Fuzzy metric and its applications in removing the image noise

  • Nebojša M. RalevićEmail author
  • Danijela Karaklić
  • Neda Pištinjat
Methodologies and Application


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.


Distance function Fuzzy metric Image filtering Triangular conorms Triangular norms 



Nebojša M. Ralević acknowledges the financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia, in the frame of Projects applied under Nos. TR 34014 and ON 174009.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Nebojša M. Ralević
    • 1
    Email author
  • Danijela Karaklić
    • 1
  • Neda Pištinjat
    • 2
  1. 1.Department of Fundamentals Sciences, Faculty of Technical SciencesUniversity of Novi SadNovi SadSerbia
  2. 2.Microsoft Development Center Serbia (MDCS)BeogradSerbia

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