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).
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References
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). IEEE
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, USA
Jalab HA, Ibrahim RW (2015) Fractional Alexander polynomials for image denoising. Sig Process 107:340–354
Jalab HA, Ibrahim RW (2014) Fractional conway polynomials for image denoising with regularized fractional power parameters. J Math Imaging Vis 51(3):1–9
Jalab H, Ibrahim R (2016) Image denoising algorithms based on fractional sinc α with the covariance of fractional Gaussian fields. Imaging Sci J 64:100–108
Yu Q et al (2013) The use of a Riesz fractional differential-based approach for texture enhancement in image processing. ANZIAM J 54:590–607
Podlubny I (1999) Fractional differential equations. Acadamic Press, London, p E2
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York
Kilbas AAA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. Elsevier, Amsterdam
Hilfer R et al (2000) Applications of fractional calculus in physics, vol 128. World Scientific, Singapore
Ortigueira MD (2006) Riesz potential operators and inverses via fractional centred derivatives. Int J Math Math Sci 2006:1–12
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):4011–4025
Tsallis C (2009) Introduction to nonextensive statistical mechanics. Springer, Berlin
Wang Z et al (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 13(4):600–612
Gonzales RC, Woods RE, Eddins SL (2004) Digital image processing using MATLAB. Pearson Prentice Hall, Englewood Cliffs
Cuesta E, Kirane M, Malik SA (2012) Image structure preserving denoising using generalized fractional time integrals. Sig Process 92(2):553–563
Zhang Y-S et al (2014) Fractional domain varying-order differential denoising method. Opt Eng 53(10):102102-1–102102-7
Hu J, Pu Y, Zhou J (2011) A novel image denoising algorithm based on riemann-liouville definition. J Comput 6(7):1332–1338
Jalab HA, Ibrahim RW (2012) Denoising algorithm based on generalized fractional integral operator with two parameters. Discrete Dyn Nat Soc 2012:1–14
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This research is funded by the Ministry of Higher Education Malaysia under the Fundamental Research Grant Scheme (FRGS), Project No.: FP073-2015A.
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All authors jointly worked on deriving the results and approved the final manuscript.
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Jalab, H.A., Ibrahim, R.W. & Ahmed, A. Image denoising algorithm based on the convolution of fractional Tsallis entropy with the Riesz fractional derivative. Neural Comput & Applic 28 (Suppl 1), 217–223 (2017). https://doi.org/10.1007/s00521-016-2331-7
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DOI: https://doi.org/10.1007/s00521-016-2331-7