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.

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  1. Astola J, Haavisto P, Neuvo Y (1990) Vector median filters. Proc IEEE 78(32):678–689

    Article  Google Scholar 

  2. Bloch I (1999) On fuzzy distances and their use in image processing under imprecision. Pattern Recogn Lett 32:1873–1895

    Article  Google Scholar 

  3. Deza MM, Deza E (2009) Encyclopedia of distances. Springer, Berlin

    Google Scholar 

  4. Gregori V, Romaguera S (2000) Some properties of fuzzy metric spaces. Fuzzy Sets Syst 115:485–489

    MathSciNet  Article  Google Scholar 

  5. Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer, Dordrecht

    Google Scholar 

  6. Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic, theory and applications. Prentice Hall, Englewood Cliffs

    Google Scholar 

  7. Morillas S, Gregori V, Peris-Fajarnes G, Latorre P (2005a) A fast impulsive noise color image filter using fuzzy metrics. Real-Time Imaging 11(5–6):417–428

    Article  Google Scholar 

  8. Morillas S, Gregori V, Peris-Fajarnes G, Latorre P (2005b) A New Vector Median Filter Based on Fuzzy Metrics. Lecture Notes in Computer Science, vol. 3656, pp. 81–90

  9. Morillas S, Gregori V, Peris-Fajarnes G, Sapena A (2007) New adaptive vector filter using fuzzy metrics. J Electron Imaging 16(3):033,007:1–15

    MathSciNet  Article  Google Scholar 

  10. Nedović L, Ralević N, Pavkov I (2017) Aggregated distance functions and their application in image processing. Soft Comput. https://doi.org/10.1007/s00500-017-2657-9

    Article  MATH  Google Scholar 

  11. Smolka B, Szczepanski M, Plataniotis KN, Venetsanopoulos AN (2001) On the fast modified vector median filter. Can Conf Electr Comput Eng 2(2):1315–1320

    MATH  Google Scholar 

  12. Valentin G, Morillas S, Sapena A (2011) Examples of fuzzy metrics and applications. Fuzzy Sets Syst 170:95–111

    MathSciNet  Article  Google Scholar 

  13. Wang Z, Bovik AC (2002) A universal image quality index. IEEE Signal Process Lett 9(3):81–84

    Article  Google Scholar 

Download references


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.

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Correspondence to Nebojša M. Ralević.

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Ralević, N.M., Karaklić, D. & Pištinjat, N. Fuzzy metric and its applications in removing the image noise. Soft Comput 23, 12049–12061 (2019). https://doi.org/10.1007/s00500-019-03762-5

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  • Distance function
  • Fuzzy metric
  • Image filtering
  • Triangular conorms
  • Triangular norms