Fuzzy metric and its applications in removing the image noise

This paper introduces the notions of the fuzzy T-metric and fuzzy S-metric and considers their duality and their links to standard metrics. Numerous examples of fuzzy T-metrics are given in the paper, and some of their properties are proved. A procedure for constructing new fuzzy metrics as a composition of the given fuzzy metrics and norms is provided. An application of the fuzzy T-metrics in image filtering is given. The proposed filtering algorithm is a modification of the algorithm presented by Valentin, Morillas and Sapena in Valentin G et al. (Fuzzy Sets Syst 170:95–111, 2011). Instead of the fuzzy metric that is used in the paper of Valentin G et al. 2011, a new fuzzy T -metric is applied in the algorithm. The aim was to improve the sharpness and the quality of the image, measured by the image quality index UIQI. It is shown that the image filtered by this modified algorithm has better quality and greater sharpness than the image filtered by the median filter. Fuzzy T-metric parameters that give the best image quality and sharpness are determined experimentally.