
-
Previous Article
A comparative study of atomistic-based stress evaluation
- DCDS-B Home
- This Issue
-
Next Article
On existence and uniqueness properties for solutions of stochastic fixed point equations
A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition
1. | Ecole Nationale des Sciences Appliquées de Marrakech, Université Cadi Ayyad, B.P. 575 Avenue Abdelkrim Al Khattabi Marrakech, Morocco |
2. | Institut de Mathématiques de Bordeaux and INRIA-Carmen Bordeaux Sud-Ouest, Université de Bordeaux, 33076 Bordeaux Cedex, France |
3. | Ecole Supérieure de Technologie d'Essaouira, Université Cadi Ayyad, B.P. 383 Essaouira El Jadida, Essaouira, Morocco |
This paper is devoted to the mathematical and numerical study of a new proposed model based on a fractional diffusion equation coupled with a nonlinear regularization of the Total Variation operator. This model is primarily intended to introduce a weak norm in the fidelity term, where this norm is considered more appropriate for capturing very oscillatory characteristics interpreted as a texture. Furthermore, our proposed model profits from the benefits of a variable exponent used to distinguish the features of the image. By using Faedo-Galerkin method, we prove the well-posedness (existence and uniqueness) of the weak solution for the proposed model. Based on the alternating direction implicit method of Peaceman-Rachford and the approximations of the Gr$ \ddot{u} $nwald-Letnikov operators, we develop the numerical discretization of our fractional diffusion equation. Experimental results claim that our model provides high-quality results in cartoon-texture-edges decomposition and image denoising. In particular, our model can successfully reduce the staircase phenomenon during the image denoising. Furthermore, small details, texture and fine structures still maintained in the restored image. Finally, we compare our numerical results with the existing models in the literature.
References:
[1] |
R. Aboulaich, D. Meskine and A. Souissi,
New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882.
doi: 10.1016/j.camwa.2008.01.017. |
[2] |
R. A. Adams, Sobolev Spaces, Ac. Press, New york, 1975.
![]() ![]() |
[3] |
L. Afraites, A. Atlas, F. Karami and D. Meskine,
Some class of parabolic systems applied to image processing, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1671-1687.
doi: 10.3934/dcdsb.2016017. |
[4] |
L. Alvarez, P.-L. Lions and J.-M. Morel,
Image selective smoothing and edge detection by nonlinear diffusion. Ⅱ, SIAM J. Numer. Anal., 29 (1992), 845-866.
doi: 10.1137/0729052. |
[5] |
F. Andreu, C. Ballester, V. Caselles and J. M. Mazón,
Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360.
|
[6] |
N. Aronszajn,
Boundary values of functions with finite Dirichlet integral, Techn. Report of Univ. of Kansas, 14 (1955), 77-94.
|
[7] |
G. Aubert and J.-F. Aujol,
Modeling very oscillating signals. Application to image processing, Appl. Math. Optim., 51 (2005), 163-182.
doi: 10.1007/s00245-004-0812-z. |
[8] |
J.-F. Aujol, G. Aubert, L. Blanc-Féraud and A. Chambolle,
Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging Vision, 22 (2005), 71-88.
doi: 10.1007/s10851-005-4783-8. |
[9] |
A. Buades, B. Coll and J. M. Morel,
A non-local algorithm for image denoising, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2 (2005), 60-65.
|
[10] |
A. Chambolle, R. A. DeVore, N.-Y. Lee and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process., 7 (1998), 319–335.
doi: 10.1109/83.661182. |
[11] |
E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, (1955). |
[12] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[13] |
K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, (2010).
doi: 10.1007/978-3-642-14574-2. |
[14] |
S. Dipierro, X. Ros-Oton and E. Valdinoci,
Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377-416.
doi: 10.4171/RMI/942. |
[15] |
D. L. Donoho,
De-noising by soft-thresholding, IEEE Transactions on Information Theory, 41 (1995), 613-627.
doi: 10.1109/18.382009. |
[16] |
A. Elmahi and D. Meskine,
Parabolic equations in Orlicz spaces, J. London Math. Soc. (2), 72 (2005), 410-428.
doi: 10.1112/S0024610705006630. |
[17] |
A. Elmoataz, X. Desquesnes and O. Lézoray,
Non-Local Morphological PDEs and $p$-Laplacian Equation on Graphs With Applications in Image Processing and Machine Learning, IEEE Journal of Selected Topics in Signal Processing, 6 (2012), 764-779.
doi: 10.1109/JSTSP.2012.2216504. |
[18] |
E. Gagliardo,
Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 8 (1959), 24-51.
|
[19] |
J. B. Garnett, P. W. Jones, T. M. Le and L. A Vese,
Modeling oscillatory components with the homogeneous spaces $B\dot MO^{-\alpha}$ and $\dot W{}^{-\alpha,p}$, Pure Appl. Math. Q., 7 (2011), 275-318.
doi: 10.4310/PAMQ.2011.v7.n2.a2. |
[20] |
J. B. Garnett, T. M. Le, Y. Meyer and L. A Vese,
Image decompositions using bounded variation and generalized homogeneous Besov spaces, Appl. Comput. Harmon. Anal., 23 (2007), 25-56.
doi: 10.1016/j.acha.2007.01.005. |
[21] |
Y. Giga, M. Muszkieta and P. Rybka,
A duality based approach to the minimizing total variation flow in the space $H^{-s}$, Jpn. J. Ind. Appl. Math., 36 (2019), 261-286.
doi: 10.1007/s13160-018-00340-4. |
[22] |
G. Gilboa and S. Osher,
Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630.
doi: 10.1137/060669358. |
[23] |
G. Gilboa and S. Osher,
Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.
doi: 10.1137/070698592. |
[24] |
J.-P. Gossez,
Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205.
doi: 10.1090/S0002-9947-1974-0342854-2. |
[25] |
Z. Guo, J. Yin and Q. Liu,
On a reaction-diffusion system applied to image decomposition and restoration, Math. Comput. Modelling, 53 (2011), 1336-1350.
doi: 10.1016/j.mcm.2010.12.031. |
[26] |
Y. Jin, J. Jost and G. Wang,
A new nonlocal variational setting for image processing, Inverse Probl. Imaging, 9 (2015), 415-430.
doi: 10.3934/ipi.2015.9.415. |
[27] |
Y. Kim and L. A. Vese,
Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability, Inverse Probl. Imaging, 3 (2009), 43-68.
doi: 10.3934/ipi.2009.3.43. |
[28] |
S. Kindermann, S. Osher and P. W. Jones,
Deblurring and denoising of images by nonlocal functionals, Multiscale Model. Simul., 4 (2005), 1091-1115.
doi: 10.1137/050622249. |
[29] |
T. M. Le and L. A. Vese,
Image decomposition using total variation and $ \rm{div} $($ \rm{BMO} $), Multiscale Model. Simul., 4 (2005), 390-423.
doi: 10.1137/040610052. |
[30] |
L. H. Lieu and L. A. Vese,
Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces, Appl. Math. Optim., 58 (2008), 167-193.
doi: 10.1007/s00245-008-9047-8. |
[31] |
X. Liu and L. Huang,
A new nonlocal total variation regularization algorithm for image denoising, Math. Comput. Simulation, 97 (2014), 224-233.
doi: 10.1016/j.matcom.2013.10.001. |
[32] |
Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, American Mathematical Society, Providence, RI, (2001).
doi: 10.1090/ulect/022. |
[33] |
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc., New York, (1993). |
[34] |
S. Osher, A. Solé and L. Vese,
Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, Multiscale Model. Simul., 1 (2003), 349-370.
doi: 10.1137/S1540345902416247. |
[35] |
P. Perona and J. Malik,
Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639.
doi: 10.1109/34.56205. |
[36] |
L. I. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithms, Phys. D., 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[37] |
L. N. Slobodeckiĭ,
Generalized Sobolev spaces and their application to boundary problems for partial differential equations, Leningrad. Gos. Ped. Inst. Učen. Zap., 197 (1958), 54-112.
|
[38] |
N. N. Tarkhanov, The Analysis of Solutions of Elliptic Equations, Kluwer Academic Publishers Group, Dordrecht, (1997).
doi: 10.1007/978-94-015-8804-1. |
[39] |
L. A. Vese and S. J. Osher,
Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), 553-572.
doi: 10.1023/A:1025384832106. |
[40] |
Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli,
Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612.
doi: 10.1109/TIP.2003.819861. |
[41] |
L. P. Yaroslavsky, Digital Picture Processing, Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-642-81929-2. |
show all references
References:
[1] |
R. Aboulaich, D. Meskine and A. Souissi,
New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882.
doi: 10.1016/j.camwa.2008.01.017. |
[2] |
R. A. Adams, Sobolev Spaces, Ac. Press, New york, 1975.
![]() ![]() |
[3] |
L. Afraites, A. Atlas, F. Karami and D. Meskine,
Some class of parabolic systems applied to image processing, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1671-1687.
doi: 10.3934/dcdsb.2016017. |
[4] |
L. Alvarez, P.-L. Lions and J.-M. Morel,
Image selective smoothing and edge detection by nonlinear diffusion. Ⅱ, SIAM J. Numer. Anal., 29 (1992), 845-866.
doi: 10.1137/0729052. |
[5] |
F. Andreu, C. Ballester, V. Caselles and J. M. Mazón,
Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360.
|
[6] |
N. Aronszajn,
Boundary values of functions with finite Dirichlet integral, Techn. Report of Univ. of Kansas, 14 (1955), 77-94.
|
[7] |
G. Aubert and J.-F. Aujol,
Modeling very oscillating signals. Application to image processing, Appl. Math. Optim., 51 (2005), 163-182.
doi: 10.1007/s00245-004-0812-z. |
[8] |
J.-F. Aujol, G. Aubert, L. Blanc-Féraud and A. Chambolle,
Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging Vision, 22 (2005), 71-88.
doi: 10.1007/s10851-005-4783-8. |
[9] |
A. Buades, B. Coll and J. M. Morel,
A non-local algorithm for image denoising, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2 (2005), 60-65.
|
[10] |
A. Chambolle, R. A. DeVore, N.-Y. Lee and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process., 7 (1998), 319–335.
doi: 10.1109/83.661182. |
[11] |
E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, (1955). |
[12] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[13] |
K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, (2010).
doi: 10.1007/978-3-642-14574-2. |
[14] |
S. Dipierro, X. Ros-Oton and E. Valdinoci,
Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377-416.
doi: 10.4171/RMI/942. |
[15] |
D. L. Donoho,
De-noising by soft-thresholding, IEEE Transactions on Information Theory, 41 (1995), 613-627.
doi: 10.1109/18.382009. |
[16] |
A. Elmahi and D. Meskine,
Parabolic equations in Orlicz spaces, J. London Math. Soc. (2), 72 (2005), 410-428.
doi: 10.1112/S0024610705006630. |
[17] |
A. Elmoataz, X. Desquesnes and O. Lézoray,
Non-Local Morphological PDEs and $p$-Laplacian Equation on Graphs With Applications in Image Processing and Machine Learning, IEEE Journal of Selected Topics in Signal Processing, 6 (2012), 764-779.
doi: 10.1109/JSTSP.2012.2216504. |
[18] |
E. Gagliardo,
Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 8 (1959), 24-51.
|
[19] |
J. B. Garnett, P. W. Jones, T. M. Le and L. A Vese,
Modeling oscillatory components with the homogeneous spaces $B\dot MO^{-\alpha}$ and $\dot W{}^{-\alpha,p}$, Pure Appl. Math. Q., 7 (2011), 275-318.
doi: 10.4310/PAMQ.2011.v7.n2.a2. |
[20] |
J. B. Garnett, T. M. Le, Y. Meyer and L. A Vese,
Image decompositions using bounded variation and generalized homogeneous Besov spaces, Appl. Comput. Harmon. Anal., 23 (2007), 25-56.
doi: 10.1016/j.acha.2007.01.005. |
[21] |
Y. Giga, M. Muszkieta and P. Rybka,
A duality based approach to the minimizing total variation flow in the space $H^{-s}$, Jpn. J. Ind. Appl. Math., 36 (2019), 261-286.
doi: 10.1007/s13160-018-00340-4. |
[22] |
G. Gilboa and S. Osher,
Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630.
doi: 10.1137/060669358. |
[23] |
G. Gilboa and S. Osher,
Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.
doi: 10.1137/070698592. |
[24] |
J.-P. Gossez,
Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205.
doi: 10.1090/S0002-9947-1974-0342854-2. |
[25] |
Z. Guo, J. Yin and Q. Liu,
On a reaction-diffusion system applied to image decomposition and restoration, Math. Comput. Modelling, 53 (2011), 1336-1350.
doi: 10.1016/j.mcm.2010.12.031. |
[26] |
Y. Jin, J. Jost and G. Wang,
A new nonlocal variational setting for image processing, Inverse Probl. Imaging, 9 (2015), 415-430.
doi: 10.3934/ipi.2015.9.415. |
[27] |
Y. Kim and L. A. Vese,
Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability, Inverse Probl. Imaging, 3 (2009), 43-68.
doi: 10.3934/ipi.2009.3.43. |
[28] |
S. Kindermann, S. Osher and P. W. Jones,
Deblurring and denoising of images by nonlocal functionals, Multiscale Model. Simul., 4 (2005), 1091-1115.
doi: 10.1137/050622249. |
[29] |
T. M. Le and L. A. Vese,
Image decomposition using total variation and $ \rm{div} $($ \rm{BMO} $), Multiscale Model. Simul., 4 (2005), 390-423.
doi: 10.1137/040610052. |
[30] |
L. H. Lieu and L. A. Vese,
Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces, Appl. Math. Optim., 58 (2008), 167-193.
doi: 10.1007/s00245-008-9047-8. |
[31] |
X. Liu and L. Huang,
A new nonlocal total variation regularization algorithm for image denoising, Math. Comput. Simulation, 97 (2014), 224-233.
doi: 10.1016/j.matcom.2013.10.001. |
[32] |
Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, American Mathematical Society, Providence, RI, (2001).
doi: 10.1090/ulect/022. |
[33] |
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc., New York, (1993). |
[34] |
S. Osher, A. Solé and L. Vese,
Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, Multiscale Model. Simul., 1 (2003), 349-370.
doi: 10.1137/S1540345902416247. |
[35] |
P. Perona and J. Malik,
Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639.
doi: 10.1109/34.56205. |
[36] |
L. I. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithms, Phys. D., 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[37] |
L. N. Slobodeckiĭ,
Generalized Sobolev spaces and their application to boundary problems for partial differential equations, Leningrad. Gos. Ped. Inst. Učen. Zap., 197 (1958), 54-112.
|
[38] |
N. N. Tarkhanov, The Analysis of Solutions of Elliptic Equations, Kluwer Academic Publishers Group, Dordrecht, (1997).
doi: 10.1007/978-94-015-8804-1. |
[39] |
L. A. Vese and S. J. Osher,
Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), 553-572.
doi: 10.1023/A:1025384832106. |
[40] |
Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli,
Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612.
doi: 10.1109/TIP.2003.819861. |
[41] |
L. P. Yaroslavsky, Digital Picture Processing, Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-642-81929-2. |














Noise level | ||||||
12.54 | 20.23 | 13.57 | 14.84 | 12.66 | 15.11 | |
28.14 | 35.82 | 27.22 | 29.16 | 28.17 | 30.60 | |
0.782 | 0.971 | 0.865 | 0.933 | 0.590 | 0.949 |
Noise level | ||||||
12.54 | 20.23 | 13.57 | 14.84 | 12.66 | 15.11 | |
28.14 | 35.82 | 27.22 | 29.16 | 28.17 | 30.60 | |
0.782 | 0.971 | 0.865 | 0.933 | 0.590 | 0.949 |
06.50 | 17.49 | 09.04 | 12.10 | 06.62 | 13.41 | |
22.10 | 33.09 | 22.02 | 25.38 | 22.13 | 28.90 | |
0.580 | 0.925 | 0.743 | 0.896 | 0.422 | 0.837 |
06.50 | 17.49 | 09.04 | 12.10 | 06.62 | 13.41 | |
22.10 | 33.09 | 22.02 | 25.38 | 22.13 | 28.90 | |
0.580 | 0.925 | 0.743 | 0.896 | 0.422 | 0.837 |
03.01 | 16.12 | 05.51 | 10.07 | 03.06 | 12.44 | |
18.58 | 31.73 | 18.56 | 23.39 | 18.54 | 27.92 | |
0.501 | 0.884 | 0.650 | 0.885 | 0.325 | 0.780 |
03.01 | 16.12 | 05.51 | 10.07 | 03.06 | 12.44 | |
18.58 | 31.73 | 18.56 | 23.39 | 18.54 | 27.92 | |
0.501 | 0.884 | 0.650 | 0.885 | 0.325 | 0.780 |
[1] |
Su-Hong Jiang, Min Li. A modified strictly contractive peaceman-rachford splitting method for multi-block separable convex programming. Journal of Industrial and Management Optimization, 2018, 14 (1) : 397-412. doi: 10.3934/jimo.2017052 |
[2] |
Michael Hintermüller, Monserrat Rincon-Camacho. An adaptive finite element method in $L^2$-TV-based image denoising. Inverse Problems and Imaging, 2014, 8 (3) : 685-711. doi: 10.3934/ipi.2014.8.685 |
[3] |
Zhiguang Zhang, Qiang Liu, Tianling Gao. A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising. Inverse Problems and Imaging, 2021, 15 (6) : 1451-1469. doi: 10.3934/ipi.2021018 |
[4] |
Wei Zhu, Xue-Cheng Tai, Tony Chan. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Problems and Imaging, 2013, 7 (4) : 1409-1432. doi: 10.3934/ipi.2013.7.1409 |
[5] |
Feishe Chen, Lixin Shen, Yuesheng Xu, Xueying Zeng. The Moreau envelope approach for the L1/TV image denoising model. Inverse Problems and Imaging, 2014, 8 (1) : 53-77. doi: 10.3934/ipi.2014.8.53 |
[6] |
Sören Bartels, Nico Weber. Parameter learning and fractional differential operators: Applications in regularized image denoising and decomposition problems. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021048 |
[7] |
Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems and Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523 |
[8] |
Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007 |
[9] |
Rongliang Chen, Jizu Huang, Xiao-Chuan Cai. A parallel domain decomposition algorithm for large scale image denoising. Inverse Problems and Imaging, 2019, 13 (6) : 1259-1282. doi: 10.3934/ipi.2019055 |
[10] |
Weina Wang, Chunlin Wu, Jiansong Deng. Piecewise constant signal and image denoising using a selective averaging method with multiple neighbors. Inverse Problems and Imaging, 2019, 13 (5) : 903-930. doi: 10.3934/ipi.2019041 |
[11] |
Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 |
[12] |
Mujibur Rahman Chowdhury, Jun Zhang, Jing Qin, Yifei Lou. Poisson image denoising based on fractional-order total variation. Inverse Problems and Imaging, 2020, 14 (1) : 77-96. doi: 10.3934/ipi.2019064 |
[13] |
Fangfang Dong, Yunmei Chen. A fractional-order derivative based variational framework for image denoising. Inverse Problems and Imaging, 2016, 10 (1) : 27-50. doi: 10.3934/ipi.2016.10.27 |
[14] |
Ming Huang, Cong Cheng, Yang Li, Zun Quan Xia. The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1885-1905. doi: 10.3934/jimo.2019034 |
[15] |
Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931 |
[16] |
Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025 |
[17] |
Jianhong (Jackie) Shen, Sung Ha Kang. Quantum TV and applications in image processing. Inverse Problems and Imaging, 2007, 1 (3) : 557-575. doi: 10.3934/ipi.2007.1.557 |
[18] |
Ying Wen, Jiebao Sun, Zhichang Guo. A new anisotropic fourth-order diffusion equation model based on image features for image denoising. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022004 |
[19] |
Weihong Guo, Jing Qin. A geometry guided image denoising scheme. Inverse Problems and Imaging, 2013, 7 (2) : 499-521. doi: 10.3934/ipi.2013.7.499 |
[20] |
Jianjun Zhang, Yunyi Hu, James G. Nagy. A scaled gradient method for digital tomographic image reconstruction. Inverse Problems and Imaging, 2018, 12 (1) : 239-259. doi: 10.3934/ipi.2018010 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]