Advertisement

Abstract

This paper introduces a generalization of self-dual marked flattenings defined in the lattice of mappings. This definition provides a way to associate a self-dual operator to every mapping that decomposes an element into sub-elements (i.e. gives a cover). Contrary to classical flattenings whose definition relies on the complemented structure of the powerset lattices, our approach uses the pseudo relative complement and supplement of the bi-Heyting algebra and a new notion of inf-structuring functions that provides a very general way to structure the space. We show that using an inf-structuring function based on connections allows to recover the original definition of marked flattenings and we provide, as an example, a simple inf-structuring function whose derived self-dual operator better preserves contrasts and does not introduce new pixel values.

Keywords

inf-structuring function self-dual operator flattening Heyting algebra connection hyper-connection image processing mathematical morphology 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Braga-Neto, U., Goutsias, J.: A theoretical tour of connectivity in image processing and analysis. JMIV 19(1), 5–31 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Crespo, J.: Adjacency stable connected operators and set levelings. IVC 28(10), 1483–1490 (2010)CrossRefGoogle Scholar
  3. 3.
    Heijmans, H.J.A.M.: Self-dual morphological operators and filters. JMIV 6, 15–36 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Heijmans, H.J.A.M., Keshet, R.: Inf-semilattice approach to self-dual morphology. JMIV 17(1), 55–80 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Keshet, R.: Shape-tree semilattice. JMIV 22(2-3), 309–331 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Lawvere, F.W.: Intrinsic co-heyting boundaries and the leiniz rule in certain toposes. In: Category Theory. LNM, vol. 1488, pp. 279–281. Springer (1991)Google Scholar
  7. 7.
    Meyer, F.: From connected operators to levelings. In: Mathematical Morphology and its Applications to Image and Signal Processing, pp. 191–198. Kluwer (1998)Google Scholar
  8. 8.
    Meyer, F.: The levelings. In: Mathematical Morphology and its Applications to Image and Signal Processing, pp. 199–206. Kluwer (1998)Google Scholar
  9. 9.
    Monasse, P., Guichard, F.: Fast computation of a contrast-invariant image representation. IEEE TIP 9(5), 860–872 (2000)Google Scholar
  10. 10.
    Perret, B., Lefèvre, S., Collet, C.: Toward a new axiomatic for hyper-connections. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 85–95. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  11. 11.
    Perret, B., Lefèvre, S., Collet, C., Slezak, É.: Hyperconnections and hierarchical representations for grayscale and multiband image processing. IEEE TIP 21(1), 14–27 (2012)zbMATHGoogle Scholar
  12. 12.
    Reyes, G.E., Zolfaghari, H.: Bi-heyting algebras, toposes and modalities. Journal of Philosophical Logic 25, 25–43 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Roerdink, J.B.T.M.: Adaptivity and group invariance in mathematical morphology. In: IEEE ICIP 2009, pp. 2253–2256 (2009)Google Scholar
  14. 14.
    Salembier, P., Oliveras, A., Garrido, L.: Anti-extensive connected operators for image and sequence processing. IEEE TIP 7(4), 555–570 (1998)Google Scholar
  15. 15.
    Serra, J.: The centre and self-dual filtering. In: Image Analysis and Mathematical Morphology. II: Theoretical Advances, pp. 159–180. Academic Press (1988)Google Scholar
  16. 16.
    Serra, J.: Mathematical morphology for boolean lattices. In: Image Analysis and Mathematical Morphology. II: Theoretical Advances, pp. 37–58. Academic Press (1988)Google Scholar
  17. 17.
    Serra, J.: Connectivity on complete lattices. JMIV 9(3), 231–251 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Serra, J.: Connections for sets and functions. Fundam. Inform. 41, 147–186 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Serra, J., Vachier, C., Meye, F.: Levelings. In: Mathematical Morphology, pp. 199–228. ISTE and John Wiley & Sons (2010)Google Scholar
  20. 20.
    Soille, P.: Beyond self-duality in morphological image analysis. IVC 23, 249–257 (2005)CrossRefGoogle Scholar
  21. 21.
    Stell, J.G.: Relations in mathematical morphology with applications to graphs and rough sets. In: Winter, S., Duckham, M., Kulik, L., Kuipers, B. (eds.) COSIT 2007. LNCS, vol. 4736, pp. 438–454. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  22. 22.
    Stell, J.G., Worboys, M.F.: The algebraic structure of sets of regions. In: Frank, A.U. (ed.) COSIT 1997. LNCS, vol. 1329, pp. 163–174. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  23. 23.
    Vickers, S.: Topology via Logic. Cambridge University Press (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Benjamin Perret
    • 1
  1. 1.Laboratoire d’Informatique Gaspard-Monge, Equipe A3SI, ESIEE ParisUniversité Paris-EstNoisy le Grand CEDEXFrance

Personalised recommendations