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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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