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Generalized Fractional Filter-Based Algorithm for Image Denoising

  • Anil K. Shukla
  • Rajesh K. PandeyEmail author
  • Swati Yadav
  • Ram Bilas Pachori
Article

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.

Keywords

Fractional calculus Difference equations Interpolation Texture Image denoising 

Notes

Acknowledgements

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

References

  1. 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. 245–250Google Scholar
  2. 2.
    O.P. Agrawal, Generalized variational problems and Euler–Lagrange equations. Comput. Math. Appl. 59(5), 1852–1864 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    R.L. Bagley, P.J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J. 23(6), 918–925 (1985)zbMATHGoogle Scholar
  4. 4.
    J. Bai, X.C. Feng, Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492–2502 (2007)MathSciNetGoogle Scholar
  5. 5.
    G. Baloch, H. Ozkaramanli, R. Yu, Residual correlation regularization based image denoising. IEEE Signal Process. Lett. 25(2), 298–302 (2018)Google Scholar
  6. 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)MathSciNetzbMATHGoogle Scholar
  7. 7.
    A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 (Springer, Berlin, 2014)zbMATHGoogle Scholar
  8. 8.
    R.H. Chan, K. Chen, A multilevel algorithm for simultaneously denoising and deblurring images. SIAM J. Sci. Comput. 32(2), 1043–1063 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Y. Chen, B.M. Vinagre, A new IIR-type digital fractional order differentiator. Signal Process. 83(11), 2359–2365 (2003)zbMATHGoogle Scholar
  10. 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)MathSciNetGoogle Scholar
  11. 11.
    L. Debnath, Fractional integral and fractional differential equations in fluid mechanics. Fract. Calc. Appl. Anal. 6, 119–155 (2003)MathSciNetzbMATHGoogle Scholar
  12. 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)Google Scholar
  13. 13.
    V. Fedorov, C. Ballester, Affine non-local means image denoising. IEEE Trans. Image Process. 26(5), 2137–2148 (2017)MathSciNetzbMATHGoogle Scholar
  14. 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)Google Scholar
  15. 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)MathSciNetzbMATHGoogle Scholar
  16. 16.
    R.C. Gonzalez, R.E. Woods, Digital Image Processing, vol. 455, 2nd edn. (Publishing House of Electronics Industry, Beijing, 2002)Google Scholar
  17. 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. 2862–2869Google Scholar
  18. 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. 107–112Google Scholar
  19. 19.
  20. 20.
  21. 21.
  22. 22.
  23. 23.
  24. 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)Google Scholar
  25. 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. 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. 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. 769–776Google Scholar
  28. 28.
    H.A. Jalab, R.W. Ibrahim, Fractional Alexander polynomials for image denoising. Signal Process. 107, 340–354 (2015)Google Scholar
  29. 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)Google Scholar
  30. 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)MathSciNetzbMATHGoogle Scholar
  31. 31.
    B. Justusson, Median filtering: Statistical properties, in Two-Dimensional Digital Signal Processing II, ed. by T.S. Huang (Springer, Berlin, 1981), pp. 161–196Google Scholar
  32. 32.
    C. Kervrann, J. Boulanger, Optimal spatial adaptation for patch-based image denoising. IEEE Trans. Image Process. 15(10), 2866–2878 (2006)Google Scholar
  33. 33.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)zbMATHGoogle Scholar
  34. 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)Google Scholar
  35. 35.
    R.L. Magin, Fractional calculus in bioengineering, part 1. Crit. Rev. Biomed. Eng. 32(1), 1–104 (2004)Google Scholar
  36. 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)zbMATHGoogle Scholar
  37. 37.
    M.D. Ortigueira, J.T. Machado, Fractional calculus applications in signals and systems. Signal Process. 10(86), 2503–2504 (2006)zbMATHGoogle Scholar
  38. 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)Google Scholar
  39. 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)Google Scholar
  40. 40.
    K. Panetta, L. Bao, S. Agaian, Sequence-to-sequence similarity-based filter for image denoising. IEEE Sens. J. 16(11), 4380–4388 (2016)Google Scholar
  41. 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. 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)MathSciNetzbMATHGoogle Scholar
  43. 43.
    H.K. Rafsanjani, M.H. Sedaaghi, S. Saryazdi, An adaptive diffusion coefficient selection for image denoising. Digit. Signal Process. 64, 71–82 (2017)MathSciNetzbMATHGoogle Scholar
  44. 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)MathSciNetzbMATHGoogle Scholar
  45. 45.
    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives (Gordon and Breach Science Publishers, Yverdon-les-Bains, Switzerland, 1993)Google Scholar
  46. 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)MathSciNetzbMATHGoogle Scholar
  47. 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)Google Scholar
  48. 48.
    H. Sheng, Y. Chen, T. Qiu, Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications (Springer, Berlin, 2011)zbMATHGoogle Scholar
  49. 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)Google Scholar
  50. 50.
    S. Somali, Implicit midpoint rule to the nonlinear degenerate boundary value problems. Int. J. Comput. Math. 79(3), 327–332 (2002)MathSciNetzbMATHGoogle Scholar
  51. 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)Google Scholar
  52. 52.
    C.C. Tseng, Design of fractional order digital FIR differentiators. IEEE Signal Process. Lett. 8(3), 77–79 (2001)MathSciNetGoogle Scholar
  53. 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)Google Scholar
  54. 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)Google Scholar
  55. 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)Google Scholar
  56. 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)MathSciNetzbMATHGoogle Scholar
  57. 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)Google Scholar
  58. 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)MathSciNetzbMATHGoogle Scholar
  59. 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)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndian Institute of Technology (BHU)VaranasiIndia
  2. 2.Discipline of Electrical EngineeringIndian Institute of Technology IndoreIndoreIndia

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