Efficient Algorithms to Implement the Confinement Tree

  • Julian Mattes
  • Jacques Demongeot
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1953)


The aim of this paper is to present a new algorithm to calculate the confinement tree of an image - also known as component tree or dendrone - for which we can prove that its worst-case complexity is O(n log n) where n is the number of pixels. More precisely, in a first part, we present an algorithm which separates the different kinds of operations - which we call scanning, fusion, propagation, and attribute operations - such that we can separately apply complexity analysis on them and such that all operations except propagation stay in O(n). The implementation of the propagation operations is presented in a second part, first in O(n2n), where nn is the number of nodes in the tree (n n = n). This is suficient if the number of pixels is much larger than the number of nodes (n n << n). Else, we show how to obtain O(n n log n n ) complexity for propagation. We construct two example images to investigate the behavior of two known algorithms for which we can show worst-case complexity of O(n2 log n) and O(n2), respectively, and we compare it to our algorithm. Finally, a practical evaluation will be opposed to the theoretical results. Several variations of the implementation will show which operations are time consuming in practice.


Grey Level Unique Representation Practical Evaluation Attribute Operation Fast Execution Time 
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.


  1. 1.
    Chassery, J.-M., Montanvert, A.: Géométrie discrète en analyse d’images. Hermes (1991) 395Google Scholar
  2. 2.
    Cormen, T. H., Leiserson, L. E., Rivest, R. L.: Introduction to algorithms. MIT Press (1998) 396Google Scholar
  3. 3.
    Couprié, M., Bertrand, G.: Topological grey scale watershed transformation. In: SPIE Vision Geometry V Proceedings, Vol. 3168 Bellingham, WA (1997) 136–146 392, 393Google Scholar
  4. 4.
    Guillataud, P.: Contribution à l’analyse dendronique des images. PhD thesis, Université de Bordeaux I (1992) 392, 393Google Scholar
  5. 5.
    Hanusse, P., Guillataud, P.: Sémantique des images par analyse dendronique. In: AFCET, 8th RFIA, Vol. 2. Lyon (1992) 577–588 392, 393, 394Google Scholar
  6. 6.
    Hartigan, J. A.: Statistical theory in clustering. J. of Classification 2 (1985) 63–76 392, 394zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Jones, R.: Connected Filtering and Segmentation Using Component Trees. Computer Vision and Image Understanding 75 (1999) 215–228 392, 393, 394, 395, 399CrossRefGoogle Scholar
  8. 8.
    Jones, R.: Connected Filtering and Segmentation Using Component Trees: Efficient Implementation Algorithms. http://www.dms.CSIRO.AU/ronaldj/pseudocode (1999) 393, 394, 395, 403
  9. 9.
    Kok-Wiles, S.L, Brady, J. M., Highnam, R.: Comparing mammogram pairs for the detection of lesions. In: Karssemeijer, N. (ed.): 4th Int. Workshop of Digital Mammography, Nijmegen, Netherlands, June 1998. Kluwer, Amsterdam, (1998) 392Google Scholar
  10. 10.
    Mattes, J., Demongeot, J.: Dynamic confinement, classification, and imaging. In: 22nd Ann. Conf. GfKl, Dresden, Germany, March 1998. Studies in Classification, Data Analysis, and Knowledge Organization. Springer-Verlag (1999) 205–214 392, 394Google Scholar
  11. 11.
    Mattes, J., Demongeot, J.: Tree representation and implicit tree matching for acoarse to fine image matching algorithm. In: MICCAI’99, C. Taylor, A. Clochester (Eds.). LNCS. Springer-Verlag (1999) 646–655 392, 393, 394, 403Google Scholar
  12. 12.
    Mattes, J., Richard, M., Demongeot, J.: Tree representation for image matching and object recognition. In: DGCI’99, G. Bertrand and M. Couprié and L. Perroton (Eds.). LNCS. Springer-Verlag (1999) 298–309 392, 403Google Scholar
  13. 13.
    Salembier, P., Oliveras, A., Garrido, L.: Antiextensive Connected Operators for Image and Sequence Processing. IEEE Trans. on Image Processing 7 (1998) 555–570 392, 393, 394, 395CrossRefGoogle Scholar
  14. 14.
    Wishart, D.: Mode analysis: A generalization of the nearest neighbor which reduces chaining effects. In: Cole, A. J. (Ed.): Numerical Taxonomy. Academic Press, London (1969) 282–319 392, 394Google Scholar
  15. 15.
    Zuck, P., Lao, Z., Skwish, S., Glickman, J. F., Yang, K., Burbaum, J., Inglese, J.: Ligand-receptor binding measured by laser-scanning imaging. PNAS 96 (1999) 11122–7 393CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Julian Mattes
    • 1
    • 2
  • Jacques Demongeot
    • 1
  1. 1.TIMC-IMAG, Faculty of MedicineLa TroncheFrance
  2. 2.iBioS, DKFZ HeidelbergHeidelbergGermany

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