Abstract
The watershed transformation is an efficient tool for segmenting grayscale images. An original approach to the watershed (Bertrand, Journal of Mathematical Imaging and Vision, Vol. 22, Nos. 2/3, pp. 217–230, 2005.; Couprie and Bertrand, Proc. SPIE Vision Geometry VI, Vol. 3168, pp. 136–146, 1997.) consists in modifying the original image by lowering some points while preserving some topological properties, namely, the connectivity of each lower cross-section. Such a transformation (and its result) is called a W-thinning, a topological watershed being an “ultimate” W-thinning. In this paper, we study algorithms to compute topological watersheds. We propose and prove a characterization of the points that can be lowered during a W-thinning, which may be checked locally and efficiently implemented thanks to a data structure called component tree. We introduce the notion of M-watershed of an image F, which is a W-thinning of F in which the minima cannot be extended anymore without changing the connectivity of the lower cross-sections. The set of points in an M-watershed of F which do not belong to any regional minimum corresponds to a binary watershed of F. We propose quasi-linear algorithms for computing M-watersheds and topological watersheds. These algorithms are proved to give correct results with respect to the definitions, and their time complexity is analyzed.
Similar content being viewed by others
References
G. Bertrand, “On topological watersheds”Journal of Mathematical Imaging and Vision, Vol. 22, Nos. 2/3, pp. ??–??, 2005.
G. Bertrand, J.C. Everat, and M. Couprie, “Image segmentation through operators based upon topology”Journal of Electronic Imaging, Vol. 6, No. 4, pp. 395-405, 1997.
S. Beucher and Ch. Lantuéjoul, “Use of watersheds in contour detection” inProc. Int. Workshop on Image Processing, Real-Time Edge and Motion Detection/Estimation, Rennes, France, 1979.
S. Beucher and F. Meyer, “The morphological approach to segmentation: The watershed transformation” inMathematical Morphology in Image Processing, Dougherty (Ed.), Chap. 12, Marcel Dekker, 1993, pp. 433–481.
M.A. Bender and M. Farach-Colton, “The LCA problem revisited” inProc. 4th Latin American Symposium on Theoretical Informatics, LNCS, Springer, Vol. 1776, 2000, pp. 88–94.
U.M. Braga-Neto and J. Goutsias, “A theoretical tour of connectivity in image processing and analysis”Journal of Mathematical Imaging and Vision, Vol. 19, pp. 5–31, 2003.
E.J. Breen and R. Jones, “Attribute openings, thinnings and granulometries”Computer Vision and Image Understanding, Vol. 64, No. 3, pp. 377–389, 1996.
T. H. Cormen, C. Leiserson, and R. Rivest,Introduction to Algorithms, McGraw-Hill, 1990.
M. Couprie and G. Bertrand, “Topological grayscale watershed transformation” inProc. SPIE Vision Geometry VI, Vol. 3168, 1997, pp. 136–146.
M. Couprie, F.N. Bezerra, and G. Bertrand, “Topological operators for grayscale image processing”Journal of Electronic Imaging, Vol. 10, No. 4, pp. 1003–1015, 2001.
V. Goetcherian, “From binary to grey tone image processing using fuzzy logic concepts”Pattern Recognition, Vol. 12, No. 12, pp. 7–15, 1980.
P. Guillataud, “Contribution á l’analyse dendroniques des images” PhD thesis of Université de Bordeaux I, 1992.
P. Hanusse and P. Guillataud, “Sémantique des images par analyse dendronique” in8th Conf. Reconnaissance des Formes et Intelligence Artificielle, AFCET Ed., Lyon, Vol. 2, 1992, pp. 577–588.
J.A. Hartigan, “Statistical theory in clustering”Journal of classification, No. 2, pp. 63–76, 1985.
D. Harel and R.E. Tarjan, “Fast algorithms for finding nearest common ancestors”SIAM J. Comput., Vol. 13, No. 2, pp. 338–355, 1984.
R. Jones, “Connected filtering and segmentation using component trees”Computer Vision and Image Understanding, Vol. 75, No. 3, pp. 215–228, 1999.
T.Y Kong and A. Rosenfeld, “Digital topology: Introduction and survey”Computer Vision, Graphics and Image Processing, Vol. 48, pp. 357–393, 1989.
J. Mattes and J. Demongeot, “Tree representation and implicit tree matching for a coarse to fine image matching algorithm” inProc. MICCAI, LNCS, Springer, Vol. 1679, 1999, pp. 646–655.
J. Mattes and J. Demongeot, “Efficient algorithms to implement the confinement tree” inProc. DGCI, LNCS, Springer, Vol. 1953, 2000, pp. 392–405.
J. Mattes, M. Richard, and J. Demongeot, “Tree representation for image matching and object recognition” inProc. DGCI, LNCS, Springer, Vol. 1568, 1999, pp. 298–309.
A. Meijster and M. Wilkinson, “A comparison of algorithms for connected set openings and closings”IEEE PAMI, Vol. 24, pp. 484–494, 2002.
F. Meyer, “Un algorithme optimal de ligne de partage des eaux” inProc. 8th Conf. Reconnaissance des Formes et Intelligence Artificielle, AFCET Ed., Lyon, Vol. 2, 1991, pp. 847–859.
L. Najman and M. Couprie, “Watershed algorithms and contrast preservation” inProc. DGCI, LNCS, Springer, Vol. 2886, 2003, pp. 62–71.
L. Najman and M. Couprie, “Quasi-linear algorithm for the component tree” inProc. SPIE Vision Geometry XII, Vol. 5300, 2004, pp. 98–107.
L. Najman, M. Couprie, and G. Bertrand, “Watersheds, mosaics, and the emergence paradigm” to appear inDiscrete Applied Mathematics, 2005.
L. Najman and M. Schmitt, “Watershed of a continuous function”Signal Processing, Vol. 38, pp. 99–112, 1994.
J.B.T.M. Roerdink and A. Meijster, “The watershed transform: Definitions, algorithms and parallelization strategies”Fundamenta Informaticae, Vol. 41, pp. 187–228, 2000.
A. Rosenfeld, “On connectivity properties of grayscale pictures”Pattern Recognition, Vol. 16, pp. 47–50, 1983.
J. Serra,Image Analysis and Mathematical Morphology, Vol. II:Theoretical Advances, Academic Press, 1988.
P. Salembier, A. Oliveras, and L. Garrido, “Antiextensive connected operators for image and sequence processing”IEEE Trans. on Image Processing, Vol. 7, No. 4, pp. 555–570, 1998.
R.E. Tarjan, “Disjoint sets”Data Structures and Network Algorithms, Chap. 2, SIAM, 1978, pp. 23–31.
M. Thorup, “On RAM priority queues” in7th ACM-SIAM Symposium on Discrete Algorithms, 1996, pp. 59–67.
C. Vachier, “Extraction de caractéristiques, segmentation d’images et Morphologie Mathématique” PhD Thesis, École des Mines, Paris, 1995.
L. Vincent and P. Soille, “Watersheds in digital spaces: An efficient algorithm based on immersion simulations”IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 13, No. 6, pp. 583–598, 1991.
D. Wishart, “Mode analysis: A generalization of the nearest neighbor which reduces chaining effects” inNumerical Taxonomy, A.J. Cole (Ed.), Academic Press, 1969, pp. 282–319.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Michel Couprie received his Ingénieur’s degree from the École Supérieure d’Ingénieurs en Électrotechnique et Électronique (Paris, France) in 1985 and the Ph.D. degree from the Pierre et Marie Curie University (Paris, France) in 1988. Since 1988 he has been working in ESIEE where he is an Associate Professor. He is a member of the Laboratoire Algorithmique et Architecture des Systémes Informatiques, ESIEE, Paris, and of the Institut Gaspard Monge, Universit é de Marne-la-Vallée. His current research interests include image analysis and discrete mathematics.
Laurent Najman received his Ph.D. of applied mathematics from Paris-Dauphine university and an Ingénieur’s degree from the Ecole des Mines de Paris. After earning his Ingénieur’s degree, he worked in the research laboratories of Thomson-CSF for three years, before joining Animation Science in 1995, as director of research and development. In 1998, he joined OcÉ Print Logic Technolgies, as senior scientist. Since 2002, he is associate professor with the A2SI laboratory of ESIEE, Paris. His current research interest is discrete mathematical morphology.
Gilles Bertrand received his Ingénieur’s degree from the École Centrale des Arts et Manufactures in 1976. Until 1983 he was with the Thomson-CSF company, where he designed image processing systems for aeronautical applications. He received his Ph. from the École Centrale in 1986. He is currently teaching and doing research with the Laboratoire Algorithmique et Architecture des Systémes Informatiques, ESIEE, Paris, and with the Institut Gaspard Monge, Université de Marne-la-Vallée. His research interests are image analysis, pattern recognition, mathematical morphology and digital topology.
Rights and permissions
About this article
Cite this article
Couprie, M., Najman, L. & Bertrand, G. Quasi-Linear Algorithms for the Topological Watershed. J Math Imaging Vis 22, 231–249 (2005). https://doi.org/10.1007/s10851-005-4892-4
Issue Date:
DOI: https://doi.org/10.1007/s10851-005-4892-4