
-
Previous Article
Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions
- DCDS-B Home
- This Issue
-
Next Article
Approximate dynamics of a class of stochastic wave equations with white noise
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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] |
Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069 |
[2] |
Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 |
[3] |
Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 |
[4] |
Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184 |
[5] |
Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 |
[6] |
Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 |
[7] |
Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017 |
[8] |
Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 |
[9] |
Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321 |
[10] |
Israa Mohammed Khudher, Yahya Ismail Ibrahim, Suhaib Abduljabbar Altamir. Individual biometrics pattern based artificial image analysis techniques. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2020056 |
[11] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[12] |
Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013 |
[13] |
Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 |
[14] |
Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021017 |
[15] |
Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021020 |
[16] |
Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021021 |
[17] |
Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 |
[18] |
Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002 |
[19] |
Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263 |
[20] |
Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]