×

Fully fractional anisotropic diffusion for image denoising. (English) Zbl 1225.94003

Summary: This paper introduces a novel Fully Fractional Anisotropic Diffusion Equation for noise removal which contains spatial as well as time fractional derivatives. It is a generalization of a method proposed by Cuesta which interpolates between the heat and the wave equation by the use of time fractional derivatives, and the method proposed by Bai and Feng, which interpolates between the second and the fourth order anisotropic diffusion equation by the use of spatial fractional derivatives. This equation has the benefits of both of these methods. For the construction of a numerical scheme, the proposed partial differential equation (PDE) has been treated as a spatially discretized Fractional Ordinary Differential Equation (FODE) model, and then the Fractional Linear Multistep Method (FLMM) combined with the discrete Fourier transform (DFT) is used. We prove that the analytical solution to the proposed FODE has certain regularity properties which are sufficient to apply a convergent and stable fractional numerical procedure. Experimental results confirm that our model manages to preserve edges, especially highly oscillatory regions, more efficiently than the baseline parabolic diffusion models.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
34A08 Fractional ordinary differential equations
35R11 Fractional partial differential equations
45K05 Integro-partial differential equations
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
68U10 Computing methodologies for image processing
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Perona, P.; Malik, J., Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern. Anal. Mach. Intell., 12, 629-639 (1990)
[2] Catte, F.; Lions, P. L.; Morel, J. M.; Coll, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29, 182-193 (1992) · Zbl 0746.65091
[3] Rudin, L. I.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268 (1992) · Zbl 0780.49028
[4] Vogel, C.; Oman, M., Iterative methods for total variation denoising, SIAM J. Sci. Statisc. Comput., 17, 227-238 (1996) · Zbl 0847.65083
[5] You, Y. L.; Kaveh, M., Fourth-order partial differential equation for noise removal, IEEE Trans. Image Process., 9, 1723-1730 (2000) · Zbl 0962.94011
[6] Lysaker, M.; Lundervold, A.; Tai, X. C., Noise removal using fourth-order partial differential equation with aplication to medical magnetic resonance images in space and time, IEEE Trans. Image Process, 12, 1579-1590 (2003) · Zbl 1286.94020
[7] Chambole, A.; Lions, P. L., Image recovery via total variation minimization and related problems, Numer. Math., 76, 167-188 (1997) · Zbl 0874.68299
[8] P. Blomgren, P. Mulet, T.F. Chan, C.K. Wong, Total variation image restoration: numerical methods and extensions, in: Proc. Int. Conf. Image Process., vol. 3, 1997, pp. 384-387.; P. Blomgren, P. Mulet, T.F. Chan, C.K. Wong, Total variation image restoration: numerical methods and extensions, in: Proc. Int. Conf. Image Process., vol. 3, 1997, pp. 384-387.
[9] Chan, T. F.; Marquina, A.; Mulet, P., High order total variation-based image restoration, SIAM J. Sci. Comput., 22, 503-516 (2000) · Zbl 0968.68175
[10] Bai, J.; Chu Feng, X., Fractional order anisotropic difusion for image denoising, IEEE Trans. Image Process., 16, 2492-2502 (2007) · Zbl 1119.76377
[11] E. Cuesta, J. Finat, Image Processing by means of linear integro-differential equation, in: Proc. Int. Conf. Vizual. Imaging and Immage Process., 10.10.2003, Benalmadena, Spain.; E. Cuesta, J. Finat, Image Processing by means of linear integro-differential equation, in: Proc. Int. Conf. Vizual. Imaging and Immage Process., 10.10.2003, Benalmadena, Spain.
[12] M. Weilber, Efficient numerical methods for fractional differential equations and their analytical background, Doctorial Dissertation, 09.06.2005.; M. Weilber, Efficient numerical methods for fractional differential equations and their analytical background, Doctorial Dissertation, 09.06.2005.
[13] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[14] Aubert, G.; Kornprobst, P., (Mathematical Problems in Image Processing: PDE’s and the Calculus of Variations. Mathematical Problems in Image Processing: PDE’s and the Calculus of Variations, Applied Mathematical Sciences, vol. 147 (2002), Springer-Verlag) · Zbl 1109.35002
[15] Didas, S.; Weickert, J.; Burgeth, B., Properties of higher order nonlinear diffusion filtering, J. Math Imaging Vis., 35, 208-226 (2009) · Zbl 1490.94011
[16] Lubich, C., Fractional linear multistep methods for abel-volterra integral equations of the second kind, Math. Comp., 45, 172, 463-469 (1985) · Zbl 0584.65090
[17] Henrici, P., Discrete variable methods in ordinary differential equations, (Encyclopedia of Mathematics and its Applications, vol. 34 (1968), Camb. Univ. Pr.: Camb. Univ. Pr. New York) · Zbl 0112.34901
[18] Dieudonné, Foundations of Modern Analysis (1960), Academic Press · Zbl 0100.04201
[19] Diethelm, K.; Ford, J. M.; Ford, N. J.; Weilbeer, M., Pitfalls in fast numerical solvers for fractional differential equations, J. Comput. Appl. Math., 186, 482-503 (2006) · Zbl 1078.65550
[20] Mordecai, Avriel, Nonlinear Programming: Analysis and Methods (2003), Dover Publishing · Zbl 1140.90002
[21] Cao, Y.; Yin, J.; Liu, G.; Li, M., A class of nonlinear parabolic-hyperbolic equations applied to image restoration, Nonlinear Analysis. RWA, 11, 1, 253-261 (2010) · Zbl 1180.35378
[22] Wang, Z.; Bovik, A. C.; Sheikh, H. R.; Simoncelli, E. P., Image quality assessment: from error to structural similarity, IEEE Trans. Image Process., 13, 4, 600-612 (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.