Skip to main content
SpringerLink
Log in

Generalized Fractional Filter-Based Algorithm for Image Denoising

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper presents a new algorithm for image denoising using a fractional integral mask of the K-operator. K-operator is the generalized fractional operator, and it reduces to Riemann–Liouville and Caputo fractional derivatives in a special case. The proposed algorithm is applied to digital images of different nature to demonstrate the performance of image denoising. Experimental results are compared with other existing filters together with block matching and 3-D filtering, and weighted nuclear norm minimization-based approaches. The obtained experimental results show that the proposed algorithm is computationally efficient and its average performance is comparatively better than other discussed methods.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26

Similar content being viewed by others

References

  1. A. Aboshosha, M. Hassan, M. Ashour, M. El Mashade, Image denoising based on spatial filters, an analytical study, in International Conference on Computer Engineering & Systems, 2009. ICCES 2009 (IEEE, 2009), pp. 245–250

  2. O.P. Agrawal, Generalized variational problems and Euler–Lagrange equations. Comput. Math. Appl. 59(5), 1852–1864 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. R.L. Bagley, P.J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J. 23(6), 918–925 (1985)

    Article  MATH  Google Scholar 

  4. J. Bai, X.C. Feng, Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492–2502 (2007)

    Article  MathSciNet  Google Scholar 

  5. G. Baloch, H. Ozkaramanli, R. Yu, Residual correlation regularization based image denoising. IEEE Signal Process. Lett. 25(2), 298–302 (2018)

    Article  Google Scholar 

  6. R. Campagna, S. Crisci, S. Cuomo, L. Marcellino, G. Toraldo, Modification of TV-ROF denoising model based on split Bregman iterations. Appl. Math. Comput. 315, 453–467 (2017)

    MathSciNet  MATH  Google Scholar 

  7. A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 (Springer, Berlin, 2014)

    MATH  Google Scholar 

  8. R.H. Chan, K. Chen, A multilevel algorithm for simultaneously denoising and deblurring images. SIAM J. Sci. Comput. 32(2), 1043–1063 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. Chen, B.M. Vinagre, A new IIR-type digital fractional order differentiator. Signal Process. 83(11), 2359–2365 (2003)

    Article  MATH  Google Scholar 

  10. K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)

    Article  MathSciNet  Google Scholar 

  11. L. Debnath, Fractional integral and fractional differential equations in fluid mechanics. Fract. Calc. Appl. Anal. 6, 119–155 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Y. Farouj, J.M. Freyermuth, L. Navarro, M. Clausel, P. Delachartre, Hyperbolic wavelet-fisz denoising for a model arising in ultrasound imaging. IEEE Trans. Comput. Imaging 3(1), 1–10 (2017)

    Article  Google Scholar 

  13. V. Fedorov, C. Ballester, Affine non-local means image denoising. IEEE Trans. Image Process. 26(5), 2137–2148 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Ghasemi, A. Tabesh, J. Askari-Marnani, Application of fractional calculus theory to robust controller design for wind turbine generators. IEEE Trans. Energy Convers. 29(3), 780–787 (2014)

    Article  Google Scholar 

  15. G. Ghimpeţeanu, T. Batard, M. Bertalmío, S. Levine, A decomposition framework for image denoising algorithms. IEEE Trans. Image Process. 25(1), 388–399 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. R.C. Gonzalez, R.E. Woods, Digital Image Processing, vol. 455, 2nd edn. (Publishing House of Electronics Industry, Beijing, 2002)

    Google Scholar 

  17. S. Gu, L. Zhang, W. Zuo, X. Feng, Weighted nuclear norm minimization with application to image denoising, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2014), pp. 2862–2869

  18. H. Guo, X. Li, C. Qing-li, W. Ming-rong, Image denoising using fractional integral, in IEEE International Conference on Computer Science and Automation Engineering (CSAE), 2012, vol. 2 (IEEE, 2012), pp. 107–112

  19. https://github.com/cszn/DnCNN/tree/master/testsets

  20. https://www.nlm.nih.gov/research/visible/image/mri.html

  21. https://www.petitcolas.net/watermarking/image_database./index.html

  22. https://sipi.usc.edu/database/database.php?volume=misc

  23. https://www.imageprocessingplace.com/root_filesv3/image_database.html

  24. N. He, J.B. Wang, L.L. Zhang, K. Lu, An improved fractional-order differentiation model for image denoising. Signal Process. 112, 180–188 (2015)

    Article  Google Scholar 

  25. J. Hu, Y.F. Pu, J. Zhou, A novel image denoising algorithm based on Riemann–Liouville definition. JCP 6(7), 1332–1338 (2011)

    Google Scholar 

  26. M.R. Islam, C. Xu, R.A. Raza, Y. Han, An effective weighted hybrid regularizing approach for image noise reduction. Circuits Syst. Signal Process. 38(1), 1–24 (2018)

    Google Scholar 

  27. V. Jain, H.S. Seung, Natural image denoising with convolutional networks. in Proceedings of the 21st International Conference on Neural Information Processing Systems (Curran Associates Inc., 2008), pp. 769–776

  28. H.A. Jalab, R.W. Ibrahim, Fractional Alexander polynomials for image denoising. Signal Process. 107, 340–354 (2015)

    Article  Google Scholar 

  29. V. Joshi, R.B. Pachori, A. Vijesh, Classification of ictal and seizure-free EEG signals using fractional linear prediction. Biomed. Signal Process. Control 9, 1–5 (2014)

    Article  Google Scholar 

  30. Z. Jun, W. Zhihui, A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising. Appl. Math. Model. 35(5), 2516–2528 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. B. Justusson, Median filtering: Statistical properties, in Two-Dimensional Digital Signal Processing II, ed. by T.S. Huang (Springer, Berlin, 1981), pp. 161–196

    Chapter  Google Scholar 

  32. C. Kervrann, J. Boulanger, Optimal spatial adaptation for patch-based image denoising. IEEE Trans. Image Process. 15(10), 2866–2878 (2006)

    Article  Google Scholar 

  33. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  34. X. Liu, X.Y. Jing, G. Tang, F. Wu, Q. Ge, Image denoising using weighted nuclear norm minimization with multiple strategies. Signal Process. 135, 239–252 (2017)

    Article  Google Scholar 

  35. R.L. Magin, Fractional calculus in bioengineering, part 1. Crit. Rev. Biomed. Eng. 32(1), 1–104 (2004)

    Article  Google Scholar 

  36. K. Nishimoto, An Essence of Nishimoto’s Fractional Calculus (Calculus in the 21st Century): Integrations and Differentiations of Arbitrary Order (Descartes Press Company, Waterloo, 1991)

    MATH  Google Scholar 

  37. M.D. Ortigueira, J.T. Machado, Fractional calculus applications in signals and systems. Signal Process. 10(86), 2503–2504 (2006)

    Article  MATH  Google Scholar 

  38. R.K. Pandey, O.P. Agrawal, Comparison of four numerical schemes for isoperimetric constraint fractional variational problems with A-operator. In: ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. V009T07A025–V009T07A025. American Society of Mechanical Engineers (2015)

  39. R.K. Pandey, O.P. Agrawal, Numerical scheme for a quadratic type generalized isoperimetric constraint variational problems with A-operator. J. Comput. Nonlinear Dyn. 10(2), 021,003 (2015)

    Article  Google Scholar 

  40. K. Panetta, L. Bao, S. Agaian, Sequence-to-sequence similarity-based filter for image denoising. IEEE Sens. J. 16(11), 4380–4388 (2016)

    Article  Google Scholar 

  41. J. Polack, Time domain solution of Kirchhoff’s equation for sound propagation in viscothermal gases: a diffusion process. J. Acoust. 4, 47–67 (1991)

    Google Scholar 

  42. Y.F. Pu, J.L. Zhou, X. Yuan, Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement. IEEE Trans. Image Process. 19(2), 491–511 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. H.K. Rafsanjani, M.H. Sedaaghi, S. Saryazdi, An adaptive diffusion coefficient selection for image denoising. Digit. Signal Process. 64, 71–82 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  44. L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom. 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  45. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives (Gordon and Breach Science Publishers, Yverdon-les-Bains, Switzerland, 1993)

  46. S. Sharma, R.K. Pandey, K. Kumar, Collocation method with convergence for generalized fractional integro-differential equations. J. Comput. Appl. Math. 342, 419–430 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  47. Y. Shen, Q. Liu, S. Lou, Y.L. Hou, Wavelet-based total variation and nonlocal similarity model for image denoising. IEEE Signal Process. Lett. 24(6), 877–881 (2017)

    Article  Google Scholar 

  48. H. Sheng, Y. Chen, T. Qiu, Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications (Springer, Berlin, 2011)

    MATH  Google Scholar 

  49. K.K. Singh, M.K. Bajpai, R.K. Pandey, A novel approach for enhancement of geometric and contrast resolution properties of low contrast images. IEEE/CAA J. Autom. Sin. 5(2), 628–638 (2018)

    Article  Google Scholar 

  50. S. Somali, Implicit midpoint rule to the nonlinear degenerate boundary value problems. Int. J. Comput. Math. 79(3), 327–332 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  51. S. Suresh, S. Lal, Two-dimensional cs adaptive fir Wiener filtering algorithm for the denoising of satellite images. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 10(12), 5245–5257 (2017)

    Article  Google Scholar 

  52. C.C. Tseng, Design of fractional order digital FIR differentiators. IEEE Signal Process. Lett. 8(3), 77–79 (2001)

    Article  Google Scholar 

  53. X. Wang, H. Wang, J. Yang, Y. Zhang, A new method for nonlocal means image denoising using multiple images. PLoS ONE 11(7), e0158,664 (2016)

    Article  Google Scholar 

  54. Z. Wang, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)

    Article  Google Scholar 

  55. S. Xu, Y. Zhou, H. Xiang, S. Li, Remote sensing image denoising using patch grouping-based nonlocal means algorithm. IEEE Geosci. Remote Sens. Lett. 14(12), 2275–2279 (2017)

    Article  Google Scholar 

  56. Q. Yang, D. Chen, T. Zhao, Y. Chen, Fractional calculus in image processing: a review. Fract. Calc. Appl. Anal. 19(5), 1222–1249 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  57. J. Yu, L. Tan, S. Zhou, L. Wang, M.A. Siddique, Image denoising algorithm based on entropy and adaptive fractional order calculus operator. IEEE Access 5, 12275–12285 (2017)

    Article  Google Scholar 

  58. H. Yue, X. Sun, J. Yang, F. Wu, Image denoising by exploring external and internal correlations. IEEE Trans. Image Process. 24(6), 1967–1982 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  59. W. Zhao, H. Lu, Medical image fusion and denoising with alternating sequential filter and adaptive fractional order total variation. IEEE Trans. Instrum. Meas. 66(9), 2283–2294 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the Editor and reviewers for their constructive comments to improve the quality of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

    Anil K. Shukla, Rajesh K. Pandey & Swati Yadav

  2. Discipline of Electrical Engineering, Indian Institute of Technology Indore, Indore, India

    Ram Bilas Pachori

Authors
  1. Anil K. Shukla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajesh K. Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Swati Yadav
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ram Bilas Pachori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rajesh K. Pandey.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shukla, A.K., Pandey, R.K., Yadav, S. et al. Generalized Fractional Filter-Based Algorithm for Image Denoising. Circuits Syst Signal Process 39, 363–390 (2020). https://doi.org/10.1007/s00034-019-01186-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-019-01186-y

Keywords

Search

Navigation