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Abstract

Mathematical morphology on binary images can be fully described by set theory. However, it is not sufficient to formulate mathematical morphology for grey scale images. This type of images requires the introduction of the notion of partial order of grey levels, together with the definition of sup and inf operators. More generally, mathematical morphology is now described within the context of the lattice theory. For a few decades, attempts are made to use mathematical morphology on multivariate images, such as color images, mainly based on the notion of vector order. However, none of these attempts has given fully satisfying results. Instead of aiming directly at the multivariate case we propose an extension of mathematical morphology to an intermediary situation: images composed of a finite number of independent unordered labels.

Keywords

Mathematical morphology Labeled images Image filtering 

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References

  1. 1.
    Busch, C., Eberle, M.: Morphological Operations for Color-Coded Images. Proc. of the EUROGRAPHICS 1995 14(3), C193–C204 (1995)Google Scholar
  2. 2.
    Ronse, C., Agnus, V.: Morphology on label images: Flat-type operators and connections. Journal of Mathematical Imaging and Vision 22(2-3), 283–307 (2005)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Hanbury, A., Serra, J.: Morphological operators on the unit circle. IEEE Trans. Image Processing 10(12), 1842–1850 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Heijmans, H.J.A.M.: Morphological image operators. Academic Press, Boston (1994)zbMATHGoogle Scholar
  5. 5.
    Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)zbMATHGoogle Scholar
  6. 6.
    Meyer, F.: Adjunctions on the lattice of hierarchies. HAL, hal-00566714, 24p (2011)Google Scholar
  7. 7.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)zbMATHGoogle Scholar
  8. 8.
    Soille, P.: Morphological Image Analysis. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  9. 9.
    Ronse, C.: Ordering Partial Partition for Image Segmentation and Filtering: Merging, Creating and Inflating Blocks. Journal of Mathematical Imaging and Vision 49(1), 202–233 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Salembier, P., Serra, J.: Flat Zones Filtering, Connected Operators, and Filters by reconstruction. IEEE Trans. on Images Processing 4(8), 1153–1160 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Emmanuel Chevallier
    • 1
    Email author
  • Augustin Chevallier
    • 2
  • Jesús Angulo
    • 1
  1. 1.CMM-Centre de Morphologie Mathématique, MINES ParisTechPSL-Research UniversityParisFrance
  2. 2.Ecole Normale Supérieur de CachanCachanFrance

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