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On the Probabilities of Hierarchical Watersheds

Conference paper
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Part of the Lecture Notes in Computer Science book series (LNCS, volume 11564)

Abstract

Hierarchical watersheds are obtained by iteratively merging the regions of a watershed segmentation. In the watershed segmentation of an image, each region contains exactly one (local) minimum of the original image. Therefore, the construction of a hierarchical watershed of any image I can be guided by a total order \(\prec \) on the set of minima of I. The regions that contain the least minima according to the order \(\prec \) are the first regions to be merged in the hierarchy. In fact, given any image I, for any hierarchical watershed \(\mathcal {H}\) of I, there exists more than one total order on the set of minima of I which could be used to obtain \(\mathcal {H}.\) In this article, we define the probability of a hierarchical watershed \(\mathcal {H}\) as the probability of \(\mathcal {H}\) to be the hierarchical watershed of I for an arbitrary total order on the set of minima of I. We introduce an efficient method to obtain the probability of hierarchical watersheds and we provide a characterization of the most probable hierarchical watersheds.

References

  1. 1.
    Angulo, J., Jeulin, D.: Stochastic watershed segmentation. In: ISMM. Springer (2007)Google Scholar
  2. 2.
    Audigier, R., Lotufo, R.: Uniquely-determined thinning of the tie-zone watershed based on label frequency. JMIV 27(2), 157–173 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beucher, S., Meyer, F.: The morphological approach to segmentation: the watershed transformation. Opt. Eng. N.Y. Marcel Dekker Inc. 34, 433–481 (1992)Google Scholar
  4. 4.
    Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: minimum spanning forests and the drop of water principle. IEEE PAMI 31(8), 1362–1374 (2009)CrossRefGoogle Scholar
  5. 5.
    Cousty, J., Najman, L.: Incremental algorithm for hierarchical minimum spanning forests and saliency of watershed cuts. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 272–283. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-21569-8_24CrossRefzbMATHGoogle Scholar
  6. 6.
    Cousty, J., Najman, L., Kenmochi, Y., Guimarães, S.: Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps. JMIV 60(4), 479–502 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cousty, J., Najman, L., Perret, B.: Constructive links between some morphological hierarchies on edge-weighted graphs. In: Hendriks, C.L.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 86–97. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38294-9_8CrossRefGoogle Scholar
  8. 8.
    Grimaud, M.: New measure of contrast: the dynamics. In: Image Algebra and Morphological Image Processing III, vol. 1769, pp. 292–306. International Society for Optics and Photonics (1992)Google Scholar
  9. 9.
    Jeulin, D.: Morphological probabilistic hierarchies for texture segmentation. Math. Morphol.-Theory Appl. 1(1), 216–324 (2016)Google Scholar
  10. 10.
    Najman, L., Cousty, J., Perret, B.: Playing with kruskal: algorithms for morphological trees in edge-weighted graphs. In: Hendriks, C.L.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 135–146. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38294-9_12CrossRefGoogle Scholar
  11. 11.
    Salembier, P., Garrido, L.: Binary partition tree as an efficient representation for image processing, segmentation, and information retrieval. IEEE Trans. Image Process. 9(4), 561–576 (2000)CrossRefGoogle Scholar
  12. 12.
    Straehle, C., Peter, S., Köthe, U., Hamprecht, F.A.: K-smallest spanning tree segmentations. In: Weickert, J., Hein, M., Schiele, B. (eds.) GCPR 2013. LNCS, vol. 8142, pp. 375–384. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40602-7_40CrossRefGoogle Scholar
  13. 13.
    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

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE Paris, UPEMNoisy-le-GrandFrance

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