Abstract
Mathematical morphology is a theory with applications in image processing and analysis. In a supervised approach to mathematical morphology, pixel values are ranked according to sets of foreground and background elements specified a priori by the user. In this paper, we introduce a supervised fuzzy color-based approach to color mathematical morphology that provides an elegant alternative to the support vector machine-based approach developed by Velasco-Forero and Angulo. Briefly, color elements are ranked according to the degree of truth of the proposition “the considered color is a foreground color but it is not a background color” in the new supervised color morphological approach. Furthermore, the vagueness and uncertainty inherent to the description of colors by humans can be naturally incorporated in the new approach using the concept of fuzzy colors.
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Notes
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The surface of a mapping h in the red-green plane is formally defined by the set \(\{(x,y,z):z=h(x,y,0),0 \le x \le 1, 0 \le y \le 1\}\).
References
Heijmans, H.J.A.M.: Mathematical morphology: a modern approach in image processing based on algebra and geometry. SIAM Rev. 37(1), 1–36 (1995)
Soille, P.: Morphological Image Analysis. Springer, Berlin (1999). https://doi.org/10.1007/978-3-662-03939-7
Braga-Neto, U., Goutsias, J.: Supremal multiscale signal analysis. SIAM J. Math. Anal. 36(1), 94–120 (2004)
Gonzalez-Hidalgo, M., Massanet, S., Mir, A., Ruiz-Aguilera, D.: On the choice of the pair conjunction-implication into the fuzzy morphological edge detector. IEEE Trans. Fuzzy Syst. 23(4), 872–884 (2015)
Rittner, L., Campbell, J., Freitas, P., Appenzeller, S., Pike, G.B., Lotufo, R.: Analysis of scalar maps for the segmentation of the corpus callosum in diffusion tensor fields. J. Math. Imaging Vis. 45, 214–226 (2013)
Serra, J.: A lattice approach to image segmentation. J. Math. Imaging Vis. 24, 83–130 (2006)
Sternberg, S.: Grayscale morphology. Comput. Vis. Graph. Image Process. 35, 333–355 (1986)
Bloch, I.: Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology. Inf. Sci. 181(10), 2002–2015 (2011)
De Baets, B.: Fuzzy morphology: a logical approach. In: Ayyub, B.M., Gupta, M.M. (eds.) Uncertainty Analysis in Engineering and Science: Fuzzy Logic, Statistics, and Neural Network Approach, pp. 53–67. Kluwer Academic Publishers, Norwell (1997)
Nachtegael, M., Kerre, E.E.: Connections between binary, gray-scale and fuzzy mathematical morphologies. Fuzzy Sets Syst. 124(1), 73–85 (2001)
Sussner, P., Valle, M.E.: Classification of fuzzy mathematical morphologies based on concepts of inclusion measure and duality. J. Math. Imaging Vis. 32(2), 139–159 (2008)
Ronse, C.: Why mathematical morphology needs complete lattices. Sig. Process. 21(2), 129–154 (1990)
Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1993)
Grätzer, G., et al.: General Lattice Theory, 2nd edn. Birkhäuser Verlag, Basel (2003)
Aptoula, E., Lefèvre, S.: A comparative study on multivariate mathematical morphology. Pattern Recogn. 40(11), 2914–2929 (2007)
Lézoray, O.: Complete lattice learning for multivariate mathematical morphology. J. Vis. Commun. Image Represent. 35, 220–235 (2016)
Aptoula, E., Lefèvre, S.: On lexicographical ordering in multivariate mathematical morphology. Pattern Recogn. Lett. 29(2), 109–118 (2008)
Serra, J.: The “false colour” problem. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 13–23. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03613-2_2
Hanbury, A., Serra, J.: Mathematical morphology in the CIELAB space. Image Anal. Stereol. 21, 201–206 (2002)
Barnett, V.: The ordering of multivariate data. J. Roy. Stat. Soc. A 3, 318–355 (1976)
Louverdis, G., Andreadis, I.: Soft morphological filtering using a fuzzy model and its application to colour image processing. Formal Pattern Anal. Appl. 6(4), 257–268 (2004)
Velasco-Forero, S., Angulo, J.: Random projection depth for multivariate mathematical morphology. IEEE J. Sel. Top. Sig. Process. 6(7), 753–763 (2012)
Sartor, L.J., Weeks, A.R.: Morphological operations on color images. J. Electron. Imaging 10(2), 548–559 (2001)
Al-Otum, H.M.: A novel set of image morphological operators using a modified vector distance measure with color pixel classification. J. Vis. Commun. Image Represent. 30, 46–63 (2015)
Angulo, J.: Morphological colour operators in totally ordered lattices based on distances: application to image filtering, enhancement and analysis. Comput. Vis. Image Underst. 107(1–2), 56–73 (2007)
Comer, M.L., Delp, E.J.: Morphological operations for color image processing. J. Electron. Imaging 8(3), 279–289 (1999)
Deborah, H., Richard, N., Hardeberg, J.Y.: Spectral ordering assessment using spectral median filters. In: Benediktsson, J.A., Chanussot, J., Najman, L., Talbot, H. (eds.) ISMM 2015. LNCS, vol. 9082, pp. 387–397. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18720-4_33
Ledoux, A., Richard, N., Capelle-Laizé, A.S., Fernandez-Maloigne, C.: Perceptual color hit-or-miss transform: application to dermatological image processing. SIViP 9(5), 1081–1091 (2015)
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)
Velasco-Forero, S., Angulo, J.: Supervised ordering in \(\mathbb{R}^p\): application to morphological processing of hyperspectral images. IEEE Trans. Image Process. 20(11), 3301–3308 (2011)
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. LNCVB, vol. 11, pp. 223–239. Springer, Dordrecht (2014). https://doi.org/10.1007/978-94-007-7584-8_7
Chamorro-Martínez, J., Soto-Hidalgo, J.M., Martínez-Jiménez, P.M., Sánchez, D.: Fuzzy color spaces: a conceptual approach to color vision. IEEE Trans. Fuzzy Syst. 25(5), 1264–1280 (2017)
Acharya, T., Ray, A.: Image Processing: Principles and Applications. Wiley, Hoboken (2005)
Pratt, W.: Digital Image Processing, 4th edn. Wiley, Hoboken (2007)
Haykin, S.: Neural Networks and Learning Machines, 3rd edn. Prentice-Hall, Upper Saddle River (2009)
Soto-Hidalgo, J.M., Martinez-Jimenez, P.M., Chamorro-Martinez, J., Sanchez, D.: JFCS: a color modeling java software based on fuzzy color spaces. IEEE Comput. Intell. Mag. 11(2), 16–28 (2016)
Nguyen, H.T., Walker, E.A.: A First Course in Fuzzy Logic, 2nd edn. Chapman & Hall/CRC, Boca Raton (2000)
Acknowledgment
This work was supported in part by CNPq under grant no 310118/2017-4.
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Sangalli, M., Valle, M.E. (2018). Color Mathematical Morphology Using a Fuzzy Color-Based Supervised Ordering. In: Barreto, G., Coelho, R. (eds) Fuzzy Information Processing. NAFIPS 2018. Communications in Computer and Information Science, vol 831. Springer, Cham. https://doi.org/10.1007/978-3-319-95312-0_24
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