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Examples of fuzzy metrics and applications. (English) Zbl 1210.94016
Summary: In this paper we present new examples of fuzzy metrics in the sense of A. George and V. P. Veeramani [Fuzzy Sets Syst. 64, No. 3, 395–399 (1994; Zbl 0843.54014); ibid. 90, No. 3, 365–368 (1977; Zbl 0917.54010)]. The examples have been classified attending to their construction and most of the well-known fuzzy metrics are particular cases of those given here. In particular, novel fuzzy metrics, by means of fuzzy and classical metrics and certain special types of functions, are introduced. We also give an extension theorem for two fuzzy metrics that agree in its nonempty intersection. Finally, we give an application of this type of fuzzy metrics to color image processing. We propose a fuzzy metric that simultaneously takes into account two different distance criteria between color image pixels and we use this fuzzy metric to filter noisy images, obtaining promising results. This application is also illustrative of how fuzzy metrics can be used in other engineering problems.

##### MSC:
 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 54E35 Metric spaces, metrizability
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##### References:
 [1] Astola, J.; Haavisto, P.; Neuvo, Y., Vector Median filters, Proceedings of the IEEE, 78, 678-689, (1990) [2] Bing, R.H., Extending a metric, American mathematical society, 511-519, (1947) · Zbl 0030.08003 [3] Camarena, J.-G.; Gregori, V.; Morillas, S.; Sapena, A., Fast detection and removal of impulsive noise using peer groups and fuzzy metrics, Journal of visual communication and image representation, 19, 20-29, (2008) [4] CIE Publication 15:2004, Colorimetry, third ed., CIE Central Bureau, Vienna, 2004. [5] Cui, G.; Luo, M.R.; Rigg, B.; Roesler, G.; Witt, K., Uniform colour spaces based on the DIN99 colour-difference formula, Color research & application, 27, 282-290, (2002) [6] George, A.; Veeramani, P., On some results in fuzzy metric spaces, Fuzzy sets and systems, 64, 395-399, (1994) · Zbl 0843.54014 [7] George, A.; Veeramani, P., On some results of analysis for fuzzy metric spaces, Fuzzy sets and systems, 90, 365-368, (1997) · Zbl 0917.54010 [8] Gregori, V.; Romaguera, S., Some properties of fuzzy metric spaces, Fuzzy sets and systems, 115, 485-489, (2000) · Zbl 0985.54007 [9] Gregori, V.; Romaguera, S., On completion of fuzzy metric spaces, Fuzzy sets and systems, 130, 399-404, (2002) · Zbl 1010.54002 [10] Gregori, V.; Romaguera, S., Characterizing completable fuzzy metric spaces, Fuzzy sets and systems, 144, 411-420, (2004) · Zbl 1057.54010 [11] Gregori, V.; Romaguera, S.; Sapena, A., On t-uniformly continuous mappings in fuzzy metric spaces, The journal of fuzzy mathematics, 12, 1, 237-243, (2004) · Zbl 1059.54008 [12] Gregori, V.; Romaguera, S.; Veeramani, P., A note on intuitionistic fuzzy metric spaces, Chaos, solitons & fractals, 28, 4, 902-905, (2006) · Zbl 1096.54003 [13] Mihet, D., On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy sets and systems, 158, 915-921, (2007) · Zbl 1117.54008 [14] S. Morillas, V. Gregori, G. Peris-Fajarnés, P. Latorre, A new vector median filter based on fuzzy metrics, in: Lecture Notes in Computer Science, vol. 3656, 2005, pp. 81-90. [15] Morillas, S.; Gregori, V.; Peris-Fajarnés, G.; Latorre, P., A fast impulsive noise color image filter using fuzzy metrics, Real-time imaging, 11, 5-6, 417-428, (2005) [16] Morillas, S.; Gregori, V.; Peris-Fajarnés, G., Isolating impulsive noise pixels in color images by peer group techniques, Computer vision and image understanding, 110, 1, 102-116, (2008) [17] Morillas, S.; Gregori, V.; Peris-Fajarnés, G.; Sapena, A., New adaptive vector filter using fuzzy metrics, Journal of electronic imaging, 16, 3, (2007), 033007:1-15 [18] Morillas, S.; Gregori, V.; Peris-Fajarnés, G.; Sapena, A., Local self-adaptive fuzzy filter for impulsive noise removal in color images, Signal processing, 88, 2, 390-398, (2008) · Zbl 1186.94244 [19] Gregori, V.; López-Crevillén, A.; Morillas, S.; Sapena, A., On convergence in fuzzy metric spaces, Topology and its applications, 156, 3002-3006, (2009) · Zbl 1181.54006 [20] Park, J.H., Intuitionistic fuzzy metric spaces, Chaos, solitons & fractals, 22, 1039-1046, (2004) · Zbl 1060.54010 [21] Rodríguez-López, J.; Romaguera, S., The Hausdorff fuzzy metric on compact sets, Fuzzy sets and systems, 147, 273-283, (2004) · Zbl 1069.54009 [22] Sapena, A., A contribution to the study of fuzzy metric spaces, Applied general topology, 2, 63-76, (2001) · Zbl 0985.54006 [23] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific journal of mathematics, 10, 314-334, (1960) · Zbl 0091.29801 [24] Veeramani, P., Best approximation in fuzzy metric spaces, Journal of fuzzy mathematics, 9, 75-80, (2001) · Zbl 0986.54006
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