
-
Previous Article
An adaptive total variational despeckling model based on gray level indicator frame
- IPI Home
- This Issue
-
Next Article
Inbetweening auto-animation via Fokker-Planck dynamics and thresholding
Image retinex based on the nonconvex TV-type regularization
1. | School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China |
2. | Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China |
3. | Department of Mathematical Sciences, University of Liverpool, L693BX, United Kingdom |
Retinex theory is introduced to show how the human visual system perceives the color and the illumination effect such as Retinex illusions, medical image intensity inhomogeneity and color shadow effect etc.. Many researchers have studied this ill-posed problem based on the framework of the variation energy functional for decades. However, to the best of our knowledge, the existing models via the sparsity of the image based on the nonconvex $ \ell^p $-quasinorm were limited. To deal with this problem, this paper considers a TV$ _p $-HOTV$ _q $-based retinex model with $ p, q\in(0, 1) $. Specially, the TV$ _p $ term based on the total variation(TV) regularization can describe the reflectance efficiently, which has the piecewise constant structure. The HOTV$ _q $ term based on the high order total variation(HOTV) regularization can penalize the smooth structure called the illumination. Since the proposed model is non-convex, non-smooth and non-Lipschitz, we employ the iteratively reweighed $ \ell_1 $ (IRL1) algorithm to solve it. We also discuss some properties of our proposed model and algorithm. Experimental experiments on the simulated and real images illustrate the effectiveness and the robustness of our proposed model both visually and quantitatively by compared with some related state-of-the-art variational models.
References:
[1] |
R. Amestoy, E. Provenzi, M. Bertalmío and V. Caselles, A perceptually inspired variational framework for color enhancement, IEEE Transactions on Pattern Analysis and Machine Intelligence, 31 (2009), 458-474. Google Scholar |
[2] |
M. Benning, F. Knoll, C. Schonlieb and T. Valkonen,
Preconditioned ADMM with nonlinear operator constraint, IFIP Conference on System Modeling and Optimization, 494 (2015), 117-126.
doi: 10.1007/978-3-319-55795-3_10. |
[3] |
M. Bertalmío, V. Caselles, E. Provenzi and A. Rizzi,
Perceptual color correction through variational techniques, IEEE Transactions on Image Processing, 16 (2007), 1058-1072.
doi: 10.1109/TIP.2007.891777. |
[4] |
M. Bertalmío, V. Caselles and E. Provenzi, Issues about retinex theory and contrast enhancement, International Journal of Computer Vision, 83 (2009), 101-119. Google Scholar |
[5] |
A. Blake,
Boundary conditions for lightness computation in Mondrian world, Computer Vision Graphics Image Processing, 32 (1985), 314-327.
doi: 10.1016/0734-189X(85)90054-4. |
[6] |
S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein,
Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2011), 1-122.
doi: 10.1561/9781601984616. |
[7] |
E. Candes, M. Wakin and S. Boyd,
Enhancing sparsity by reweighted $l_1$ minimization, Journal of Fourier Analysis and Applications, 14 (2008), 877-905.
doi: 10.1007/s00041-008-9045-x. |
[8] |
T. Cooper and F. Baqai,
Analysis and extensions of the frankle-mccann retinex algorithm, Journal of Electronic Imaging, 13 (2004), 85-93.
doi: 10.1117/1.1636182. |
[9] |
X. Chen, F. Xu and Y. Ye,
Lower bound theory of nonzero entries in solutions of $\ell^2-\ell^p$ minimization, SIAM Journal on Scientific Computing, 32 (2010), 2832-2852.
doi: 10.1137/090761471. |
[10] |
X. Chen and W. Zhou,
Convergence of the reweighted $\ell_1$ minimization algorithm for $\ell_2-\ell^p$ minimization, Journal Computational Optimization and Application, 59 (2014), 47-61.
doi: 10.1007/s10589-013-9553-8. |
[11] |
J. Douglas and H. Rachford,
On the numerical solution of heat conduction problems in two and three space variables, Transactions of the American Mathematical Society, 82 (1956), 421-439.
doi: 10.1090/S0002-9947-1956-0084194-4. |
[12] |
Y. Duan, H. Chang, W. Huang, J. Zhou, Z. Lu and C. Wu,
The $L_0$ regularized Mumford-Shah model for bias correction and segmentation of medical images, IEEE Transactions on Image Processing, 24 (2015), 3927-3938.
doi: 10.1109/TIP.2015.2451957. |
[13] |
D. Gabay and B. Mercier,
A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Computers and Mathematics with Applications, 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1. |
[14] |
N. Galatsanos and A. Katsaggelos,
Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, IEEE Transactions on Image Processing, 1 (1992), 322-336.
doi: 10.1109/83.148606. |
[15] |
D. Ghilli and K. Kunisch,
On monotone and primal-dual active set schemes for $\ell^p$-type problems, $p\in(0, 1]$, Computational Optimizationand Applications, 72 (2019), 45-85.
doi: 10.1007/s10589-018-0036-9. |
[16] |
R. Glowinski, S. Luo and X. Tai,
Fast operator-splitting algorithms for variational imaging models: Some recent developments, Handbook of Numerical Analysis, 20 (2019), 191-232.
|
[17] |
R. Glowinski, S. Osher and W. Yin, Splitting Methods in Communication, Imaging, Science, and Engineering, Springer, Cham, 2016.
doi: 10.1007/978-3-319-41589-5. |
[18] |
Z. Gu, F. Li and X. Lv,
A detail preserving variational model for image Retinex, Applied Mathematical Modelling, 68 (2019), 643-661.
doi: 10.1016/j.apm.2018.11.052. |
[19] |
P. Hansen and D. Leary,
The use of the L-curve in the regularization of discrete ill-posed problems, SIAM Journal on Scientific Computing, 14 (1993), 1487-1503.
doi: 10.1137/0914086. |
[20] |
B. Horn,
Determining lightness from an image, Computer Graphics Image Processing, 3 (1974), 277-299.
doi: 10.1016/0146-664X(74)90022-7. |
[21] |
B. Horn,
Understanding image intensities, Artificial Intelligence, 8 (1977), 201-231.
doi: 10.1016/0004-3702(77)90020-0. |
[22] |
D. Jobson, Z. Rahman and G. Woodell,
Properties and performance of a center/surround retinex, IEEE Transactions on Image Processing, 6 (1997), 451-462.
doi: 10.1109/83.557356. |
[23] |
D. Jobson, Z. Rahman and G. Woodell,
A multiscale Retinex for bridging the gap between color image and the human observation of scenes, IEEE Transactions on Image Processing, 6 (2002), 965-976.
doi: 10.1109/83.597272. |
[24] |
Y. Jung, T. Jeong and S. Yun,
Non-convex TV denoising corrupted by impulse noise, Inverse Problems and Imaging, 11 (2017), 689-702.
doi: 10.3934/ipi.2017032. |
[25] |
R. Kimmel, M. Elad, D. Shaked, R. Keshet and I. Sobel, A variational framework for retinex, International Journal of Computer Vision, 52 (2003), 7-23. Google Scholar |
[26] |
M. Lai, Y. Xu and W. Yin,
Improved iteratively reweighted least squares for unconstrained smoothed $\ell_q$ minimization, SIAM Journal on Numerical Analysis, 51 (2013), 927-957.
doi: 10.1137/110840364. |
[27] |
E. Land and J. Mccann, Lightness and Retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11. Google Scholar |
[28] |
E. Land, The Retinex theory of color vision, Scientific American, 237 (1977), 108-128. Google Scholar |
[29] |
E. Land, An alternative technique for the computation of the designator in the retinex theory of color vision, Proceedings of the National Academy of Sciences, 83 (1986), 3078-3080. Google Scholar |
[30] |
M. Langer and S. Zucker,
Spatially varying illumination: A computational model of converging and diverging sources, European Conference on Computer Vision, 801 (1994), 226-232.
doi: 10.1007/BFb0028356. |
[31] |
A. Lanza1, S. Morigi1 and F. Sgallari,
Constrained $TV_p-\ell_2$ model for image restoration, Journal of Scientific Computing, 68 (2016), 64-91.
doi: 10.1007/s10915-015-0129-x. |
[32] |
L. Liu, Z. Pang and Y. Duan,
Retinex based on exponent-type total variation scheme, Inverse Problems and Imaging, 12 (2018), 1199-1217.
doi: 10.3934/ipi.2018050. |
[33] |
Z. Liu, C. Wu and Y. Zhao,
A new globally convergent algorithm for non-Lipschitz $\ell^p-\ell^q$ minimization, Advances in Computational Mathematics, 45 (2019), 1369-1399.
doi: 10.1007/s10444-019-09668-y. |
[34] |
J. Liang and X. Zhang,
Retinex by higher order total variation ${L}^1$ decomposition, Journal of Mathematical Imaging and Vision, 52 (2015), 345-355.
doi: 10.1007/s10851-015-0568-x. |
[35] |
Z. Lu,
Iterative reweighted minimization methods for $\ell^p$ regularized unconstrained nonlinear programming, Mathematical Programming: Series A and B, 147 (2014), 277-307.
doi: 10.1007/s10107-013-0722-4. |
[36] |
D. Marini and A. Rizzi, A computational approach to color adaptation effects, Image and Vision Computing, 18 (2000), 1005-1014. Google Scholar |
[37] |
W. Ma and S. Osher,
A TV bregman iterative model of retinex theory, Inverse Problems and Imaging, 6 (2012), 697-708.
doi: 10.3934/ipi.2012.6.697. |
[38] |
J. Mccann, Lessons learned from mondrians applied to real images and color gamuts, Proceedings of the IST/SID 7th Color Imaging Conference, 1999, 1–8. Google Scholar |
[39] |
J. Morel, A. Petro and C. Sbert, Fast implementation of color constancy algorithms, Proceedings of SPIE, 7241, 2009. Google Scholar |
[40] |
J. Morel, A. Petro and C. Sbert,
A PDE formalization of retinex theory, IEEE Transactions on Image Processing, 19 (2010), 2825-2837.
doi: 10.1109/TIP.2010.2049239. |
[41] |
V. Morozov, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5280-1. |
[42] |
M. Ng and W. Wang,
A total variation model for retinex, SIAM Journal on Imaging Sciences, 4 (2011), 345-365.
doi: 10.1137/100806588. |
[43] |
P. Ochs, A. Dosovitskiy, T. Brox and T. Pock,
On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision, SIAM Journal on Imaging Sciences, 8 (2015), 331-372.
doi: 10.1137/140971518. |
[44] |
J. Oliveira, J. Dias and M. Figueiredo, Adaptive total variation image deblurring: A majorizationCminimization approach, Signal Processing, 89 (2009), 1683-1693. Google Scholar |
[45] |
H. Pan, Y. Wen and H. Zhu,
A regularization parameter selection model for total variation based image noise removal, Applied Mathematical Modelling, 68 (2019), 353-367.
doi: 10.1016/j.apm.2018.11.032. |
[46] |
E. Provenzi, D. Marini, L. De Carli and A. Rizzi,
Mathematical definition and analysis of the retinex algorithm, Journal of the Optical Society of America A, 22 (2005), 2613-2621.
doi: 10.1364/JOSAA.22.002613. |
[47] |
S. Sabacha and M. Teboulle,
Lagrangian methods for composite optimization, Handbook of Numerical Analysis, 20 (2019), 401-436.
|
[48] |
A. Theljani and K. Chen, A Nash game based variational model for joint image intensity correction and registration to deal with varying illumination, Inverse Problems, 36 (2020), 034002. Google Scholar |
[49] |
Y. Wen and R. Chan,
Using generalized cross validation to select regularization parameter for total variation regularization problems, Inverse Problems and Imaging, 12 (2018), 1103-1120.
doi: 10.3934/ipi.2018046. |
[50] |
W. Wang and C. He,
A variational model with barrier functionals for Retinex, SIAM Journal on Imaging Sciences, 8 (2015), 1955-1980.
doi: 10.1137/15M1006908. |
[51] |
J. Zhang, R. Chen, C. Deng and S. Wang,
Fast linearized augmented method for Euler's elastica model, Numerical Mathematics:Theory, Methods and Applications, 10 (2017), 98-115.
doi: 10.4208/nmtma.2017.m1611. |
[52] |
X. Zhang, Y. Shi, Z. Pang and Y. Zhu,
Fast algorithm for image denoising with different boundary conditions, Journal of the Franklin Institute, 354 (2017), 4595-4614.
doi: 10.1016/j.jfranklin.2017.04.011. |
[53] |
D. Zosso, G. Tran and S. Osher, A unifying retinex model based on non-local differential operators, Computational Imaging XI, 865702, 2013.. Google Scholar |
[54] |
W. Zuo, D. Meng, L. Zhang, X. Feng and D. Zhang, A generalized iterated shrinkage algorithm for non-convex sparse coding, IEEE International Conference on Computer Vision, 2013,217–224. Google Scholar |
show all references
References:
[1] |
R. Amestoy, E. Provenzi, M. Bertalmío and V. Caselles, A perceptually inspired variational framework for color enhancement, IEEE Transactions on Pattern Analysis and Machine Intelligence, 31 (2009), 458-474. Google Scholar |
[2] |
M. Benning, F. Knoll, C. Schonlieb and T. Valkonen,
Preconditioned ADMM with nonlinear operator constraint, IFIP Conference on System Modeling and Optimization, 494 (2015), 117-126.
doi: 10.1007/978-3-319-55795-3_10. |
[3] |
M. Bertalmío, V. Caselles, E. Provenzi and A. Rizzi,
Perceptual color correction through variational techniques, IEEE Transactions on Image Processing, 16 (2007), 1058-1072.
doi: 10.1109/TIP.2007.891777. |
[4] |
M. Bertalmío, V. Caselles and E. Provenzi, Issues about retinex theory and contrast enhancement, International Journal of Computer Vision, 83 (2009), 101-119. Google Scholar |
[5] |
A. Blake,
Boundary conditions for lightness computation in Mondrian world, Computer Vision Graphics Image Processing, 32 (1985), 314-327.
doi: 10.1016/0734-189X(85)90054-4. |
[6] |
S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein,
Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2011), 1-122.
doi: 10.1561/9781601984616. |
[7] |
E. Candes, M. Wakin and S. Boyd,
Enhancing sparsity by reweighted $l_1$ minimization, Journal of Fourier Analysis and Applications, 14 (2008), 877-905.
doi: 10.1007/s00041-008-9045-x. |
[8] |
T. Cooper and F. Baqai,
Analysis and extensions of the frankle-mccann retinex algorithm, Journal of Electronic Imaging, 13 (2004), 85-93.
doi: 10.1117/1.1636182. |
[9] |
X. Chen, F. Xu and Y. Ye,
Lower bound theory of nonzero entries in solutions of $\ell^2-\ell^p$ minimization, SIAM Journal on Scientific Computing, 32 (2010), 2832-2852.
doi: 10.1137/090761471. |
[10] |
X. Chen and W. Zhou,
Convergence of the reweighted $\ell_1$ minimization algorithm for $\ell_2-\ell^p$ minimization, Journal Computational Optimization and Application, 59 (2014), 47-61.
doi: 10.1007/s10589-013-9553-8. |
[11] |
J. Douglas and H. Rachford,
On the numerical solution of heat conduction problems in two and three space variables, Transactions of the American Mathematical Society, 82 (1956), 421-439.
doi: 10.1090/S0002-9947-1956-0084194-4. |
[12] |
Y. Duan, H. Chang, W. Huang, J. Zhou, Z. Lu and C. Wu,
The $L_0$ regularized Mumford-Shah model for bias correction and segmentation of medical images, IEEE Transactions on Image Processing, 24 (2015), 3927-3938.
doi: 10.1109/TIP.2015.2451957. |
[13] |
D. Gabay and B. Mercier,
A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Computers and Mathematics with Applications, 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1. |
[14] |
N. Galatsanos and A. Katsaggelos,
Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, IEEE Transactions on Image Processing, 1 (1992), 322-336.
doi: 10.1109/83.148606. |
[15] |
D. Ghilli and K. Kunisch,
On monotone and primal-dual active set schemes for $\ell^p$-type problems, $p\in(0, 1]$, Computational Optimizationand Applications, 72 (2019), 45-85.
doi: 10.1007/s10589-018-0036-9. |
[16] |
R. Glowinski, S. Luo and X. Tai,
Fast operator-splitting algorithms for variational imaging models: Some recent developments, Handbook of Numerical Analysis, 20 (2019), 191-232.
|
[17] |
R. Glowinski, S. Osher and W. Yin, Splitting Methods in Communication, Imaging, Science, and Engineering, Springer, Cham, 2016.
doi: 10.1007/978-3-319-41589-5. |
[18] |
Z. Gu, F. Li and X. Lv,
A detail preserving variational model for image Retinex, Applied Mathematical Modelling, 68 (2019), 643-661.
doi: 10.1016/j.apm.2018.11.052. |
[19] |
P. Hansen and D. Leary,
The use of the L-curve in the regularization of discrete ill-posed problems, SIAM Journal on Scientific Computing, 14 (1993), 1487-1503.
doi: 10.1137/0914086. |
[20] |
B. Horn,
Determining lightness from an image, Computer Graphics Image Processing, 3 (1974), 277-299.
doi: 10.1016/0146-664X(74)90022-7. |
[21] |
B. Horn,
Understanding image intensities, Artificial Intelligence, 8 (1977), 201-231.
doi: 10.1016/0004-3702(77)90020-0. |
[22] |
D. Jobson, Z. Rahman and G. Woodell,
Properties and performance of a center/surround retinex, IEEE Transactions on Image Processing, 6 (1997), 451-462.
doi: 10.1109/83.557356. |
[23] |
D. Jobson, Z. Rahman and G. Woodell,
A multiscale Retinex for bridging the gap between color image and the human observation of scenes, IEEE Transactions on Image Processing, 6 (2002), 965-976.
doi: 10.1109/83.597272. |
[24] |
Y. Jung, T. Jeong and S. Yun,
Non-convex TV denoising corrupted by impulse noise, Inverse Problems and Imaging, 11 (2017), 689-702.
doi: 10.3934/ipi.2017032. |
[25] |
R. Kimmel, M. Elad, D. Shaked, R. Keshet and I. Sobel, A variational framework for retinex, International Journal of Computer Vision, 52 (2003), 7-23. Google Scholar |
[26] |
M. Lai, Y. Xu and W. Yin,
Improved iteratively reweighted least squares for unconstrained smoothed $\ell_q$ minimization, SIAM Journal on Numerical Analysis, 51 (2013), 927-957.
doi: 10.1137/110840364. |
[27] |
E. Land and J. Mccann, Lightness and Retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11. Google Scholar |
[28] |
E. Land, The Retinex theory of color vision, Scientific American, 237 (1977), 108-128. Google Scholar |
[29] |
E. Land, An alternative technique for the computation of the designator in the retinex theory of color vision, Proceedings of the National Academy of Sciences, 83 (1986), 3078-3080. Google Scholar |
[30] |
M. Langer and S. Zucker,
Spatially varying illumination: A computational model of converging and diverging sources, European Conference on Computer Vision, 801 (1994), 226-232.
doi: 10.1007/BFb0028356. |
[31] |
A. Lanza1, S. Morigi1 and F. Sgallari,
Constrained $TV_p-\ell_2$ model for image restoration, Journal of Scientific Computing, 68 (2016), 64-91.
doi: 10.1007/s10915-015-0129-x. |
[32] |
L. Liu, Z. Pang and Y. Duan,
Retinex based on exponent-type total variation scheme, Inverse Problems and Imaging, 12 (2018), 1199-1217.
doi: 10.3934/ipi.2018050. |
[33] |
Z. Liu, C. Wu and Y. Zhao,
A new globally convergent algorithm for non-Lipschitz $\ell^p-\ell^q$ minimization, Advances in Computational Mathematics, 45 (2019), 1369-1399.
doi: 10.1007/s10444-019-09668-y. |
[34] |
J. Liang and X. Zhang,
Retinex by higher order total variation ${L}^1$ decomposition, Journal of Mathematical Imaging and Vision, 52 (2015), 345-355.
doi: 10.1007/s10851-015-0568-x. |
[35] |
Z. Lu,
Iterative reweighted minimization methods for $\ell^p$ regularized unconstrained nonlinear programming, Mathematical Programming: Series A and B, 147 (2014), 277-307.
doi: 10.1007/s10107-013-0722-4. |
[36] |
D. Marini and A. Rizzi, A computational approach to color adaptation effects, Image and Vision Computing, 18 (2000), 1005-1014. Google Scholar |
[37] |
W. Ma and S. Osher,
A TV bregman iterative model of retinex theory, Inverse Problems and Imaging, 6 (2012), 697-708.
doi: 10.3934/ipi.2012.6.697. |
[38] |
J. Mccann, Lessons learned from mondrians applied to real images and color gamuts, Proceedings of the IST/SID 7th Color Imaging Conference, 1999, 1–8. Google Scholar |
[39] |
J. Morel, A. Petro and C. Sbert, Fast implementation of color constancy algorithms, Proceedings of SPIE, 7241, 2009. Google Scholar |
[40] |
J. Morel, A. Petro and C. Sbert,
A PDE formalization of retinex theory, IEEE Transactions on Image Processing, 19 (2010), 2825-2837.
doi: 10.1109/TIP.2010.2049239. |
[41] |
V. Morozov, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5280-1. |
[42] |
M. Ng and W. Wang,
A total variation model for retinex, SIAM Journal on Imaging Sciences, 4 (2011), 345-365.
doi: 10.1137/100806588. |
[43] |
P. Ochs, A. Dosovitskiy, T. Brox and T. Pock,
On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision, SIAM Journal on Imaging Sciences, 8 (2015), 331-372.
doi: 10.1137/140971518. |
[44] |
J. Oliveira, J. Dias and M. Figueiredo, Adaptive total variation image deblurring: A majorizationCminimization approach, Signal Processing, 89 (2009), 1683-1693. Google Scholar |
[45] |
H. Pan, Y. Wen and H. Zhu,
A regularization parameter selection model for total variation based image noise removal, Applied Mathematical Modelling, 68 (2019), 353-367.
doi: 10.1016/j.apm.2018.11.032. |
[46] |
E. Provenzi, D. Marini, L. De Carli and A. Rizzi,
Mathematical definition and analysis of the retinex algorithm, Journal of the Optical Society of America A, 22 (2005), 2613-2621.
doi: 10.1364/JOSAA.22.002613. |
[47] |
S. Sabacha and M. Teboulle,
Lagrangian methods for composite optimization, Handbook of Numerical Analysis, 20 (2019), 401-436.
|
[48] |
A. Theljani and K. Chen, A Nash game based variational model for joint image intensity correction and registration to deal with varying illumination, Inverse Problems, 36 (2020), 034002. Google Scholar |
[49] |
Y. Wen and R. Chan,
Using generalized cross validation to select regularization parameter for total variation regularization problems, Inverse Problems and Imaging, 12 (2018), 1103-1120.
doi: 10.3934/ipi.2018046. |
[50] |
W. Wang and C. He,
A variational model with barrier functionals for Retinex, SIAM Journal on Imaging Sciences, 8 (2015), 1955-1980.
doi: 10.1137/15M1006908. |
[51] |
J. Zhang, R. Chen, C. Deng and S. Wang,
Fast linearized augmented method for Euler's elastica model, Numerical Mathematics:Theory, Methods and Applications, 10 (2017), 98-115.
doi: 10.4208/nmtma.2017.m1611. |
[52] |
X. Zhang, Y. Shi, Z. Pang and Y. Zhu,
Fast algorithm for image denoising with different boundary conditions, Journal of the Franklin Institute, 354 (2017), 4595-4614.
doi: 10.1016/j.jfranklin.2017.04.011. |
[53] |
D. Zosso, G. Tran and S. Osher, A unifying retinex model based on non-local differential operators, Computational Imaging XI, 865702, 2013.. Google Scholar |
[54] |
W. Zuo, D. Meng, L. Zhang, X. Feng and D. Zhang, A generalized iterated shrinkage algorithm for non-convex sparse coding, IEEE International Conference on Computer Vision, 2013,217–224. Google Scholar |















PSNR | MSSIM | ||
TVH1 | 30.9323 | 0.9812 | |
Shape1 | L0MS | 31.7962 | 0.9837 |
HOTVL | 18.9389 | 0.9374 | |
ETV | 36.2657 | 0.9967 | |
OUR | 36.8019 | 0.9973 | |
TVH1 | 31.5509 | 0.9514 | |
Shape2 | L0MS | 31.6024 | 0.9463 |
HoTVL1 | 15.6438 | 0.8137 | |
ETV | 36.6105 | 0.9972 | |
OUR | 37.6838 | 0.9975 |
PSNR | MSSIM | ||
TVH1 | 30.9323 | 0.9812 | |
Shape1 | L0MS | 31.7962 | 0.9837 |
HOTVL | 18.9389 | 0.9374 | |
ETV | 36.2657 | 0.9967 | |
OUR | 36.8019 | 0.9973 | |
TVH1 | 31.5509 | 0.9514 | |
Shape2 | L0MS | 31.6024 | 0.9463 |
HoTVL1 | 15.6438 | 0.8137 | |
ETV | 36.6105 | 0.9972 | |
OUR | 37.6838 | 0.9975 |
PSNR | MSSIM | PSNR | MSSIM | PSNR | MSSIM | PSNR | MSSIM | ||
TVH1 | 26.8509 | 0.9436 | 25.3513 | 0.9176 | 24.5925 | 0.8939 | 23.4850 | 0.8642 | |
Test | HoTVL1 | 30.2055 | 0.9515 | 27.7438 | 0.9288 | 26.4877 | 0.9069 | 24.7440 | 0.8774 |
Image1 | L0MS | 29.7718 | 0.9227 | 27.8711 | 0.9088 | 26.1986 | 0.8936 | 24.3809 | 0.8649 |
ETV | 32.6699 | 0.9904 | 31.1178 | 0.9844 | 29.1294 | 0.9767 | 28.4238 | 0.9686 | |
OUR | 33.0513 | 0.9909 | 31.2246 | 0.9846 | 30.1037 | 0.9781 | 28.4666 | 0.9686 | |
TVH1 | 27.4061 | 0.9230 | 26.5258 | 0.8935 | 25.5879 | 0.8661 | 24.0629 | 0.8348 | |
Test | HoTVL1 | 30.0791 | 0.9317 | 29.0495 | 0.9065 | 27.0113 | 0.8795 | 25.1211 | 0.8481 |
Image2 | L0MS | 32.0987 | 0.9246 | 29.0807 | 0.9016 | 27.1344 | 0.8760 | 24.9108 | 0.8382 |
ETV | 33.4363 | 0.9911 | 31.4108 | 0.9842 | 29.8199 | 0.9764 | 28.6496 | 0.9698 | |
OUR | 33.9694 | 0.9916 | 31.4609 | 0.9841 | 29.8766 | 0.9760 | 29.0947 | 0.9703 |
PSNR | MSSIM | PSNR | MSSIM | PSNR | MSSIM | PSNR | MSSIM | ||
TVH1 | 26.8509 | 0.9436 | 25.3513 | 0.9176 | 24.5925 | 0.8939 | 23.4850 | 0.8642 | |
Test | HoTVL1 | 30.2055 | 0.9515 | 27.7438 | 0.9288 | 26.4877 | 0.9069 | 24.7440 | 0.8774 |
Image1 | L0MS | 29.7718 | 0.9227 | 27.8711 | 0.9088 | 26.1986 | 0.8936 | 24.3809 | 0.8649 |
ETV | 32.6699 | 0.9904 | 31.1178 | 0.9844 | 29.1294 | 0.9767 | 28.4238 | 0.9686 | |
OUR | 33.0513 | 0.9909 | 31.2246 | 0.9846 | 30.1037 | 0.9781 | 28.4666 | 0.9686 | |
TVH1 | 27.4061 | 0.9230 | 26.5258 | 0.8935 | 25.5879 | 0.8661 | 24.0629 | 0.8348 | |
Test | HoTVL1 | 30.0791 | 0.9317 | 29.0495 | 0.9065 | 27.0113 | 0.8795 | 25.1211 | 0.8481 |
Image2 | L0MS | 32.0987 | 0.9246 | 29.0807 | 0.9016 | 27.1344 | 0.8760 | 24.9108 | 0.8382 |
ETV | 33.4363 | 0.9911 | 31.4108 | 0.9842 | 29.8199 | 0.9764 | 28.6496 | 0.9698 | |
OUR | 33.9694 | 0.9916 | 31.4609 | 0.9841 | 29.8766 | 0.9760 | 29.0947 | 0.9703 |
PSNR | MSSIM | ||
TVH1 | 31.7514 | 0.8887 | |
Checkboard | L0MS | 33.1312 | 0.9602 |
HOTVL1 | 29.5090 | 0.8725 | |
ETV | 38.7053 | 0.9936 | |
OUR | 39.2105 | 0.9934 | |
TVH1 | 30.7779 | 0.9799 | |
L0MS | 30.0935 | 0.9774 | |
Cube | HOTVL1 | 27.9950 | 0.9692 |
ETV | 29.6898 | 0.9866 | |
OUR | 33.3660 | 0.9896 |
PSNR | MSSIM | ||
TVH1 | 31.7514 | 0.8887 | |
Checkboard | L0MS | 33.1312 | 0.9602 |
HOTVL1 | 29.5090 | 0.8725 | |
ETV | 38.7053 | 0.9936 | |
OUR | 39.2105 | 0.9934 | |
TVH1 | 30.7779 | 0.9799 | |
L0MS | 30.0935 | 0.9774 | |
Cube | HOTVL1 | 27.9950 | 0.9692 |
ETV | 29.6898 | 0.9866 | |
OUR | 33.3660 | 0.9896 |
[1] |
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 |
[2] |
Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265 |
[3] |
Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617 |
[4] |
Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020133 |
[5] |
W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 |
[6] |
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 |
[7] |
Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161 |
[8] |
Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781 |
[9] |
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 |
[10] |
Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 |
[11] |
Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021006 |
[12] |
Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011 |
[13] |
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 |
[14] |
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 |
[15] |
Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1 |
[16] |
Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907 |
[17] |
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 |
[18] |
Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379 |
[19] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[20] |
Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044 |
2019 Impact Factor: 1.373
Tools
Article outline
Figures and Tables
[Back to Top]