Incremental Algorithm for Hierarchical Minimum Spanning Forests and Saliency of Watershed Cuts

  • Jean Cousty
  • Laurent Najman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6671)


We study hierarchical segmentations that are optimal in the sense of minimal spanning forests of the original image. We introduce a region-merging operation called uprooting, and we prove that optimal hierarchical segmentations are equivalent to the ones given by uprooting a watershed-cut based segmentation. Based on those theoretical results, we propose an efficient algorithm to compute such hierarchies, as well as the first saliency map algorithm compatible with the morphological filtering framework.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allène, C., Audibert, J.-Y., Couprie, M., Keriven, R.: Some links between extremum spanning forests, watersheds and min-cuts. IVC 28(10), 1460–1471 (2010)CrossRefGoogle Scholar
  2. 2.
    Arbeláez, P.A., Cohen, L.D.: A metric approach to vector-valued image segmentation. IJCV 69(1), 119–126 (2006)CrossRefGoogle Scholar
  3. 3.
    Audigier, R., Lotufo, R.: Seed-relative segmentation robustness of watershed and fuzzy connectedness approaches. In: IEEE SIBGRAPI 2007, pp. 61–70 (2007)Google Scholar
  4. 4.
    Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Latin American Theoretical Informatics, pp. 88–94 (2000)Google Scholar
  5. 5.
    Beucher, S.: Watershed, hierarchical segmentation and waterfall algorithm. In: ISMM 1994, pp. 69–76 (1994)Google Scholar
  6. 6.
    Couprie, C., Grady, L., Najman, L., Talbot, H.: Power Watersheds: A Unifying Graph Based Optimization Framework. PAMI (to appear, 2011)Google Scholar
  7. 7.
    Couprie, M., Najman, L., Bertrand, G.: Quasi-linear algorithms for the topological watershed. JMIV 22(2-3), 231–249 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cousty, J., Najman, L., Serra, J.: Raising in watershed lattices. In: 15th IEEE ICIP 2008, pp. 2196–2199 (2008)Google Scholar
  9. 9.
    Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed Cuts: Minimum Spanning Forests and the Drop of Water Principle. PAMI 31(8), 1362–1374 (2009)CrossRefGoogle Scholar
  10. 10.
    Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: thinnings, shortest-path forests and topological watersheds. PAMI 32(5), 925–939 (2010)CrossRefGoogle Scholar
  11. 11.
    Cousty, J., Najman, L., Bertrand, G., Couprie, M.: Weighted fusion graphs: merging properties and watersheds. DAM 156(15 ), 3011–3027 (2008)MathSciNetMATHGoogle Scholar
  12. 12.
    Cousty, J., Najman, L., Serra, J.: Some morphological operators in graph spaces. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 149–160. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Felzenszwalb, P., Huttenlocher, D.: Efficient graph-based image segmentation. International Journal of Computer Vision 59, 167–181 (2004)CrossRefGoogle Scholar
  14. 14.
    Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34, 596–615 (1987)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gower, J.C., Ross, G.J.S.: Minimum spanning tree and single linkage cluster analysis. Appl. Stats. 18, 54–64 (1969)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Guigues, L., Cocquerez, J.P., Men, H.L.: Scale-sets image analysis. IJCV 68(3), 289–317 (2006)CrossRefGoogle Scholar
  17. 17.
    Jardine, N., Sibson, R.: Mathematical taxonomy. Wiley, Chichester (1971)MATHGoogle Scholar
  18. 18.
    Marcotegui, B., Beucher, S.: Fast implementation of waterfall based on graphs. In: ISMM 2005, pp. 177–186 (2005)Google Scholar
  19. 19.
    Meyer, F.: Minimum spanning forests for morphological segmentation. In: ISMM 1994, pp. 77–84 (1994)Google Scholar
  20. 20.
    Meyer, F.: The dynamics of minima and contours. In: ISMM, pp. 329–336 (1996)Google Scholar
  21. 21.
    Meyer, F., Najman, L.: Segmentation, minimum spanning tree and hierarchies. In: Mathematical Morphology, ch. 9, pp. 229–261. ISTE-Wiley (2010)Google Scholar
  22. 22.
    Morris, O.J., Lee, M.d.J., Constantinides, A.G.: Graph theory for image analysis: an approach based on the shortest spanning tree. IEE Proc. on Communications, Radar and Signal 133(2), 146–152 (1986)CrossRefGoogle Scholar
  23. 23.
    Najman, L., Schmitt, M.: Geodesic saliency of watershed contours and hierarchical segmentation. PAMI 18(12), 1163–1173 (1996)CrossRefGoogle Scholar
  24. 24.
    Najman, L.: On the equivalence between hierarchical segmentations and ultrametric watersheds. JMIV 40(3), 231–247 (2011)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Najman, L., Couprie, M.: Building the component tree in quasi-linear time. IEEE TIP 15(11), 3531–3539 (2006)Google Scholar
  26. 26.
    Najman, L., Couprie, M., Bertrand, G.: Watersheds, mosaics and the emergence paradigm. DAM 147(2-3), 301–324 (2005)MathSciNetMATHGoogle Scholar
  27. 27.
    Philipp-Foliguet, S., Jordan, M., Najman, L., Cousty, J.: Artwork 3D Model Database Indexing and Classification. Patt. Recogn. 44(3), 588–597 (2011)CrossRefMATHGoogle Scholar
  28. 28.
    Salembier, P., Oliveras, A., Garrido, L.: Anti-extensive connected operators for image and sequence processing. IEEE TIP 7(4), 555–570 (1998)Google Scholar
  29. 29.
    Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. J. ACM 22, 215–225 (1975)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Vachier, C., Meyer, F.: Extinction value: a new measurement of persistence. In: IEEE Workshop on Nonlinear Signal and Image Processing, pp. 254–257 (1995)Google Scholar
  31. 31.
    Zahn, C.T.: Graph-theoretical methods for detecting and descibing gestalt clusters. IEEE Transactions on Computers C-20(1), 99–112 (1971)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jean Cousty
    • 1
  • Laurent Najman
    • 1
  1. 1.Laboratoire d’Informatique Gaspard-MongeUniversité Paris-Est, A3SI, ESIEEFrance

Personalised recommendations