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Reaction-diffusion equation based on fractional-time anisotropic diffusion for textured images recovery. (English) Zbl 1513.35313


MSC:

35K57 Reaction-diffusion equations
35R11 Fractional partial differential equations
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Full Text: DOI

References:

[1] Hakim, A.; Ben-Loghfyry, A., A total variable-order variation model for image denoising, AIMS MATH., 4, 5, 1320-1335 (2019) · Zbl 1486.94014 · doi:10.3934/math.2019.5.1320
[2] Morfu, S., On some applications of diffusion processes for image processing, Phys. Letters A, 373, 29, 2438-2444 (2009) · Zbl 1231.68270 · doi:10.1016/j.physleta.2009.04.076
[3] Perona, P.; Malik, J., Scale-space and edge detection using anisotropic diffusion, IEEE Trans. on pattern anal. and machine intell., 12, 7, 629-639 (1990) · doi:10.1109/34.56205
[4] Lhachemi, H.; Prieur, C., Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a dirichlet or neumann boundary measurement, Automatica, 135 (2022) · Zbl 1480.93199 · doi:10.1016/j.automatica.2021.109955
[5] Tajadodi, H., A numerical approach of fractional advection-diffusion equation with atangana-baleanu derivative, Chaos, Solitons & Fractals, 130 (2020) · Zbl 1489.65125 · doi:10.1016/j.chaos.2019.109527
[6] Singh, A., Das, S., Ong, S.H., Jafari, H.: Numerical solution of nonlinear reaction-advection-diffusion equation. Journal of Computational and Nonlinear Dynamics 14(4) (2019)
[7] Elliott, C.; Smitheman, S., Numerical analysis of the tv regularization and h- 1 fidelity model for decomposing an image into cartoon plus texture, IMA J. of Numerical Anal., 29, 3, 651-689 (2009) · Zbl 1169.94003 · doi:10.1093/imanum/drn025
[8] Guo, Z.; Yin, J.; Liu, Q., On a reaction-diffusion system applied to image decomposition and restoration, Math. and Comput. Modelling, 53, 5-6, 1336-1350 (2011) · Zbl 1217.65172 · doi:10.1016/j.mcm.2010.12.031
[9] Rudin, LI; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear phenomena, 60, 1-4, 259-268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[10] Weickert, J.: Scale-space properties of nonlinear diffusion filtering with a diffusion tensor (1994)
[11] Weickert, J.: Anisotropic Diffusion in Image Processing, vol. 1. Teubner, Stuttgart (1998) · Zbl 0886.68131
[12] Weickert, J., Applications of nonlinear diffusion in image processing and computer vision, Acta Math. Univ. Comenianae, 70, 1, 33-50 (2001) · Zbl 0988.35070
[13] Cuesta, E.; Kirane, M.; Malik, SA, Image structure preserving denoising using generalized fractional time integrals, Signal Process., 92, 2, 553-563 (2012) · doi:10.1016/j.sigpro.2011.09.001
[14] Bai, J.; Feng, X-C, Fractional-order anisotropic diffusion for image denoising, IEEE trans. on image process., 16, 10, 2492-2502 (2007) · Zbl 1119.76377 · doi:10.1109/TIP.2007.904971
[15] Janev, M.; Pilipović, S.; Atanacković, T.; Obradović, R.; Ralević, N., Fully fractional anisotropic diffusion for image denoising, Math. and Comput. Modelling, 54, 1-2, 729-741 (2011) · Zbl 1225.94003 · doi:10.1016/j.mcm.2011.03.017
[16] Ben-loghfyry, A., Hakim, A.: Robust time-fractional diffusion filtering for noise removal. Mathematical Methods in the Applied Sciences (2022)
[17] Ben-loghfyry, A.; Hakim, A., Time-fractional diffusion equation for signal and image smoothing, Math. Modeling and Comput., 9, 2, 351-364 (2022) · doi:10.23939/mmc2022.02.351
[18] Oliveira, DS; de Oliveira, EC, On a caputo-type fractional derivative, Adv. in Pure and Appl. Math., 10, 2, 81-91 (2019) · Zbl 1414.26018 · doi:10.1515/apam-2017-0068
[19] Li, C., Qian, D., Chen, Y.: On riemann-liouville and caputo derivatives. Discrete Dynamics in Nature and Society 2011 (2011) · Zbl 1213.26008
[20] Aubin, J-P, Analyse mathematique-un theoreme de compacite, Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sci., 256, 24, 5042 (1963) · Zbl 0195.13002
[21] Li, L.; Liu, J-G, Some compactness criteria for weak solutions of time fractional pdes, SIAM J. on Math. Anal., 50, 4, 3963-3995 (2018) · Zbl 1403.35318 · doi:10.1137/17M1145549
[22] Luchko, Y., Operational method in fractional calculus, Fract. Calc. Appl. Anal, 2, 4, 463-488 (1999) · Zbl 1030.26009
[23] Sabatier, J.; Agrawal, OP; Machado, JT, Advances in Fractional Calculus (2007), Dordrecht: Springer, Dordrecht · Zbl 1116.00014 · doi:10.1007/978-1-4020-6042-7
[24] Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Academic Press (1998) · Zbl 0924.34008
[25] Machado, JT; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Commun. in Nonlinear Sci. and Numerical Simulation, 16, 3, 1140-1153 (2011) · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027
[26] Alikhanov, A., A priori estimates for solutions of boundary value problems for fractional-order equations, Differential equations, 46, 5, 660-666 (2010) · Zbl 1208.35161 · doi:10.1134/S0012266110050058
[27] Evans, LC, Partial differential equations, Graduate studies in math., 19, 4, 7 (1998) · Zbl 0902.35002
[28] Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. SIAM (2014) · Zbl 1311.49001
[29] Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker guide to the fractional sobolev spaces, Bulletin des sci. mathématiques, 136, 5, 521-573 (2012) · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004
[30] Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (2010), New York: Springer, New York · Zbl 1218.46002
[31] Schwartz, J.T., Karcher, H.: Nonlinear Functional Analysis. New York, Gordon and Breach (1969) · Zbl 0203.14501
[32] Mu, J.; Ahmad, B.; Huang, S., Existence and regularity of solutions to time-fractional diffusion equations, Computers & Math. with Appl., 73, 6, 985-996 (2017) · Zbl 1409.35223 · doi:10.1016/j.camwa.2016.04.039
[33] Kilbas, A.: Theory and Applications of Fractional Differential Equations · Zbl 1210.35276
[34] Murio, DA, Implicit finite difference approximation for time fractional diffusion equations, Computers & Math. with Appl., 56, 4, 1138-1145 (2008) · Zbl 1155.65372 · doi:10.1016/j.camwa.2008.02.015
[35] Zhang, J.; Chen, K., A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution, SIAM J. on Imaging Sci., 8, 4, 2487-2518 (2015) · Zbl 1327.62388 · doi:10.1137/14097121X
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