Shape-Tree Semilattices

Variations and Implementation Schemes
  • Renato Keshet
Conference paper
Part of the Computational Imaging and Vision book series (CIVI, volume 30)


The shape-tree semilattice is a new framework for quasi-self-dual morphological processing, where eroded images have all shapes shrunk in a contrast-invariant way. This approach was recently introduced, and is further investigated here. Apart of reviewing their original definition, different algorithms for computing the shape-tree morphological operators are presented.


Complete inf-semilattices self-dual operators tree of shapes fillhole 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Keshet, “Shape-Tree Semilattice,” Journal of Mathematical Imaging and Vision, special issue of mathematical morphology after 40 years, scheduled to March 2005.Google Scholar
  2. [2]
    J. Serra, Image Analysis and Mathematical Morphology, Vol. 1, London: Academic Press, 1982.Google Scholar
  3. [3]
    H.J.A.M. Heijmans, “Connected Morphological Operators for Binary Images,” Computer Vision and Image Understanding, Vol. 73, No. 1, pp. 99–120, 1999.CrossRefGoogle Scholar
  4. [4]
    P. Soille, Morphological Image Analysis: Principles and Applications, NY: Springer, 2nd edition, 2003.Google Scholar
  5. [5]
    R. Keshet (Kresch), “Mathematical Morphology on Complete Semilattices and its Applications to Image Processing,” Fundamenta Informaticae, Vol. 41, Nos. 1–2, pp. 33–56, January 2000.Google Scholar
  6. [6]
    H.J.A.M. Heijmans and R. Keshet, “Inf-semilattice approach to self-dual morphology,” Journal of Mathematical Imaging Vision, Vol. 17, No. 1, pp. 55–80, July 2002.CrossRefGoogle Scholar
  7. [7]
    P. Monasse and F. Guichard, “Fast computation of a contrast-invariant image representation,” IEEE Trans. on Image Processing, No. 9, Vol. 5, pp. 860–872, May 2000.CrossRefGoogle Scholar
  8. [8]
    P. Monasse and F. Guichard, “Scale-space from a level lines tree,” Journal of Visual Communication and Image Representation, Vol. 11, pp. 224–236, 2000.CrossRefGoogle Scholar
  9. [9]
    V. Caselles and P. Monasse, “Grain filters,” Journal of Mathematical Imaging and Vision, Vol. 17, No. 3, pp. 249–270, November 2002.CrossRefGoogle Scholar
  10. [10]
    C. Ballester, V. Caselles and P. Monasse, “The Tree of Shapes of an Image,” ESAIM: Control, Optimisation and Calculus of Variations, Vol. 9, pp. 1–18, 2003.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Renato Keshet
    • 1
  1. 1.Hewlett-Packard Labs—IsraelTechnion City, HaifaIsrael

Personalised recommendations