Fractional Masks Based On Generalized Fractional Differential Operator for Image Denoising

Authors: Hamid A. Jalab, Rabha W. Ibrahim

Abstract:

This paper introduces an image denoising algorithm based on generalized Srivastava-Owa fractional differential operator for removing Gaussian noise in digital images. The structures of nxn fractional masks are constructed by this algorithm. Experiments show that, the capability of the denoising algorithm by fractional differential-based approach appears efficient to smooth the Gaussian noisy images for different noisy levels. The denoising performance is measured by using peak signal to noise ratio (PSNR) for the denoising images. The results showed an improved performance (higher PSNR values) when compared with standard Gaussian smoothing filter.

Keywords: Fractional calculus, fractional differential operator, fractional mask, fractional filter.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1336074

References:

[1] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: John Wiley & Sons, New York, 1993.
[2] I. Podlubny, Fractional Differential Equations. Academic. New York: Academic Press, New York, 1999.
[3] R. Hilfer, Ed., Application of Fractional Calculus in Physics. Singapore: World Scientific, Singapore, p.463, 2000.
[4] B. J. West, M. Bologna, P. Grigolini,, Physics of Fractal Operators. New York Springer, 2003.
[5] A. A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland: Mathematics Studies, Elsevier 2006.
[6] J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado, Advances in fractional calculus: Theoretical developments and applications in physics and engineering: Springer, 2007.
[7] H. A. Jalab and R. W. Ibrahim, "Denoising algorithm based on generalized fractional integral operator with two parameters,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 529849, 14 pages, 2012.
[8] H. A. Jalab and R. W. Ibrahim, "Texture Enhancement for Medical Images Based on Fractional Differential Masks,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID618536, 10 pages, 2013.
[9] H. A. Jalab and R. W. Ibrahim, "Texture Enhancement Based on the Savitzky-Golay Fractional Differential Operator,” Mathematical Problems in Engineering Volume 2013, Article ID 149289, 8 pages.
[10] R. W. Ibrahim and M. Darus, "Subordination and superordination for univalent solutions for fractional differential equations," Journal of Mathematical Analysis and Applications, vol. 345, pp. 871-879, 2008.
[11] R. W. Ibrahim, "Existence and uniqueness of holomorphic solutions for fractional Cauchy problem," Journal of Mathematical Analysis and Applications, 2011.
[12] R. W. Ibrahim, "Solutions of fractional diffusion problems," Electronic Journal of Differential Equations, vol. 2010, pp. 1-11, 2010.
[13] R. W. Ibrahim, "On the existence for diffeo-integral inclusion of Sobolev-type of fractional order with applications," ANZIAM Journal, vol. 52, pp. E1--E21, 2010.
[14] 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, pp. 1-10, 2011.
[15] R. W. Ibrahim, "On holomorphic solutions for nonlinear singular fractional differential equations," Computers & Mathematics with Applications,62(3),p 1084-1090, 2011.
[16] S. Owa. H.M. Srivastava, Univalent Functions, Fractional Calculus, and Their Applications: Halsted Press, John Wiley and Sons, 1989.
[17] R. W. Ibrahim, "On generalized Srivastava-Owa fractional operators in the unit disk," Advances in Difference Equations, vol. 2011, p. 55, 2011.
[18] V.R.Vijaykumar, P.T.Vanathi, P.Kanagasabapathy, "Fast and Efficient Algorithm to Remove Gaussian Noise in Digital Images," IAENG International Journal of Computer Science, vol. 37, 2010.
[19] R. H. Chan, W. H. Chung, M. Nikolova, "Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization," Image Processing, IEEE Transactions on, vol. 14, pp. 1479-1485, 2005.
[20] J. Harikiran, B.Saichandana and B.Divakar, "Impulse Noise Removal in Digital Images," International Journal of Computer Applications IJCA, vol. 10, pp. 24- 28, 2010.
[21] S. D. Ruikar and D. D. Doye, "Wavelet Based Image Denoising Technique," IJACSA) International Journal of Advanced Computer Science and Applications, vol. 2, 2011.
[22] G. Vijaya and V. Vasudevan, "Image denoising based on soft computing techniques, volume 7, Issue 1, 2011.
[23] Y. F. Pu and J. L. Zhou, "A novel approach for multi-scale texture segmentation based on fractional differential," International Journal of Computer Mathematics, vol. 88, pp. 58-78, 2011.
[24] H. Jinrong,P Yifei,Z. Jiliu, "A Novel Image Denoising Algorithm Based on Riemann-Liouville Definition," Journal of Computers, vol. 6. 2011.
[25] R. W. Schafer, "What Is a Savitzky-Golay Filter (Lecture Notes)," Signal Processing Magazine, IEEE, vol. 28, pp. 111-117, 2011.
[26] J. Hu, Y. Pu, J.Zhou, A novel image denoising algorithm based on Riemann-Liouville definition, J. Computers 6(7)(2011) 1332-1338.