Soft Color Morphology: A Fuzzy Approach for Multivariate Images

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. 1.

    Angulo, J., Serra, J.: Modelling and segmentation of colour images in polar representations. Image Vis. Comput. 25(4), 475–495 (2007)

    Article  Google Scholar 

  2. 2.

    Aptoula, E., Lefevre, S.: A comparative study on multivariate mathematical morphology. Pattern Recognit. 40(11), 2914–2929 (2007)

    Article  MATH  Google Scholar 

  3. 3.

    Aptoula, E., Lefevre, S.: On lexicographical ordering in multivariate mathematical morphology. Pattern Recognit. Lett. 29(2), 109–118 (2008)

    Article  Google Scholar 

  4. 4.

    Baczyński, M., Jayaram, B.: Fuzzy Implications, Studies in Fuzziness and Soft Computing, vol. 231. Springer, Berlin (2008)

    MATH  Google 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)

    Article  MATH  Google 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). https://doi.org/10.1007/s11554-018-0748-1

    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)

  8. 8.

    Bloch, I., Maître, H.: Fuzzy mathematical morphologies: a comparative study. Pattern Recognit. 28(9), 1341–1387 (1995)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

  10. 10.

    Chanussot, J., Lambert, P.: Total ordering based on space filling curves for multivalued morphology. Comput. Imaging Vis. 12, 51–58 (1998)

    MATH  Google 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). https://doi.org/10.1007/s10851-015-0607-7

  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)

    Google Scholar 

  14. 14.

    Gonzalez, R.C., Woods, R.E.: Digital image processing, 3rd edn. Prentice Hall (2007)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Goutsias, J., Heijmans, H.J., Sivakumar, K.: Morphological operators for image sequences. Comput. Vis. Image Underst. 62(3), 326–346 (1995)

    Article  Google Scholar 

  17. 17.

    Gu, C.: Multivalued Morphology and Its Application in Moving Object Segmentation and Tracking, pp. 345–352. Springer, Berlin (1996)

    MATH  Google 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)

    Article  Google 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)

    Book  MATH  Google Scholar 

  22. 22.

    Lézoray, O.: Complete lattice learning for multivariate mathematical morphology. J. Vis. Commun. Image Represent. 35, 220–235 (2016)

    Article  Google 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)

    Article  MATH  Google Scholar 

  24. 24.

    Sartor, L.J., Weeks, A.R.: Morphological operations on color images. J. Electron. Imaging 10(2), 548–559 (2001)

    Article  Google Scholar 

  25. 25.

    Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic, London (1982)

    MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    Article  Google 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)

    Google 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 

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Pedro Bibiloni.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bibiloni, P., González-Hidalgo, M. & Massanet, S. Soft Color Morphology: A Fuzzy Approach for Multivariate Images. J Math Imaging Vis 61, 394–410 (2019). https://doi.org/10.1007/s10851-018-0849-2

Download citation

Keywords

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