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Processing Multispectral Images via Mathematical Morphology

  • Andreas KleefeldEmail author
  • Bernhard Burgeth
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

In this chapter, we illustrate how to process multispectral and hyperspectral images via mathematical morphology. First, according to the number of channels the data are embeded into a sufficiently high dimensional space. This transformation utilizes a special geometric structure, namely double hypersimplices, for further processing the data. For example, RGB-color images are transformed into points within a specific double hypersimplex. It is explained in detail how to calculate the supremum and infimum of samples of those transformed data to allow for the meaningful definition of morphological operations such as dilation and erosion and in a second step top hats, gradients, and morphological Laplacian. Finally, numerical results are presented to explore the advantages and shortcomings of the new proposed approach.

Keywords

Hyperspectral Image Multispectral Image Mathematical Morphology Morphological Operation Base Triangle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Angula, J., Lefèvre, S., Lezoray, O.: Color representation and processing in polar color spaces. In: Fernandez-Maloigne, C., Robert-Inacio, F., Macaire, L. (eds.) Digital Color Imaging, pp. 1–40. Wiley-ISTE, Hoboken, New Jersey (2013)Google Scholar
  2. 2.
    Aptoula, E., Lefèvre, S.: A comparative study on multivariate mathematical morphology. Pattern Recognit. 40(11), 2914–2929 (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    Banon, G.J.F., Barrera, J., Braga-Neto, U.d.M., Hirata, N.S.T. (eds.): Proceedings of the 8th International Symposium on Mathematical Morphology: Volume 1 - Full Papers. Computational Imaging and Vision, vol. 1. Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos (2007)Google Scholar
  4. 4.
    Braun, K.M., Balasubramanian, R., Eschbach, R.: Development and evaluation of six gamut-mapping algorithms for pictorial images. In: Color Imaging Conference, pp. 144–148. IS&T - The Society for Imaging Science and Technology, Springfield (1999)Google Scholar
  5. 5.
    Burgeth, B., Kleefeld, A.: Morphology for color images via Loewner order for matrix fields. In: Luengo Hendriks, C.L., Borgefors, G., Strand, R. (eds.) Mathematical Morphology and Its Applications to Signal and Image Processing (Proceedings of the 11th International Symposium on Mathematical Morphology, Uppsala, 27–29 May). Lecture Notes in Computer Science, vol. 7883, pp. 243–254. Springer, Berlin (2013)CrossRefGoogle Scholar
  6. 6.
    Burgeth, B., Kleefeld, A.: An approach to color-morphology based on einstein addition and loewner order. Pattern Recognit. Lett. 47, 29–39 (2014)CrossRefGoogle Scholar
  7. 7.
    Burgeth, B., Papenberg, N., Bruhn, A., Welk, M., Feddern, C., Weickert, J.: Mathematical morphology based on the loewner ordering for tensor data. In: Ronse, C., Najman, L., Decencière, E. (eds.) Mathematical Morphology: 40 Years On, Computational Imaging and Vision, vol. 30, pp. 407–418. Springer, Dordrecht (2005)CrossRefGoogle Scholar
  8. 8.
    Burgeth, B., Welk, M., Feddern, C., Weickert, J.: Mathematical morphology on tensor data using the loewner ordering. In: Weickert, H.H.J. (ed.) Visualization and Processing of Tensor Fields. Springer, Berlin (2006)Google Scholar
  9. 9.
    Comer, M.L., Delp, E.J.: Morphological operations for color image processing. J. Electron. Imaging 8(3), 279–289 (1999)CrossRefGoogle Scholar
  10. 10.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic, Boston (1994)zbMATHGoogle Scholar
  11. 11.
    Luengo Hendriks, C.L., Borgefors, G., Strand, R. (eds.): Mathematical morphology and its applications to image and signal processing. In: Proceedings of the 11th International Symposium on Mathematical Morphology. Lecture Notes in Computer Science, vol. 7883. Springer, Berlin (2013)Google Scholar
  12. 12.
    Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)Google Scholar
  13. 13.
    Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)zbMATHGoogle Scholar
  14. 14.
    Ostwald, W.: Die Farbenfibel. Unesma, Leipzig (1916)Google Scholar
  15. 15.
    Ronse, C., Serra, J.: Algebraic foundations of morphology. In: Najman, L., Talbot, H. (eds.) Mathematical Morphology: From Theory to Applications, Chap. 2, pp. 35–80. ISTE/Wiley, London (2010)Google Scholar
  16. 16.
    Ronse, C., Najman, L., Decencière, E. (eds.): Mathematical Morphology: 40 Years On, Computational Imaging and Vision, vol. 30. Springer, Dordrecht (2005)Google Scholar
  17. 17.
    Serra, J.: Echantillonnage et estimation des phénomènes de transition minier. Ph.D. thesis, University of Nancy, France (1967)Google Scholar
  18. 18.
    Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic, London (1982)Google Scholar
  19. 19.
    Serra, J.: The false colour problem. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing. Proceedings of the 9th International Symposium on Mathematical Morphology. Lecture Notes in Computer Science, vol. 5720, Chap. 2, pp. 13–23. Springer, Heidelberg (2009)Google Scholar
  20. 20.
    Serra, J., Soille, P. (eds.): Mathematical Morphology and Its Applications to Image Processing. Computational Imaging and Vision, vol. 2. Kluwer, Dordrecht (1994)Google Scholar
  21. 21.
    Soille, P.: Morphological Image Analysis, 2nd edn. Springer, Berlin (2003)zbMATHGoogle Scholar
  22. 22.
    Soille, P., Pesaresi, M., Ouzounis, G. (eds.): Mathematical Morphology and Its Applications to Image and Signal Processing. Proceedings of the 10th International Symposium on Mathematical Morphology. Lecture Notes in Computer Science, vol. 6671. Springer, Berlin (2011)Google Scholar
  23. 23.
    Ungar, A.A.: Einstein’s special relativity: the hyperbolic geometric viewpoint. In: Conference on Mathematics, Physics and Philosophy on the Interpretations of Relativity, II. Budapest (2009)Google Scholar
  24. 24.
    van de Gronde, J.J., Roerdink, J.B.T.M.: Group-invariant frames for colour morphology. In: Luengo Hendriks, C.L., Borgefors, G., Strand, R. (eds.) Mathematical Morphology and Its Applications to Signal and Image Processing (Proceedings of the 11th International Symposium on Mathematical Morphology, Uppsala, 27–29 May). Lecture Notes in Computer Science, vol. 7883, pp. 267–278. Springer, Berlin (2013)CrossRefGoogle Scholar
  25. 25.
    Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.): Mathematical Morphology and Its Application to Signal and Image Processing. Proceedings of the 9th International Symposium on Mathematical Morphology. Lecture Notes in Computer Science, vol. 5720. Springer, Heidelberg (2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics, Natural Sciences and Computer ScienceBrandenburg University of Technology CottbusCottbusGermany
  2. 2.Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

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