Journal of Mathematical Imaging and Vision

, Volume 61, Issue 3, pp 394–410 | Cite as

Soft Color Morphology: A Fuzzy Approach for Multivariate Images

  • Pedro BibiloniEmail author
  • Manuel González-Hidalgo
  • Sebastia Massanet


Mathematical morphology is a framework composed by a set of well-known image processing techniques, widely used for binary and grayscale images, but less commonly used to process color or multivariate images. In this paper, we generalize fuzzy mathematical morphology to process multivariate images in such a way that overcomes the problem of defining an appropriate order among colors. We introduce the soft color erosion and the soft color dilation, which are the foundations of the rest of operators. Besides studying their theoretical properties, we analyze their behavior and compare them with the corresponding morphological operators from other frameworks that deal with color images. The soft color morphology outstands when handling images in the CIEL\({}^*a{}^*b{}^*\) color space, where it guarantees that no colors with different chromatic values to the original ones are created. The soft color morphological operators prove to be easily customizable but also highly interpretable. Besides, they are fast operators and provide smooth outputs, more visually appealing than the crisp color transitions provided by other approaches.


Mathematical morphology Color image processing Fuzzy mathematical morphology CIEL\({}^*a{}^*b{}^*\) 



This work was partially supported by the Project TIN 2016-75404-P AEI/FEDER, UE. P. Bibiloni also benefited from the fellowship FPI/1645/2014 of the Conselleria d’Educació, Cultura i Universitats of the Govern de les Illes Balears under an operational program co-financed by the European Social Fund.


  1. 1.
    Angulo, J., Serra, J.: Modelling and segmentation of colour images in polar representations. Image Vis. Comput. 25(4), 475–495 (2007)CrossRefGoogle Scholar
  2. 2.
    Aptoula, E., Lefevre, S.: A comparative study on multivariate mathematical morphology. Pattern Recognit. 40(11), 2914–2929 (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    Aptoula, E., Lefevre, S.: On lexicographical ordering in multivariate mathematical morphology. Pattern Recognit. Lett. 29(2), 109–118 (2008)CrossRefGoogle Scholar
  4. 4.
    Baczyński, M., Jayaram, B.: Fuzzy Implications, Studies in Fuzziness and Soft Computing, vol. 231. Springer, Berlin (2008)zbMATHGoogle Scholar
  5. 5.
    Benavent, X., Dura, E., Vegara, F., Domingo, J.: Mathematical morphology for color images: an image-dependent approach. Math. Probl. Eng. 2012, 18 (2012)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bibiloni, P., González-Hidalgo, M., Massanet, S.: A real-time fuzzy morphological algorithm for retinal vessel segmentation. J. Real Time Image Process. (2017). Google Scholar
  7. 7.
    Bibiloni, P., González-Hidalgo, M., Massanet, S.: Soft color morphology. In: 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 1–6. IEEE (2017)Google Scholar
  8. 8.
    Bloch, I., Maître, H.: Fuzzy mathematical morphologies: a comparative study. Pattern Recognit. 28(9), 1341–1387 (1995)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bouchet, A., Alonso, P., Pastore, J.I., Montes, S., Díaz, I.: Fuzzy mathematical morphology for color images defined by fuzzy preference relations. Pattern Recognit. 60, 720–733 (2016)CrossRefGoogle Scholar
  10. 10.
    Chanussot, J., Lambert, P.: Total ordering based on space filling curves for multivalued morphology. Comput. Imaging Vis. 12, 51–58 (1998)zbMATHGoogle Scholar
  11. 11.
    Chevallier, E., & Angulo, J.J: The irregularity issue of total orders on metric spaces and its consequences for mathematical morphology. J Math Imaging Vis. 54, 344–357 (2016).
  12. 12.
    De Baets, B.: A fuzzy morphology: a logical approach. In: Ayyub, B.M., Gupta, M.M. (eds.) Uncertainty Analysis in Engineering and Sciences: Fuzzy Logic, Statistics, and Neural Network Approach, pp. 53–67. Springer, Berlin (1998)Google Scholar
  13. 13.
    De Witte, V., Schulte, S., Nachtegael, M., Mélange, T., Kerre, E.E.: A lattice-based approach to mathematical morphology for greyscale and colour images. In: Kaburlasos, V.G., Ritter, G.X. (eds.) Computational Intelligence Based on Lattice Theory, pp. 129–148. Springer, Berlin (2007)CrossRefGoogle Scholar
  14. 14.
    Gonzalez, R.C., Woods, R.E.: Digital image processing, 3rd edn. Prentice Hall (2007)Google Scholar
  15. 15.
    González-Hidalgo, M., Massanet, S., Mir, A., Ruiz-Aguilera, D.: A fuzzy morphological hit-or-miss transform for grey-level images: a new approach. Fuzzy Sets Syst. 286, 30–65 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Goutsias, J., Heijmans, H.J., Sivakumar, K.: Morphological operators for image sequences. Comput. Vis. Image Underst. 62(3), 326–346 (1995)CrossRefGoogle Scholar
  17. 17.
    Gu, C.: Multivalued Morphology and Its Application in Moving Object Segmentation and Tracking, pp. 345–352. Springer, Berlin (1996)zbMATHGoogle Scholar
  18. 18.
    Haas, A., Matheron, G., Serra, J.: Morphologie mathématique et granulométries en place. Ann. Mines 11, 736–753 (1967)Google Scholar
  19. 19.
    Haralick, R.M., Sternberg, S.R., Zhuang, X.: Image analysis using mathematical morphology. IEEE Trans. Pattern Anal. Mach. Intell. 9(4), 532–550 (1987)CrossRefGoogle Scholar
  20. 20.
    Kerre, E.E., Nachtegael, M.: Classical and fuzzy approaches towards mathematical morphology. Physica 52, 3–56 (2013). (Chap. 1)Google Scholar
  21. 21.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms, vol. 8. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  22. 22.
    Lézoray, O.: Complete lattice learning for multivariate mathematical morphology. J. Vis. Commun. Image Represent. 35, 220–235 (2016)CrossRefGoogle Scholar
  23. 23.
    Louverdis, G., Vardavoulia, M.I., Andreadis, I., Tsalides, P.: A new approach to morphological color image processing. Pattern Recognit. 35(8), 1733–1741 (2002)CrossRefzbMATHGoogle Scholar
  24. 24.
    Sartor, L.J., Weeks, A.R.: Morphological operations on color images. J. Electron. Imaging 10(2), 548–559 (2001)CrossRefGoogle Scholar
  25. 25.
    Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic, London (1982)zbMATHGoogle Scholar
  26. 26.
    Serra, J.: Image Analysis and Mathematical Morphology: Theoretical Advances, vol. 2. Academic, London (1988)Google Scholar
  27. 27.
    Valle, M.E., Valente, R.A.: Mathematical morphology on the spherical CIELab quantale with an application in color image boundary detection. J. Math. Imaging Vis. 57(2), 183–201 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    van de Gronde, J.J., Roerdink, J.B.: Group-invariant colour morphology based on frames. IEEE Trans. Image Process. 23(3), 1276–1288 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Velasco-Forero, S., Angulo, J.: Random projection depth for multivariate mathematical morphology. IEEE J. Sel. Top. Signal Process. 6(7), 753–763 (2012)CrossRefGoogle Scholar
  30. 30.
    Velasco-Forero, S., Angulo, J.: Vector ordering and multispectral morphological image processing. In: Celebi, M.E., Smolka, B. (eds.) Advances in Low-Level Color Image Processing, pp. 223–239. Springer, Berlin (2014)CrossRefGoogle Scholar
  31. 31.
    Wyszecki, G., Stiles, W.S.: Color Science: Concepts and Methods, Quantitative Data and Formulae, Wiley Series in Pure and Applied Optics, 2nd edn. Wiley, New York (2000)Google Scholar

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Authors and Affiliations

  1. 1.Soft Computing, Image Processing and Aggregation (SCOPIA) Research Group, Department of Mathematics and Computer ScienceUniversity of the Balearic IslandsPalmaSpain
  2. 2.Balearic Islands Health Research Institute (IdISBa)PalmaSpain

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