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Denoising algorithm based on generalized fractional integral operator with two parameters. (English) Zbl 1244.94007
Summary: A novel digital image denoising algorithm called generalized fractional integral filter is introduced based on the generalized Srivastava-Owa fractional integral operator. The structures of \(n \times n\) fractional masks of this algorithm are constructed. The denoising performance is measured by employing experiments according to visual perception and PSNR values. The results demonstrate that apart from enhancing the quality of filtered image, the proposed algorithm also reserves the textures and edges present in the image. Experiments also prove that the improvements achieved are competent with the Gaussian smoothing filter.

MSC:
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
34A08 Fractional ordinary differential equations and fractional differential inclusions
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[1] K. S. Miller and B. Ross, An Introduction to the Fractional Integrals and Derivatives-Theory and Application, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002
[2] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[3] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. · Zbl 1111.93302 · doi:10.1142/9789812817747
[4] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003. · Zbl 1225.62144 · doi:10.1111/j.0006-341X.2003.00121.x
[5] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. · Zbl 1206.26007 · doi:10.1016/S0304-0208(06)80001-0
[6] J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007. · Zbl 1196.35069 · doi:10.1134/S1064562407030209
[7] A. C. Sparavigna, “Using fractional differentiation in astronomy,” Computer Vision and Pattern Recognition (2010), http://arxiv.org/abs/0910.2381v3.
[8] R. Marazzato and A. C. Sparavigna, “Astronomical image processing based on fractional calculus: the AstroFracTool,” Instrumentation and Methods for Astrophysics (2009), http://arxiv.org/abs/0910.4637v2.
[9] C. C. Tseng, “Design of variable and adaptive fractional order fir differentiators,” Signal Processing, vol. 86, no. 10, pp. 2554-2566, 2006. · Zbl 1172.94495 · doi:10.1016/j.sigpro.2006.02.004
[10] J. A. T. Machado, M. F. Silva, R. S. Barbosa, et al., “Some applications of fractional calculus in engineering,” Mathematical Problems in Engineering, vol. 2010, Article ID 639801, 34 pages, 2010. · Zbl 1191.26004 · doi:10.1155/2010/639801 · eudml:226365
[11] J. Hu, Y. Pu, and J. Zhou, “A novel image denoising algorithm based on riemann-liouville definition,” Journal of Computers, vol. 6, no. 7, pp. 1332-1338, 2011. · doi:10.4304/jcp.6.7.1332-1338
[12] R. W. Ibrahim and M. Darus, “Subordination and superordination for analytic functions involving fractional integral operator,” Complex Variables and Elliptic Equations. An International Journal of Elliptic Equations and Complex Analysis, vol. 53, no. 11, pp. 1021-1031, 2008. · Zbl 1155.30006 · doi:10.1080/17476930802429131
[13] R. W. Ibrahim and M. Darus, “Subordination and superordination for univalent solutions for fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 871-879, 2008. · Zbl 1147.30009 · doi:10.1016/j.jmaa.2008.05.017
[14] S. M. Momani and R. W. Ibrahim, “On a fractional integral equation of periodic functions involving Weyl-Riesz operator in Banach algebras,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1210-1219, 2008. · Zbl 1136.45010 · doi:10.1016/j.jmaa.2007.08.001
[15] R. W. Ibrahim, “Solutions of fractional diffusion problems,” Electronic Journal of Differential Equations, vol. 2010, no. 147, pp. 1-11, 2010. · Zbl 1205.35333 · emis:journals/EJDE/Volumes/2010/147/abstr.html · eudml:232917
[16] R. W. Ibrahim, “On the existence for diffeo-integral inclusion of Sobolev-type of fractional order with applications,” ANZIAM Journal, vol. 52, no. (E), pp. E1-E21, 2010.
[17] D. B\ualeanu, O. G. Mustafa, and R. P. Agarwal, “On the solution set for a class of sequential fractional differential equations,” Journal of Physics A, vol. 43, no. 38, Article ID 385209, 2010. · Zbl 1216.34004 · doi:10.1088/1751-8113/43/38/385209
[18] R. W. Ibrahim and M. Darus, “On analytic functions associated with the Dziok-Srivastava linear operator and Srivastava-Owa fractional integral operator,” Arabian Journal for Science and Engineering, vol. 36, no. 3, pp. 441-450, 2011. · Zbl 1218.30031 · doi:10.1007/s13369-011-0043-y
[19] R. W. Ibrahim, “Existence and uniqueness of holomorphic solutions for fractional cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 380, no. 1, pp. 232-240, 2011. · Zbl 1214.30027 · doi:10.1016/j.jmaa.2011.03.001
[20] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0818.26003
[21] J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140-1153, 2011. · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027
[22] H. M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics and Its Applications, John Wiley & Sons, New York, NY, USA, 1989. · Zbl 0683.00012
[23] R. W. Ibrahim, “On generalized Srivastava-Owa fractional operators in the unit disk,” Advances in Difference Equations, vol. 2011, article no. 55, 2011. · Zbl 1273.35295 · doi:10.1186/1687-1847-2011-55
[24] E. Cuesta, M. Kirane, and S. Malik, “Image structure preserving denoising using generalized fractional time integrals,” Signal Processing, vol. 92, no. 2, pp. 553-563, 2012. · doi:10.1016/j.sigpro.2011.09.001
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