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Neural Computing and Applications

, Volume 28, Supplement 1, pp 217–223 | Cite as

Image denoising algorithm based on the convolution of fractional Tsallis entropy with the Riesz fractional derivative

  • Hamid A. JalabEmail author
  • Rabha W. Ibrahim
  • Amr Ahmed
Original Article

Abstract

Image denoising is an important component of image processing. The interest in the use of Riesz fractional order derivative has been rapidly growing for image processing recently. This paper mainly introduces the concept of fractional calculus and proposes a new mathematical model in using the convolution of fractional Tsallis entropy with the Riesz fractional derivative for image denoising. The structures of n × n fractional mask windows in the x and y directions of this algorithm are constructed. The image denoising performance is assessed using the visual perception, and the objective image quality metrics, such as peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM). The proposed algorithm achieved average PSNR of 28.92 dB and SSIM of 0.8041. The experimental results prove that the improvements achieved are compatible with other standard image smoothing filters (Gaussian, Kuan, and Homomorphic Wiener).

Keywords

Fractional calculus Fractional mask Fractional Tsallis entropy Riesz fractional derivative 

Notes

Acknowledgments

This research is funded by the Ministry of Higher Education Malaysia under the Fundamental Research Grant Scheme (FRGS), Project No.: FP073-2015A.

Author contributions

All authors jointly worked on deriving the results and approved the final manuscript.

Compliance with ethical standards

Conflict of interest

The authors declare that there are no conflict of interests regarding the publication of this article.

References

  1. 1.
    Tseng C-C, Lee S-L (2014) Digital image sharpening using Riesz fractional order derivative and discrete Hartley transform. In: 2014 IEEE Asia Pacific conference on circuits and systems (APCCAS). IEEEGoogle Scholar
  2. 2.
    Ibrahim RW, Jalab HA (2013) Time-space fractional heat equation in the unit disk. In: Trujillo JJ (ed) Abstract and applied analysis. Hindawi Publishing Corporation, New York, USAGoogle Scholar
  3. 3.
    Jalab HA, Ibrahim RW (2015) Fractional Alexander polynomials for image denoising. Sig Process 107:340–354CrossRefGoogle Scholar
  4. 4.
    Jalab HA, Ibrahim RW (2014) Fractional conway polynomials for image denoising with regularized fractional power parameters. J Math Imaging Vis 51(3):1–9Google Scholar
  5. 5.
    Jalab H, Ibrahim R (2016) Image denoising algorithms based on fractional sinc α with the covariance of fractional Gaussian fields. Imaging Sci J 64:100–108Google Scholar
  6. 6.
    Yu Q et al (2013) The use of a Riesz fractional differential-based approach for texture enhancement in image processing. ANZIAM J 54:590–607MathSciNetCrossRefGoogle Scholar
  7. 7.
    Podlubny I (1999) Fractional differential equations. Acadamic Press, London, p E2Google Scholar
  8. 8.
    Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New YorkzbMATHGoogle Scholar
  9. 9.
    Kilbas AAA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. Elsevier, AmsterdamCrossRefzbMATHGoogle Scholar
  10. 10.
    Hilfer R et al (2000) Applications of fractional calculus in physics, vol 128. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  11. 11.
    Ortigueira MD (2006) Riesz potential operators and inverses via fractional centred derivatives. Int J Math Math Sci 2006:1–12Google Scholar
  12. 12.
    Mathai AM, Haubold HJ (2013) On a generalized entropy measure leading to the pathway model with a preliminary application to solar neutrino data. Entropy 15(10):4011–4025MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Tsallis C (2009) Introduction to nonextensive statistical mechanics. Springer, BerlinzbMATHGoogle Scholar
  14. 14.
    Wang Z et al (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 13(4):600–612CrossRefGoogle Scholar
  15. 15.
    Gonzales RC, Woods RE, Eddins SL (2004) Digital image processing using MATLAB. Pearson Prentice Hall, Englewood CliffsGoogle Scholar
  16. 16.
    Cuesta E, Kirane M, Malik SA (2012) Image structure preserving denoising using generalized fractional time integrals. Sig Process 92(2):553–563CrossRefGoogle Scholar
  17. 17.
    Zhang Y-S et al (2014) Fractional domain varying-order differential denoising method. Opt Eng 53(10):102102-1–102102-7Google Scholar
  18. 18.
    Hu J, Pu Y, Zhou J (2011) A novel image denoising algorithm based on riemann-liouville definition. J Comput 6(7):1332–1338Google Scholar
  19. 19.
    Jalab HA, Ibrahim RW (2012) Denoising algorithm based on generalized fractional integral operator with two parameters. Discrete Dyn Nat Soc 2012:1–14Google Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  1. 1.Faculty of Computer Science and Information TechnologyUniversity MalayaKuala LumpurMalaysia
  2. 2.Lincoln School of Computer ScienceUniversity of LincolnLincolnUK

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