CSDL Home IEEE Transactions on Pattern Analysis & Machine Intelligence 2009 vol.31 Issue No.08 - August

Subscribe

Issue No.08 - August (2009 vol.31)

pp: 1362-1374

Jean Cousty , Université Paris-Est, ESIEE, and INRIA Sophia Antipolis, France

Gilles Bertrand , Université Paris-Est, ESIEE, France

Laurent Najman , Université Paris-Est, ESIEE, France

Michel Couprie , Université Paris-Est, ESIEE, France

ABSTRACT

We study the watersheds in edge-weighted graphs. We define the watershed cuts following the intuitive idea of drops of water flowing on a topographic surface. We first establish the consistency of these watersheds: They can be equivalently defined by their "catchment basins” (through a steepest descent property) or by the "dividing lines” separating these catchment basins (through the drop of water principle). Then, we prove, through an equivalence theorem, their optimality in terms of minimum spanning forests. Afterward, we introduce a linear-time algorithm to compute them. To the best of our knowledge, similar properties are not verified in other frameworks and the proposed algorithm is the most efficient existing algorithm, both in theory and in practice. Finally, the defined concepts are illustrated in image segmentation, leading to the conclusion that the proposed approach improves, on the tested images, the quality of watershed-based segmentations.

INDEX TERMS

Watershed, minimum spanning forest, minimum spanning tree, graph, mathematical morphology, image segmentation.

CITATION

Jean Cousty, Gilles Bertrand, Laurent Najman, Michel Couprie, "Watershed Cuts: Minimum Spanning Forests and the Drop of Water Principle",

*IEEE Transactions on Pattern Analysis & Machine Intelligence*, vol.31, no. 8, pp. 1362-1374, August 2009, doi:10.1109/TPAMI.2008.173REFERENCES

- [1] J. Maxwell, “On Hills and Dales,”
Philosophical Magazine, vol. 4/40, pp. 421-427, 1870.- [2] C. Jordan, “Nouvelles Observations sur les Lignes de Faîtes et de Thalweg,”
Comptes Rendus des Séances de l'Académie des Sciences, vol. 75, pp. 1023-1025, 1872.- [3] H. Digabel and C. Lantuéjoul, “Iterative Algorithms,”
Proc. Second European Symp. Quantitative Analysis of Microstructures in Material Science, Biology and Medicine, pp. 85-89, 1978.- [4] S. Beucher and C. Lantuéjoul, “Use of Watersheds in Contour Detection,”
Proc. Int'l Workshop Image Processing Real-Time Edge and Motion Detection/Estimation, 1979.- [6] F. Meyer, “Un Algorithme Optimal de Ligne de Partage des Eaux,”
Proc. Actes du Huitéme Congrés, pp. 847-859, 1991.- [8] L. Najman and M. Schmitt, “Watershed of a Continuous Function,”
Signal Processing, vol. 38, no. 1, pp. 68-86, 1993.- [9] F. Lemonnier, “Architecture Electronique Dédiée aux Algorithmes Rapides de Segmentation Basés sur la Morphologie Mathématique,” PhD dissertation, Ecole des Mines de Paris, Dec. 1996.
- [10] M. Couprie and G. Bertrand, “Topological Grayscale Watershed Transform,”
Proc SPIE Vision Geometry, pp. 136-146, 1997.- [11] A. Meijster and J. Roerdink, “A Disjoint Set Algorithm for the Watershed Transform,”
Procs. European Signal Processing Conf., pp.1669-1672, 1998.- [12] A. Bieniek and A. Moga, “A Connected Component Approach to the Watershed Segmentation,”
Proc. Int'l Symp. Math. Morphology Math. Morphology and Its Applications to Image and Signal Processing, pp. 215-222, 1998.- [13] S. Stoev, “RAFSI—A Fast Watershed Algorithm Based on Rain-Falling Simulation,”
Proc. Eighth Int'l Conf. Computer Graphics, Visualization, and Interactive Digital Media, 2000.- [15] S. Beucher and F. Meyer, “The Morphological Approach to Segmentation: The Watershed Transformation,”
Math. Morphology in Image Processing, E. Dougherty, ed., pp. 443-481, Marcel Dekker, 1993.- [19] F. Meyer, “Minimum Spanning Forests for Morphological Segmentation,”
Proc. Second Int'l Conf. Math. Morphology and Its Applications to Image Processing, pp. 77-84, Sept. 1994.- [22] Y. Zhuge, J.K. Udupa, and P.K. Saha, “Vectorial Scale-Based Fuzzy-Connected Image Segmentation,”
Computer Visualization and Image Understanding, vol. 101, pp. 177-193, 2006.- [23] R. Englert and W. Kropatsch, “Image Structure from Monotonic Dual Graph,”
Proc. Applications of Graph Transformations with Industrial Relevance, International Workshop, pp. 297-308, 2000.- [26] R. Lotufo and A. Falcão, “The Ordered Queue and the Optimality of the Watershed Approaches,”
Proc. Fifth Int'l Symp. Math. Morphology, pp. 341-350, 2000.- [27] J. Cousty, G. Bertrand, L. Najman, and M. Couprie, “Watershed Cuts,”
Procs. Int'l Symp. Math. Morphology, pp. 301-312, 2007.- [28] R. Diestel,
Graph Theory. Springer, 1997.- [30] E. Dahlhaus, D. Johnson, C. Papadimitriou, P. Seymour, and M. Yannakakis, “Complexity of Multiway Cuts,”
Proc. 24th Ann. ACM Symp. Theory of Computing, pp. 241-251, 1992.- [31] C. Allène, J.-Y. Audibert, M. Couprie, J. Cousty, and R. Keriven, “Some Links between Min-Cuts, Optimal Spanning Forests and Watersheds,”
Proc. Int'l Symp. Math. Morphology, pp. 253-264, 2007.- [33] J.B.T.M. Roerdink and A. Meijster, “The Watershed Transform: Definitions, Algorithms and Parallelization Strategies,”
Fundamenta Informaticae, vol. 41, nos. 1-2, pp. 187-228, 2001.- [34] J. Cousty, “Lignes de Partage des eaux Discrètes: Théorie et Application à la Segmentation d'Images Cardiaques,” PhD dissertation, Université de Marne-la-Vallée, 2007.
- [36] R. Prim, “Shortest Connection Networks and Some Generalizations,”
Bell Systems Technical J., vol. 36, pp. 1389-1401, 1957.- [37] R.L. Graham and P. Hell, “On the History of the Minimum Spanning Tree Problem,”
Annals of the History of Computing, vol. 7, pp. 43-57, 1985.- [41] M. Thorup, “On Ram Priority Queues,”
Proc. Seventh ACM-SIAM Symposium on Discrete Algorithms, pp. 59-67, 1996.- [43] J. Cousty, G. Bertrand, L. Najman, and M. Couprie, “Watershed Cuts: Thinnings, Topological Watersheds, and Shortest Path Forests,”
IEEE Trans. Pattern Analysis and Machine Intelligence, preprint,http://igm.univ-mlv.fr/LabInfo/rapportsInternes/ 2007/09.v2.pdfhttp://doi.ieeecomputersociety.org/ 10.1109TPAMI.2009.71. - [44] M. Grimaud, “New Measure of Contrast: Dynamics,”
Image Algebra and Morphological Image Processing III, pp. 292-305, 1992.- [46] R. Duda and P. Hart,
Pattern Classification and Scene Analysis. Wiley-Interscience, 1973.- [48] P. Soille,
Morphological Image Analysis. Springer, 1999.- [50] F. Meyer and L. Najman, “Segmentation, Arbre de Poids Minimum et Hiérarchies,”
Morphologie Mathématique 1: Approaches Déterministes, Traité IC2, Série Signal et Image, Hermès Science Publications, pp. 201-232, 2008. - [52] J. Cousty, G. Bertrand, L. Najman, and M. Couprie, “On Watershed Cuts and Thinnings,”
Discrete Geometry for Computer Imagery, D. Coeurjolly, I. Sivignon, L. Tougne, and F. Dupont, eds., pp. 434-445, 2008.- [53] J. Serra,
Image Analysis and Mathematical Morphology. Academic Press, 1988.- [55] E. Breen and R. Jones, “Attribute Openings, Thinnings and Granulometries,”
Computer Vision and Image Understanding, vol. 64, no. 3, pp. 377-389, 1996.- [57] J. Cousty, L. Najman, G. Bertrand, and M. Couprie, “Minimum Spanning Tree by Watershed,” in preparation.
- [58] S. Beucher, “Watershed, Hierarchical Segmentation and Waterfall Algorithm,”
Proc. Second Int'l Symp. Math. Morphology and Its Applications to Image Processing, pp. 69-76, 1994.- [59] F. Meyer, “The Dynamics of Minima and Contours,”
Proc. Int'l Symp. Math. Morphology and Its Application to Image and Signal Processing, pp. 329-336, 1996. |