds.) WADGMM 2010. LNCS, vol. 7346, pp. 140–153. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Digital imaging: A unified topological framework. Journal of Mathematical Imaging and Vision 44(1), 19–37 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Meijster, A., Wilkinson, M.H.F.: A comparison of algorithms for connected set openings and closings. IEEE Transactions on Pattern Analysis and Machine Intelligence 24(4), 484–494 (2002)CrossRefGoogle Scholar
  14. 14.
    Meinhardt-Llopis, E.: Morphological and Statistical Techniques for the Analysis of 3D Images. Ph.D. thesis, Universitat Pompeu Fabra, Spain (March 2011)Google Scholar
  15. 15.
    Meyer, F.: Un algorithme optimal de ligne de partage des eaux. In: Actes du 8e congrès AFCET, pp. 847–859 (1991)Google Scholar
  16. 16.
    Monasse, P., Guichard, F.: Fast computation of a contrast invariant image representation. IEEE Transactions on Image Processing 9(5), 860–872 (2000)CrossRefGoogle Scholar
  17. 17.
    Najman, L., Géraud, T.: Discrete set-valued continuity and interpolation. In: Luengo Hendriks, C.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 37–48. Springer, Heidelberg (2013)Google Scholar
  18. 18.
    Song, Y.: A topdown algorithm for computation of level line trees. IEEE Transactions on Image Processing 16(8), 2107–2116 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. Journal of the ACM 22(2), 215–225 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Xu, Y., Géraud, T., Najman, L.: Context-based energy estimator: Application to object segmentation on the tree of shapes. In: Proceedings of ICIP (2012)Google Scholar
  21. 21.
    Xu, Y., Géraud, T., Najman, L.: Morphological filtering in shape spaces: Applications using tree-based image representations. In: Proceedings of ICPR (2012)Google Scholar
  22. 22.
    Xu, Y., Géraud, T., Najman, L.: Two applications of shape-based morphology: Blood vessel segmentation and generalisation of constrained connectivity. In: Luengo Hendriks, C.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 390–401. Springer, Heidelberg (2013)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Thierry Géraud
    • 1
    • 2
  • Edwin Carlinet
    • 1
    • 2
  • Sébastien Crozet
    • 1
  • Laurent Najman
    • 2
  1. 1.EPITA Research and Development Laboratory (LRDE)France
  2. 2.LIGM, Équipe A3SI, ESIEEUniversité Paris-EstFrance

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