The use of a Riesz fractional differential-based approach for texture enhancement in image processing

Abstract

Texture enhancement is an important component of image processing that finds extensive application in science and engineering. The quality of medical images, quantified using the imaging texture, plays a significant role in the routine diagnosis performed by medical practitioners. Most image texture enhancement is performed using classical integral order differential mask operators. Recently, first order fractional differential operators were used to enhance images. Experimentation with these methods led to the conclusion that fractional differential operators not only maintain the low frequency contour features in the smooth areas of the image, but they also nonlinearly enhance edges and textures corresponding to high frequency image components. However, whilst these methods perform well in particular cases, they are not routinely useful across all applications. To this end, we apply the second order Riesz fractional differential operator to improve upon existing approaches of texture enhancement. Compared with the classical integral order differential mask operators and other first order fractional differential operators, we find that our new algorithms provide higher signal to noise values and superior image quality. References R. C. Gonzalez, R. E. Woods, Digital Image Processing . Third Edition. Prentice Hall, New Jersey, USA, 2007. C. S. Panda, S. Patnaik, Filtering corrupted image and edge detection in restored grayscale image using derivative filters, International Journal of Image Processing , 3(3):105–119, 2009. http://www.cscjournals.org/csc/manuscript/Journals/IJIP/volume3/Issue3/IJIP-28.pdf Y. Zhang, Y. Pu, J. Zhou, Construction of Fractional differential Masks Based on Riemann–Liouville Definition, Journal of Computational Information Systems , 6(10):3191–3199, 2010. http://www.jofcis.com/publishedpapers/2010_6_10_3191_3199.pdf F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation, J. Appl. Math. Comput. , 191:12–21, 2007. doi:10.1016/j.amc.2006.08.162 Q. Yu, F. Liu, I. Turner and K. Burrage, Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation, Phil Trans R Soc A , 371(1990):20120150, 2013. doi:10.1098/rsta.2012.0150 E. Sejdic, I. Djurovic, L. Stankovic, Fractional Fourier transform as a signal processing tool: An overview of recent developments, Signal Processing , 91(6):1351–1369, 2011. doi:10.1016/j.sigpro.2010.10.008 B. Pesquet-Popescu and J. L. Vehel, Stochastic fractal models for image processing, IEEE Signal Processing Magazine , 19(5):48–62, 2002. doi:10.1109/MSP.2002.1028352 B. Mathieu, P. Melchior, A. Oustaloup, Ch. Ceyral, Fractional differentiation for edge detection, Signal Processing , 83(11):2421–2432, 2003. doi:10.1016/S0165-1684(03)00194-4 C. B. Gao, J. L. Zhou, J. R. Hu, F. N. Lang, Edge detection of colour image based on quaternion fractional differential, IET Image Processing , 5(3):261–272, 2011. http://digital-library.theiet.org/content/journals/10.1049/iet-ipr.2009.0409 C. B. Gao, J. L. Zhou, X. Q. Zheng, F. N. Lang, Image enhancement based on improved fractional differentiation, Journal of Computational Information Systems , 7(1):257–264, 2011. http://www.jofcis.com/publishedpapers/2011_7_1_257_264.pdf Y. Pu, J. Zhou and X. Yuan, Fractional differential mask: A fractional differential–based approach for multiscale texture enhancement, IEEE Transactions on Image Processing , 19(2):491–511, 2010. doi:10.1109/TIP.2009.2035980 M. D. Ortigueira, Riesz potential operators and inverses via fractional centred derivatives, International Journal of Mathematics and Mathematical Sciences , 2006:48391, 2006. doi:10.1155/IJMMS/2006/48391