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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
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.
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.
H. Heijmans and R. Keshet. Inf-semilattice approach to self-dual morphology. Journal of Mathematical Imaging and Vision, 17(1):55–80, 2002.
W. H. Hesselink. Salembier’s min-tree algorithm turned into breadth first search. Information Processing Letters, 88(1–2):225–229, 2003.
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.
A. Meijster and J. Roerdink. A disjoint set algorithm for the watershed transform. In EUSIPCO IX European Signal Processing Conference, pages 1665–1668, 1998.
A. Meijster and M. Wilkinson. A comparison of algorithms for connected set openings and closings. IEEE Trans. on PAMI, 24(4):484–494, 2002.
F. Meyer. From connected operators to levelings. In Mathematical Morphology and its Applications to Image and Signal Processing, pages 191–198. Kluwer, 1998.
F. Meyer. The levelings. In Mathematical Morphology and its Applications to Image and Signal Processing, pages 199–206. Kluwer, 1998.
F. Meyer. Levelings, image simplification filters for segmentation. Journal of Mathematical Imaging and Vision, 20(1–2):59–72, 2004.
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.
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.
J. B. Roerdink and A. Meijster. The watershed transform: Definitions, algorithms and parallelization strategies. Fundamenta Informaticae, 41(1–2):187–228, 2000.
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.
P. Soille. Morphological Image Analysis. Springer-Verlag, 1999.
R. E. Tarjan. Efficiency of a good but not linear set union algorithm. Journal of the ACM, 22(2):215–225, 1975.
C. Vachier. Morphological Scale-Space Analysis and Feature Extraction. In IEEE Intl. Conf. on Image Processing, volume 3, pages 676–679, October 2001.
L. Vincent. Morphological grayscale reconstruction in image analysis: Applications and efficient algorithms. IEEE Trans. on Image Processing, 2(2):176–201, 1993.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer
About this paper
Cite this paper
Géraud, T. (2005). Ruminations on Tarjan’s Union-Find Algorithm and Connected Operators. In: Ronse, C., Najman, L., Decencière, E. (eds) Mathematical Morphology: 40 Years On. Computational Imaging and Vision, vol 30. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3443-1_11
Download citation
DOI: https://doi.org/10.1007/1-4020-3443-1_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-3442-8
Online ISBN: 978-1-4020-3443-5
eBook Packages: Computer ScienceComputer Science (R0)