On the Equivalence Between Hierarchical Segmentations and Ultrametric Watersheds

Abstract

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.

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References

  1. 1.

    Guigues, L., Cocquerez, J.P., Men, H.L.: Scale-sets image analysis. Int. J. Comput. Vis. 68(3), 289–317 (2006)

    Article  Google Scholar 

  2. 2.

    Barthélemy, J.P., Brucker, F., Osswald, C.: Combinatorial optimization and hierarchical classifications. 4OR 2(3), 179–219 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Soille, P.: Constrained connectivity for hierarchical image decomposition and simplification. IEEE Trans. Pattern Anal. Mach. Intell. 30(7), 1132–1145 (2008)

    Article  Google 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)

    Google Scholar 

  6. 6.

    Johnson, S.: Hierarchical clustering schemes. Psychometrika 32, 241–254 (1967)

    Article  Google Scholar 

  7. 7.

    Jardine, N., Sibson, R.: Mathematical Taxonomy. Wiley, New York (1971)

    Google Scholar 

  8. 8.

    Diday, E.: Spatial classification. Discrete Appl. Math. 156(8), 1271–1294 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Serra, J.: A lattice approach to image segmentation. J. Math. Imaging Vis. 24(1), 83–130 (2006)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Ronse, C.: Partial partitions, partial connections and connective segmentation. J. Math. Imaging Vis. 32(2), 97–105 (2008)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Pavlidis, T.: Hierarchies in structural pattern recognition. Proc. IEEE 67(5), 737–744 (1979)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  16. 16.

    Meyer, F.: Morphological segmentation revisited. In: Space, Structure and Randomness, pp. 315–347. Springer, Berlin (2005)

    Google 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

    Google 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)

    MathSciNet  Google Scholar 

  20. 20.

    Bertrand, G.: On topological watersheds. J. Math. Imaging Vis. 22(2–3), 217–230 (2005)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Najman, L., Couprie, M., Bertrand, G.: Watersheds, mosaics and the emergence paradigm. Discrete Appl. Math. 147(2–3), 301–324 (2005)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    Article  Google 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)

  27. 27.

    Diestel, R.: Graph Theory. Graduate Texts in Mathematics. Springer, Berlin (1997)

    Google Scholar 

  28. 28.

    Kong, T., Rosenfeld, A.: Digital topology: Introduction and survey. Comput. Vis. Graph. Image Process. 48(3), 357–393 (1989)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  32. 32.

    Najman, L., Couprie, M.: Building the component tree in quasi-linear time. IEEE Trans. Image Process. 15(11), 3531–3539 (2006)

    Article  Google 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)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Krasner, M.: Espaces ultramétriques. C. R. Math. 219, 433–435 (1944)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Leclerc, B.: Description combinatoire des ultramétriques. Math. Sci. Hum. 73, 5–37 (1981)

    MathSciNet  Google Scholar 

  36. 36.

    Gower, J., Ross, G.: Minimum spanning tree and single linkage cluster analysis. Appl. Stat. 18, 54–64 (1969)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Khalimsky, E., Kopperman, R., Meyer, P.: Computer graphics and connected topologies on finite ordered sets. Topol. Appl. 36, 1–17 (1990)

    MathSciNet  MATH  Article  Google 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)

    MATH  Google Scholar 

  41. 41.

    Bertrand, G.: On critical kernels. C. R. Acad. Sci., Sér. 1 Math. 345 363–367 (2007)

    MathSciNet  MATH  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  45. 45.

    Bender, M., Farach-Colton, M.: The lca problem revisited. In: Latin Amer. Theor. INformatics, pp. 88–94 (2000)

    Google 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 

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Correspondence to Laurent Najman.

Additional information

This work was partially supported by ANR grant SURF-NT05-2_45825.

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Najman, L. On the Equivalence Between Hierarchical Segmentations and Ultrametric Watersheds. J Math Imaging Vis 40, 231–247 (2011). https://doi.org/10.1007/s10851-011-0259-1

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Keywords

  • Image segmentation
  • Classification
  • Hierarchy
  • Watershed-based segmentation