Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

An adaptive fuzzy filter for image denoising


This study considers the problem of fuzzy modeling of the images in pixel domain. A zero-order Takagi–Sugeno type fuzzy model provides fuzzy smoothing to the image intensities for removing the additive noise from an image. An adaptive fuzzy filtering algorithm is suggested for estimating the parameters of the fuzzy model with noisy image data. The mathematical analysis of the proposed filtering algorithm has been provided in both deterministic and stochastic framework. The deterministic robustness of the filtering algorithm was shown by deriving an upper bound on the magnitude of estimation errors. The fuzzy filtering algorithm doesn’t demand Gaussian assumption of the noise and is also optimal in the “sense” of variation Bayes towards Student-t distributed noises.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. 1.

    Choi, Y., Krishnapuram, R.: A robust approach to image enhancement based on fuzzy logic. IEEE Trans. Image Process. 6(6), 808–825 (1997)

  2. 2.

    Lee, C.S., Kuo, Y.H., Yu, P.T.: Weighted fuzzy mean filters for image processing. Fuzzy Sets Syst. 89(2), 157–180 (1997)

  3. 3.

    Russo, F., Ramponi, G.: A fuzzy operator for the enhancement of blurred and noisy images. IEEE Trans. Image Process. 4(8), 1169–1174 (1995)

  4. 4.

    Russo, F., Ramponi, G.: A fuzzy filter for images corrupted by impulse noise. IEEE Signal Process. Lett. 3(6), 168–170 (1996)

  5. 5.

    Schulte, S., Witte, V.D., Kerre, E.E.: A fuzzy noise reduction method for color images. IEEE Trans. Image Process. 16(5), 1425–1436 (2007)

  6. 6.

    Ville, D.V.D., Nachtegael, M., Weken, D.V., Kerre, E.E., Philips, W., Lemahieu, I.: Noise reduction by fuzzy image filtering. IEEE Trans. Fuzzy Syst. 11(4), 429–436 (2003)

  7. 7.

    Wong, A., Mishra, A., Bizheva, K., Clausi, D.A.: General bayesian estimation for speckle noise reduction in optical coherence tomography retinal imagery. Optics Express 18(8), 8338–8352 (2010)

  8. 8.

    Pižurica, A., Philips, W., Lemahieu, I., Acheroy, M.: A joint inter- and intrascale statistical model for bayesian wavelet based image denoising. IEEE Trans. Image Process. 11(5), 545–557 (2002)

  9. 9.

    Portilla, J., Strela, V., Wainwright, M.J., Simoncelli, E.P.: Image denosing using scale mixtures of gaussians in the Wavelet domain. IEEE Trans. Image Process. 12(11), 1338–1351 (2003)

  10. 10.

    Şendur, L., Selesnick, I.W.: Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency. IEEE Trans. Signal Process. 50(11), 2744–2756 (2002)

  11. 11.

    Barash, D., Comaniciu, D.: A common framework for nonlinear diffusion, adaptive smoothing, bilateral filtering and mean shift. Image Vis. Comput. 22(1), 73–81 (2004)

  12. 12.

    Awate, S.P., Whitaker, R.T.: Unsupervised, information-theoretic, adaptive image filtering for image restoration. IEEE Trans. Pattern Anal. Mach. Intell. 28(3), 364–376 (2006)

  13. 13.

    Bezdek, J.C.: Pattern recognition with fuzzy objective function algorithms. Plenum, New York (1981)

  14. 14.

    Burger, M., Haslinger, J., Bodenhofer, U.: Regularized data-driven construction of fuzzy controllers. J. Inverse Ill-Posed Probl. 10, 319–344 (2002)

  15. 15.

    Chuang, C.C., Su, S.F., Chen, S.S.: Robust TSK fuzzy modeling for function approximation with outliers. IEEE Trans. Fuzzy Syst. 9(6), 810–821 (2001)

  16. 16.

    Hong, X., Harris, C.J., Chen, S.: Robust neurofuzzy rule base knowledge extraction and estimation using subspace decomposition combined with regularization and D-optimality. IEEE Trans. Syst. Man. Cybern. B 34(1), 598–608 (2004)

  17. 17.

    Johansen, T.: Robust identification of Takagi-Sugeno-Kang fuzzy models using regularization. In: Proceedings of IEEE Conference on Fuzzy Systems, pp. 180–186. New Orleans, USA (1996)

  18. 18.

    Juang, C.F., Hsieh, C.D.: TS-fuzzy system-based support vector regression. Fuzzy Sets Syst. 160(17), 2486–2504 (2009)

  19. 19.

    Juang, C.F., Huang, R.B., Cheng, W.Y.: An interval type-2 fuzzy-neural network with support-vector regression for noisy regression problems. IEEE Trans. Fuzzy Syst. 18(4), 686–699 (2010)

  20. 20.

    Kim, J., Suga, Y., Won, S.: A new approach to fuzzy modeling of nonlinear dynamic systems with noise: relevance vector learning mechanism. IEEE Trans. Fuzzy Syst. 14(2), 222–231 (2006)

  21. 21.

    Kumar, M., Stoll, N., Stoll, R.: An energy-gain bounding approach to robust fuzzy identification. Automatica 42(5), 711–721 (2006)

  22. 22.

    Kumar, M., Stoll, R., Stoll, N.: SDP and SOCP for outer and robust fuzzy approximation. In: Proceedings of 7th IASTED International Conference on Artificial Intelligence and Soft Computing. Banff, Canada (2003)

  23. 23.

    Kumar, M., Stoll, R., Stoll, N.: Robust adaptive identification of fuzzy systems with uncertain data. Fuzzy Optim. Decis. Mak. 3(3), 195–216 (2004)

  24. 24.

    Kumar, M., Stoll, R., Stoll, N.: Robust solution to fuzzy identification problem with uncertain data by regularization. Fuzzy approximation to physical fitness with real world medical data: An application. Fuzzy Optim. Decis. Mak. 3(1), 63–82 (2004)

  25. 25.

    Kumar, M., Stoll, R., Stoll, N.: Deterministic approach to robust adaptive learning of fuzzy models. IEEE Trans. Syst. Man. Cybern. B 36(4), 767–780 (2006)

  26. 26.

    Kumar, M., Stoll, R., Stoll, N.: A min-max approach to fuzzy clustering, estimation, and identification. IEEE Trans. Fuzzy Syst. 14(2), 248–262 (2006)

  27. 27.

    Kumar, M., Stoll, R., Stoll, N.: A robust design criterion for interpretable fuzzy models with uncertain data. IEEE Trans. Fuzzy Syst. 14(2), 314–328 (2006)

  28. 28.

    Kumar, M., Weippert, M., Arndt, D., Kreuzfeld, S., Thurow, K., Stoll, N., Stoll, R.: Fuzzy filtering for physiological signal analysis. IEEE Trans. Fuzzy Syst. 18(1), 208–216 (2010).

  29. 29.

    Leski, J.M.: TSK-fuzzy modeling based on \(\epsilon \)-insensitive learning. IEEE Trans. Fuzzy Syst. 13(2), 181–193 (2005)

  30. 30.

    Wang, L., Mu, Z., Guo, H.: Fuzzy rule-based support vector regression system. J. Control Theory Appl. 3(3), 230–234 (2005)

  31. 31.

    Wang, W.Y., Lee, T.T., Liu, C.L., Wang, C.H.: Function approximation using fuzzy neural networks with robust learning algorithm. IEEE Trans. Syst. Man. Cybern. B 27, 740–747 (1997)

  32. 32.

    Yu, W., Li, X.: Fuzzy identification using fuzzy neural networks with stable learning algorithms. IEEE Trans. Fuzzy Syst. 12(3), 411–420 (2004)

  33. 33.

    Kumar, M., Stoll, N., Stoll, R.: Adaptive fuzzy filtering in a deterministic setting. IEEE Trans. Fuzzy Syst. 17(4), 763–776 (2009)

  34. 34.

    Kumar, M., Stoll, N., Stoll, R.: On the estimation of parameters of takagi-sugeno fuzzy filters. IEEE Trans. Fuzzy Syst. 17(1), 150–166 (2009)

  35. 35.

    Kumar, M., Stoll, N., Stoll, R., Thurow, K.: A stochastic framework for robust fuzzy filtering and analysis of signals—Part I. IEEE Trans. Cybern. 46(5), 1118–1131 (2016).

  36. 36.

    Kumar, M., Insan, A., Stoll, N., Thurow, K., Stoll, R.: Stochastic fuzzy modeling for ear imaging based child identification. IEEE Trans. Syst. Man Cybern. Syst. 46(9), 1265–1278 (2016).

  37. 37.

    Kumar, M., Mao, Y., Wang, Y., Chenggen, Y., Zhang, W.: Fuzzy theoretic approach to signals and systems: static systems. Inf. Sci. 418, 668–702 (2017).

  38. 38.

    Attias, H.: A variational Bayesian framework for graphical models. In: In Advances in Neural Information Processing Systems 12, pp. 209–215. MIT Press, Cambridge (2000)

  39. 39.

    Beal, M.J.: Variational algorithms for approximate Bayesian inference. Ph.D. thesis, The Gatsby Computational Neuroscience Unit, University College London, London, UK (2003)

  40. 40.

    Lappalainen, H., Miskin, J.W.: Ensemble learning. Advances in Independent Component Analysis. Springer, Berlin (2000)

  41. 41.

    Christmas, J., Everson, R.: Robust autoregression: Student-t innovations using variational Bayes. IEEE Trans. Signal Process. 59(1), 48–57 (2011)

  42. 42.

    Tipping, M.E., Lawrence, N.D.: Variational inference for Student-t models: Robust Bayesian interpolation and generalised component analysis. Neurocomputing 69, 123–141 (2005)

Download references


Funding was provided by National Natural Science Foundation of China (ID0EQOAE4397).

Author information

Correspondence to Yihua Mao.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, W., Kumar, M., Yang, J. et al. An adaptive fuzzy filter for image denoising. Cluster Comput 22, 14107–14124 (2019).

Download citation


  • Adaptive fuzzy filtering
  • Robustness
  • Image denoising
  • Variational Bayes
  • Student-t distribution