Journal of Mathematical Imaging and Vision

, Volume 40, Issue 3, pp 231–247 | Cite as

On the Equivalence Between Hierarchical Segmentations and Ultrametric Watersheds

  • Laurent Najman


We study hierarchical segmentation in the framework of edge-weighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical segmentations. We end this paper by showing how to use the proposed framework in practice on the example of constrained connectivity; in particular it allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.


Image segmentation Classification Hierarchy Watershed-based segmentation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Guigues, L., Cocquerez, J.P., Men, H.L.: Scale-sets image analysis. Int. J. Comput. Vis. 68(3), 289–317 (2006) CrossRefGoogle Scholar
  2. 2.
    Barthélemy, J.P., Brucker, F., Osswald, C.: Combinatorial optimization and hierarchical classifications. 4OR 2(3), 179–219 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Soille, P.: Constrained connectivity for hierarchical image decomposition and simplification. IEEE Trans. Pattern Anal. Mach. Intell. 30(7), 1132–1145 (2008) CrossRefGoogle Scholar
  4. 4.
    Najman, L.: Ultrametric watersheds. In: ISMM 09. LNCS, vol. 5720, pp. 181–192. Springer, Berlin (2009) Google Scholar
  5. 5.
    Benzécri, J.: L’Analyse des Données: La Taxinomie, vol. 1. Dunod, Paris (1973) zbMATHGoogle Scholar
  6. 6.
    Johnson, S.: Hierarchical clustering schemes. Psychometrika 32, 241–254 (1967) CrossRefGoogle Scholar
  7. 7.
    Jardine, N., Sibson, R.: Mathematical Taxonomy. Wiley, New York (1971) zbMATHGoogle Scholar
  8. 8.
    Diday, E.: Spatial classification. Discrete Appl. Math. 156(8), 1271–1294 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Serra, J.: A lattice approach to image segmentation. J. Math. Imaging Vis. 24(1), 83–130 (2006) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ronse, C.: Partial partitions, partial connections and connective segmentation. J. Math. Imaging Vis. 32(2), 97–105 (2008) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pavlidis, T.: Hierarchies in structural pattern recognition. Proc. IEEE 67(5), 737–744 (1979) CrossRefGoogle Scholar
  12. 12.
    Najman, L., Schmitt, M.: Geodesic saliency of watershed contours and hierarchical segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 18(12), 1163–1173 (1996) CrossRefGoogle Scholar
  13. 13.
    Arbeláez, P.A., Cohen, L.D.: A metric approach to vector-valued image segmentation. Int. J. Comput. Vis. 69(1), 119–126 (2006) CrossRefGoogle Scholar
  14. 14.
    Pavlidis, T.: Segmentation techniques. In: Structural Pattern Recognition. Springer Series in Electrophysics, vol. 1, pp. 90–123. Springer, Berlin (1977). Chaps. 4–5 Google Scholar
  15. 15.
    Meyer, F., Beucher, S.: Morphological segmentation. J. Vis. Commun. Image Represent. 1(1), 21–46 (1990) CrossRefGoogle Scholar
  16. 16.
    Meyer, F.: Morphological segmentation revisited. In: Space, Structure and Randomness, pp. 315–347. Springer, Berlin (2005) CrossRefGoogle Scholar
  17. 17.
    Meyer, F., Najman, L.: Segmentation, minimum spanning tree and hierarchies. In: Najman, L., Talbot, H. (eds.) Mathematical Morphology: from Theory to Application, pp. 229–261. ISTE-Wiley, London (2010) Google Scholar
  18. 18.
    Najman, L., Talbot, H. (eds.): Mathematical Morphology: from Theory to Applications, p. 507. ISTE-Wiley, London (2010). ISBN:9781848212152 zbMATHGoogle Scholar
  19. 19.
    Roerdink, J.B.T.M., Meijster, A.: The watershed transform: Definitions, algorithms and parallelization strategies. Fundam. Inform. 41(1–2), 187–228 (2001) MathSciNetGoogle Scholar
  20. 20.
    Bertrand, G.: On topological watersheds. J. Math. Imaging Vis. 22(2–3), 217–230 (2005) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Najman, L., Couprie, M., Bertrand, G.: Watersheds, mosaics and the emergence paradigm. Discrete Appl. Math. 147(2–3), 301–324 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Cousty, J., Bertrand, G., Couprie, M., Najman, L.: Fusion graphs: merging properties and watersheds. J. Math. Imaging Vis. 30(1), 87–104 (2008) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Cousty, J., Najman, L., Bertrand, G., Couprie, M.: Weighted fusion graphs: merging properties and watersheds. Discrete Appl. Math. 156(15), 3011–3027 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: minimum spanning forests and the drop of water principle. IEEE Trans. Pattern Anal. Mach. Intell. 31(8), 1362–1374 (2009) CrossRefGoogle Scholar
  25. 25.
    Couprie, C., Grady, L., Najman, L., Talbot, H.: Power watersheds: A new image segmentation framework extending graph cuts, random walker and optimal spanning forest. In: International Conference on Computer Vision (ICCV’09), Kyoto, Japan, October, 2009. IEEE Press, New York (2009) Google Scholar
  26. 26.
    Couprie, C., Grady, L., Najman, L., Talbot, H.: Power Watersheds: A Unifying Graph Based Optimization Framework. IEEE Trans. Pattern Anal. Mach. Intell. (2010, to appear) Google Scholar
  27. 27.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics. Springer, Berlin (1997) zbMATHGoogle Scholar
  28. 28.
    Kong, T., Rosenfeld, A.: Digital topology: Introduction and survey. Comput. Vis. Graph. Image Process. 48(3), 357–393 (1989) CrossRefGoogle Scholar
  29. 29.
    Cousty, J., Najman, L., Serra, J.: Some morphological operators in graph spaces. In: ISMM 09. LNCS, vol. 5720, pp. 49–160 (2009) Google Scholar
  30. 30.
    Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: thinnings, shortest-path forests and topological watersheds. IEEE Trans. Pattern Anal. Mach. Intell. 32(5), 925–939 (2010) CrossRefGoogle Scholar
  31. 31.
    Salembier, P., Oliveras, A., Garrido, L.: Anti-extensive connected operators for image and sequence processing. IEEE Trans. Image Process. 7(4), 555–570 (1998) CrossRefGoogle Scholar
  32. 32.
    Najman, L., Couprie, M.: Building the component tree in quasi-linear time. IEEE Trans. Image Process. 15(11), 3531–3539 (2006) CrossRefGoogle Scholar
  33. 33.
    Couprie, M., Najman, L., Bertrand, G.: Quasi-linear algorithms for the topological watershed. J. Math. Imaging Vis. 22(2–3), 231–249 (2005) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Krasner, M.: Espaces ultramétriques. C. R. Math. 219, 433–435 (1944) MathSciNetzbMATHGoogle Scholar
  35. 35.
    Leclerc, B.: Description combinatoire des ultramétriques. Math. Sci. Hum. 73, 5–37 (1981) MathSciNetGoogle Scholar
  36. 36.
    Gower, J., Ross, G.: Minimum spanning tree and single linkage cluster analysis. Appl. Stat. 18, 54–64 (1969) MathSciNetCrossRefGoogle Scholar
  37. 37.
    Kruskal, J.B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 7, 48–50 (1956) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Khalimsky, E., Kopperman, R., Meyer, P.: Computer graphics and connected topologies on finite ordered sets. Topol. Appl. 36, 1–17 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Alexandroff, P., Hopf, H.: Topology. Springer, Berlin (1937) Google Scholar
  40. 40.
    Alexandroff, P.: Diskrete Räume. Math. USSR Sb. 2(3), 501–518 (1937) zbMATHGoogle Scholar
  41. 41.
    Bertrand, G.: On critical kernels. C. R. Acad. Sci., Sér. 1 Math. 345 363–367 (2007) MathSciNetzbMATHGoogle Scholar
  42. 42.
    Cousty, J., Bertrand, G., Couprie, M., Najman, L.: Collapses and watersheds in pseudomanifolds. In: Proceedings of the 13th IWCIA, pp. 397–410. Springer, Berlin (2009) Google Scholar
  43. 43.
    Nagao, M., Matsuyama, T., Ikeda, Y.: Region extraction and shape analysis in aerial photographs. Comput. Graph. Image Process. 10(3), 195–223 (1979) CrossRefGoogle Scholar
  44. 44.
    Mattiussi, C.: The finite volume, finite difference, and finite elements methods as numerical methods for physical field problems. Adv. Imaging Electron Phys. 113, 1–146 (2000) CrossRefGoogle Scholar
  45. 45.
    Bender, M., Farach-Colton, M.: The lca problem revisited. In: Latin Amer. Theor. INformatics, pp. 88–94 (2000) CrossRefGoogle Scholar
  46. 46.
    Soille, P., Grazzini, J.: Constrained connectivity and transition regions. In: ISMM 09. LNCS, vol. 5720, pp. 59–69. Springer, Berlin (2009) Google Scholar
  47. 47.
    Cousty, J., Najman, L., Serra, J.: Raising in watershed lattices. In: 15th IEEE ICIP’08, San Diego, USA, October, 2008, pp. 2196–2199. (2008) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Laboratoire d’Informatique Gaspard-Monge, Equipe A3SI, ESIEEUniversité Paris-EstParisFrance

Personalised recommendations