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Ruminations on Tarjan’s Union-Find Algorithm and Connected Operators

  • Thierry Géraud
Part of the Computational Imaging and Vision book series (CIVI, volume 30)

Abstract

This papers presents a comprehensive and general form of the Tarjan’s union-find algorithm dedicated to connected operators. An interesting feature of this form is to introduce the notion of separated domains. The properties of this form and its flexibility are discussed and highlighted with examples. In particular, we give clues to handle correctly the constraint of domain-disjointness preservation and, as a consequence, we show how we can rely on “union-find” to obtain algorithms for self-dual filters approaches and levelings with a marker function.

Keywords

Union-find algorithm reconstructions algebraic openings and closings domain-disjointness preservation self-dual filters levelings 

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References

  1. [1]
    J. Darbon, T. Géraud, and A. Duret-Lutz. Generic implementation of morphological image operators. In Mathematical Morphology, Proc. of ISMM, pages 175–184. Sciro, 2002.Google Scholar
  2. [2]
    M. Dillencourt, H. Samet, and M. Tamminen. A general approach to connected-components labeling for arbitrary image representations. Journal of the ACM, 39(2):253–280, 1992.CrossRefGoogle Scholar
  3. [3]
    H. Heijmans and R. Keshet. Inf-semilattice approach to self-dual morphology. Journal of Mathematical Imaging and Vision, 17(1):55–80, 2002.CrossRefGoogle Scholar
  4. [4]
    W. H. Hesselink. Salembier’s min-tree algorithm turned into breadth first search. Information Processing Letters, 88(1–2):225–229, 2003.CrossRefGoogle Scholar
  5. [5]
    A. Mehnert and P Jackway. Folding induced self-dual filters. In Mathematical Morphology and its Applications to Image and Signal Processing, pages 99–108, 2000.Google Scholar
  6. [6]
    A. Meijster and J. Roerdink. A disjoint set algorithm for the watershed transform. In EUSIPCO IX European Signal Processing Conference, pages 1665–1668, 1998.Google Scholar
  7. [7]
    A. Meijster and M. Wilkinson. A comparison of algorithms for connected set openings and closings. IEEE Trans. on PAMI, 24(4):484–494, 2002.Google Scholar
  8. [8]
    F. Meyer. From connected operators to levelings. In Mathematical Morphology and its Applications to Image and Signal Processing, pages 191–198. Kluwer, 1998.Google Scholar
  9. [9]
    F. Meyer. The levelings. In Mathematical Morphology and its Applications to Image and Signal Processing, pages 199–206. Kluwer, 1998.Google Scholar
  10. [10]
    F. Meyer. Levelings, image simplification filters for segmentation. Journal of Mathematical Imaging and Vision, 20(1–2):59–72, 2004.CrossRefGoogle Scholar
  11. [11]
    L. Najman and M. Couprie. Quasi-linear algorithm for the component tree. In IS&T/SPIE Symposium on Electronic Imaging, In Vision Geometry XII, pages 18–22, 2004.Google Scholar
  12. [12]
    Olena. Generic C++ image processing library, http://olena.lrde.epita.fr, free software available under GNU Public Licence, EPITA Research and Development Laboratory, France, 2005.Google Scholar
  13. [13]
    J. B. Roerdink and A. Meijster. The watershed transform: Definitions, algorithms and parallelization strategies. Fundamenta Informaticae, 41(1–2):187–228, 2000.Google Scholar
  14. [14]
    P. Salembier and J. Ruiz. On filters by reconstruction for size and motion simplification. In Mathematical Morphology, Proc. of ISMM, pages 425–434. Sciro Publishing, 2002.Google Scholar
  15. [15]
    P. Soille. Morphological Image Analysis. Springer-Verlag, 1999.Google Scholar
  16. [16]
    R. E. Tarjan. Efficiency of a good but not linear set union algorithm. Journal of the ACM, 22(2):215–225, 1975.CrossRefGoogle Scholar
  17. [17]
    C. Vachier. Morphological Scale-Space Analysis and Feature Extraction. In IEEE Intl. Conf. on Image Processing, volume 3, pages 676–679, October 2001.Google Scholar
  18. [18]
    L. Vincent. Morphological grayscale reconstruction in image analysis: Applications and efficient algorithms. IEEE Trans. on Image Processing, 2(2):176–201, 1993.CrossRefGoogle Scholar
  19. [19]
    M. Wilkinson and J. Roerdink. Fast morphological attribute operations using tarjan’s union-find algorithm. In Mathematical Morphology and its Applications to Image and Signal Processing, Proc. of ISMM, pages 311–320, 2000.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Thierry Géraud
    • 1
  1. 1.EPITA Research and Development Laboratory (LRDE)Le Kremlin-BicêtreFrance

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