
-
Previous Article
Edge detection with mixed noise based on maximum a posteriori approach
- IPI Home
- This Issue
-
Next Article
Boundary determination of electromagnetic and Lamé parameters with corrupted data
An efficient multi-grid method for TV minimization problems
1. | Center for Applied Mathematics, Tianjin University, Tianjin, China |
2. | Center for Mathematical Sciences, Huazhong University of Technology, Wuhan, China |
3. | Department of Mathematics, Hong Kong Bapist University, Kowloon Tong, Hong Kong, China |
We propose an efficient multi-grid domain decomposition method for solving the total variation (TV) minimization problems. Our multi-grid scheme is developed based on the piecewise constant function spanned subspace correction rather than the piecewise linear one in [
References:
[1] |
R. Acar and C. R. Vogel,
Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), 1217-1229.
doi: 10.1088/0266-5611/10/6/003. |
[2] |
S. T. Acton,
Multigrid anisotropic diffusion, IEEE Transactions on Image Processing: A Publication of the IEEE Signal Processing Society, 7 (1998), 280-291.
|
[3] |
X. Bresson and T. F. Chan,
Fast dual minimization of the vectorial total variation norm and applications to color image processing, Inverse Probl. Imaging, 2 (2008), 455-484.
doi: 10.3934/ipi.2008.2.455. |
[4] |
C. Brito-Loeza and K. Chen,
On high-order denoising models and fast algorithms for vector-valued images, IEEE Trans. Image Process., 19 (2010), 1518-1527.
doi: 10.1109/TIP.2010.2042655. |
[5] |
A. Chambolle,
An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89-97.
|
[6] |
A. Chambolle and T. Pock,
A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[7] |
A. Chambolle and T. Pock,
An introduction to continuous optimization for imaging, Acta Numer., 25 (2016), 161-319.
doi: 10.1017/S096249291600009X. |
[8] |
A. Chambolle and T. Pock,
On the ergodic convergence rates of a first-order primal-dual algorithm, Math. Program., 159 (2016), 253-287.
doi: 10.1007/s10107-015-0957-3. |
[9] |
R. H. Chan and K. Chen,
A multilevel algorithm for simultaneously denoising and deblurring images, SIAM J. Sci. Comput., 32 (2010), 1043-1063.
doi: 10.1137/080741410. |
[10] |
T. F. Chan and K. Chen,
On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation, Numer. Algorithms, 41 (2006), 387-411.
doi: 10.1007/s11075-006-9020-z. |
[11] |
T. F. Chan and K. Chen,
An optimization based multilevel algorithm for total variation image denoising, Multiscale Model. Simul., 5 (2006), 615-645.
doi: 10.1137/050644999. |
[12] |
T. F. Chan and J. Shen,
Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math., 62 (2002), 1019-1043.
doi: 10.1137/S0036139900368844. |
[13] |
T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266–277.
doi: 10.1109/83.902291. |
[14] |
H. Chang, X.-C. Tai, L.-L. Wang and D. Yang,
Convergence rate of overlapping domain decomposition methods for the Rudin-Osher-Fatemi model based on a dual formulation, SIAM J. Imaging Sci., 8 (2015), 564-591.
doi: 10.1137/140965016. |
[15] |
R. Chen, J. Huang and X.-C. Cai,
A parallel domain decomposition algorithm for large scale image denoising, Inverse Probl. Imaging, 13 (2019), 1259-1282.
doi: 10.3934/ipi.2019055. |
[16] |
K. Chen and J. Savage,
An accelerated algebraic multigrid algorithm for total-variation denoising, BIT, 47 (2007), 277-296.
doi: 10.1007/s10543-007-0123-2. |
[17] |
K. Chen and X.-C. Tai,
A nonlinear multigrid method for total variation minimization from image restoration, J. Sci. Comput., 33 (2007), 115-138.
doi: 10.1007/s10915-007-9145-9. |
[18] |
Y. Duan, H. Chang and X.-C. Tai,
Convergent non-overlapping domain decomposition methods for variational image segmentation, J. Sci. Comput., 69 (2016), 532-555.
doi: 10.1007/s10915-016-0207-8. |
[19] |
E. Esser, X. Zhang and T. F. Chan,
A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM J. Imaging Sci., 3 (2010), 1015-1046.
doi: 10.1137/09076934X. |
[20] |
D. Goldfarb and W. Yin,
Parametric maximum flow algorithms for fast total variation minimization, SIAM J. Sci. Comput., 31 (2009), 3712-3743.
doi: 10.1137/070706318. |
[21] |
T. Goldstein and S. Osher,
The split Bregman method for ${L}_1$-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.
doi: 10.1137/080725891. |
[22] |
Y. Gu, L.-L. Wang and X.-C. Tai,
A direct approach toward global minimization for multiphase labeling and segmentation problems, IEEE Trans. Image Process., 21 (2012), 2399-2411.
doi: 10.1109/TIP.2011.2182522. |
[23] |
M. Hintermüller and A. Langer,
Non-overlapping domain decomposition methods for dual total variation based image denoising, J. Sci. Comput., 62 (2015), 456-481.
doi: 10.1007/s10915-014-9863-8. |
[24] |
A. Langer and F. Gaspoz,
Overlapping domain decomposition methods for total variation denoising, SIAM J. Numer. Anal., 57 (2019), 1411-1444.
doi: 10.1137/18M1173782. |
[25] |
C.-O. Lee and C. Nam, Primal domain decomposition methods for the total variation minimization, based on dual decomposition, SIAM J. Sci. Comput., 39 (2017), B403–B423.
doi: 10.1137/15M1049919. |
[26] |
C.-O. Lee, C. Nam and J. Park,
Domain decomposition methods using dual conversion for the total variation minimization with $L^1$ fidelity term, J. Sci. Comput., 78 (2019), 951-970.
doi: 10.1007/s10915-018-0791-x. |
[27] |
T. Lu, P. Neittaanmäki and X.-C. Tai,
A parallel splitting up method and its application to Navier-Stokes equations, Appl. Math. Lett., 4 (1991), 25-29.
doi: 10.1016/0893-9659(91)90161-N. |
[28] |
A. Marquina and S. Osher,
Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM J. Sci. Comput., 22 (2000), 387-405.
doi: 10.1137/S1064827599351751. |
[29] |
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. |
[30] |
J. Savage and K. Chen,
An improved and accelerated non-linear multigrid method for total-variation denoising, Int. J. Comput. Math., 82 (2005), 1001-1015.
doi: 10.1080/00207160500069904. |
[31] |
E. Y. Sidky, J. H. Jørgensen and X. Pan,
Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm, Physics in Medicine & Biology, 57 (2012), 3065-3091.
doi: 10.1088/0031-9155/57/10/3065. |
[32] |
E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Physics in Medicine & Biology, 53 (2008), 4777.
doi: 10.1088/0031-9155/53/17/021. |
[33] |
X.-C. Tai and Y. Duan,
Domain decomposition methods with graph cuts algorithms for image segmentation., Int. J. Numer. Anal. Model., 8 (2011), 137-155.
|
[34] |
X.-C. Tai and J. Xu,
Global and uniform convergence of subspace correction methods for some convex optimization problems, Math. Comp., 71 (2002), 105-124.
doi: 10.1090/S0025-5718-01-01311-4. |
[35] |
C. R. Vogel and M. E. Oman,
Iterative methods for total variation denoising, SIAM J. Sci. Comput., 17 (1996), 227-238.
doi: 10.1137/0917016. |
[36] |
C. R. Vogel and M. E. Oman,
Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Trans. Image Process., 7 (1998), 813-824.
doi: 10.1109/83.679423. |
[37] |
Y. Wang, J. Yang, W. Yin and Y. Zhang,
A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1 (2008), 248-272.
doi: 10.1137/080724265. |
[38] |
C. Wu and X.-C. Tai,
Augmented Lagrangian method, dual methods, and split bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), 300-339.
doi: 10.1137/090767558. |
[39] |
J. Xu,
Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), 581-613.
doi: 10.1137/1034116. |
[40] |
J. Xu, H. B. Chang and J. Qin,
Domain decomposition method for image deblurring, J. Comput. Appl. Math., 271 (2014), 401-414.
doi: 10.1016/j.cam.2014.03.030. |
[41] |
J. Xu, X.-C. Tai and L.-L. Wang,
A two-level domain decomposition method for image restoration, Inverse Probl. Imaging, 4 (2010), 523-545.
doi: 10.3934/ipi.2010.4.523. |
show all references
References:
[1] |
R. Acar and C. R. Vogel,
Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), 1217-1229.
doi: 10.1088/0266-5611/10/6/003. |
[2] |
S. T. Acton,
Multigrid anisotropic diffusion, IEEE Transactions on Image Processing: A Publication of the IEEE Signal Processing Society, 7 (1998), 280-291.
|
[3] |
X. Bresson and T. F. Chan,
Fast dual minimization of the vectorial total variation norm and applications to color image processing, Inverse Probl. Imaging, 2 (2008), 455-484.
doi: 10.3934/ipi.2008.2.455. |
[4] |
C. Brito-Loeza and K. Chen,
On high-order denoising models and fast algorithms for vector-valued images, IEEE Trans. Image Process., 19 (2010), 1518-1527.
doi: 10.1109/TIP.2010.2042655. |
[5] |
A. Chambolle,
An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89-97.
|
[6] |
A. Chambolle and T. Pock,
A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[7] |
A. Chambolle and T. Pock,
An introduction to continuous optimization for imaging, Acta Numer., 25 (2016), 161-319.
doi: 10.1017/S096249291600009X. |
[8] |
A. Chambolle and T. Pock,
On the ergodic convergence rates of a first-order primal-dual algorithm, Math. Program., 159 (2016), 253-287.
doi: 10.1007/s10107-015-0957-3. |
[9] |
R. H. Chan and K. Chen,
A multilevel algorithm for simultaneously denoising and deblurring images, SIAM J. Sci. Comput., 32 (2010), 1043-1063.
doi: 10.1137/080741410. |
[10] |
T. F. Chan and K. Chen,
On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation, Numer. Algorithms, 41 (2006), 387-411.
doi: 10.1007/s11075-006-9020-z. |
[11] |
T. F. Chan and K. Chen,
An optimization based multilevel algorithm for total variation image denoising, Multiscale Model. Simul., 5 (2006), 615-645.
doi: 10.1137/050644999. |
[12] |
T. F. Chan and J. Shen,
Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math., 62 (2002), 1019-1043.
doi: 10.1137/S0036139900368844. |
[13] |
T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266–277.
doi: 10.1109/83.902291. |
[14] |
H. Chang, X.-C. Tai, L.-L. Wang and D. Yang,
Convergence rate of overlapping domain decomposition methods for the Rudin-Osher-Fatemi model based on a dual formulation, SIAM J. Imaging Sci., 8 (2015), 564-591.
doi: 10.1137/140965016. |
[15] |
R. Chen, J. Huang and X.-C. Cai,
A parallel domain decomposition algorithm for large scale image denoising, Inverse Probl. Imaging, 13 (2019), 1259-1282.
doi: 10.3934/ipi.2019055. |
[16] |
K. Chen and J. Savage,
An accelerated algebraic multigrid algorithm for total-variation denoising, BIT, 47 (2007), 277-296.
doi: 10.1007/s10543-007-0123-2. |
[17] |
K. Chen and X.-C. Tai,
A nonlinear multigrid method for total variation minimization from image restoration, J. Sci. Comput., 33 (2007), 115-138.
doi: 10.1007/s10915-007-9145-9. |
[18] |
Y. Duan, H. Chang and X.-C. Tai,
Convergent non-overlapping domain decomposition methods for variational image segmentation, J. Sci. Comput., 69 (2016), 532-555.
doi: 10.1007/s10915-016-0207-8. |
[19] |
E. Esser, X. Zhang and T. F. Chan,
A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM J. Imaging Sci., 3 (2010), 1015-1046.
doi: 10.1137/09076934X. |
[20] |
D. Goldfarb and W. Yin,
Parametric maximum flow algorithms for fast total variation minimization, SIAM J. Sci. Comput., 31 (2009), 3712-3743.
doi: 10.1137/070706318. |
[21] |
T. Goldstein and S. Osher,
The split Bregman method for ${L}_1$-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.
doi: 10.1137/080725891. |
[22] |
Y. Gu, L.-L. Wang and X.-C. Tai,
A direct approach toward global minimization for multiphase labeling and segmentation problems, IEEE Trans. Image Process., 21 (2012), 2399-2411.
doi: 10.1109/TIP.2011.2182522. |
[23] |
M. Hintermüller and A. Langer,
Non-overlapping domain decomposition methods for dual total variation based image denoising, J. Sci. Comput., 62 (2015), 456-481.
doi: 10.1007/s10915-014-9863-8. |
[24] |
A. Langer and F. Gaspoz,
Overlapping domain decomposition methods for total variation denoising, SIAM J. Numer. Anal., 57 (2019), 1411-1444.
doi: 10.1137/18M1173782. |
[25] |
C.-O. Lee and C. Nam, Primal domain decomposition methods for the total variation minimization, based on dual decomposition, SIAM J. Sci. Comput., 39 (2017), B403–B423.
doi: 10.1137/15M1049919. |
[26] |
C.-O. Lee, C. Nam and J. Park,
Domain decomposition methods using dual conversion for the total variation minimization with $L^1$ fidelity term, J. Sci. Comput., 78 (2019), 951-970.
doi: 10.1007/s10915-018-0791-x. |
[27] |
T. Lu, P. Neittaanmäki and X.-C. Tai,
A parallel splitting up method and its application to Navier-Stokes equations, Appl. Math. Lett., 4 (1991), 25-29.
doi: 10.1016/0893-9659(91)90161-N. |
[28] |
A. Marquina and S. Osher,
Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM J. Sci. Comput., 22 (2000), 387-405.
doi: 10.1137/S1064827599351751. |
[29] |
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. |
[30] |
J. Savage and K. Chen,
An improved and accelerated non-linear multigrid method for total-variation denoising, Int. J. Comput. Math., 82 (2005), 1001-1015.
doi: 10.1080/00207160500069904. |
[31] |
E. Y. Sidky, J. H. Jørgensen and X. Pan,
Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm, Physics in Medicine & Biology, 57 (2012), 3065-3091.
doi: 10.1088/0031-9155/57/10/3065. |
[32] |
E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Physics in Medicine & Biology, 53 (2008), 4777.
doi: 10.1088/0031-9155/53/17/021. |
[33] |
X.-C. Tai and Y. Duan,
Domain decomposition methods with graph cuts algorithms for image segmentation., Int. J. Numer. Anal. Model., 8 (2011), 137-155.
|
[34] |
X.-C. Tai and J. Xu,
Global and uniform convergence of subspace correction methods for some convex optimization problems, Math. Comp., 71 (2002), 105-124.
doi: 10.1090/S0025-5718-01-01311-4. |
[35] |
C. R. Vogel and M. E. Oman,
Iterative methods for total variation denoising, SIAM J. Sci. Comput., 17 (1996), 227-238.
doi: 10.1137/0917016. |
[36] |
C. R. Vogel and M. E. Oman,
Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Trans. Image Process., 7 (1998), 813-824.
doi: 10.1109/83.679423. |
[37] |
Y. Wang, J. Yang, W. Yin and Y. Zhang,
A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1 (2008), 248-272.
doi: 10.1137/080724265. |
[38] |
C. Wu and X.-C. Tai,
Augmented Lagrangian method, dual methods, and split bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), 300-339.
doi: 10.1137/090767558. |
[39] |
J. Xu,
Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), 581-613.
doi: 10.1137/1034116. |
[40] |
J. Xu, H. B. Chang and J. Qin,
Domain decomposition method for image deblurring, J. Comput. Appl. Math., 271 (2014), 401-414.
doi: 10.1016/j.cam.2014.03.030. |
[41] |
J. Xu, X.-C. Tai and L.-L. Wang,
A two-level domain decomposition method for image restoration, Inverse Probl. Imaging, 4 (2010), 523-545.
doi: 10.3934/ipi.2010.4.523. |









10 | 15 | 20 | ||||||||
MML | Iter | Energy | CPU(s) | Iter | Energy | CPU(s) | Iter | Energy | CPU(s) | |
#1 | 1 | 54 | 2.32E7 | 1.10 | 86 | 3.12E7 | 2.05 | 128 | 3.80E7 | 3.12 |
2 | 23 | 2.31E7 | 1.21 | 50 | 3.05E7 | 2.54 | 80 | 3.62E7 | 3.49 | |
3 | 24 | 2.29E7 | 1.48 | 40 | 3.03E7 | 2.46 | 31 | 3.53E7 | 3.67 | |
4 | 22 | 2.29E7 | 1.53 | 32 | 3.03E7 | 3.35 | 29 | 3.52E7 | 3.21 | |
5 | 24 | 2.29E7 | 1.85 | 32 | 3.03E7 | 4.88 | 29 | 3.52E7 | 4.51 | |
6 | 22 | 2.29E7 | 2.08 | 23 | 3.03E7 | 5.73 | 29 | 3.52E7 | 5.71 | |
#3 | 1 | 35 | 8.60E7 | 5.64 | 82 | 1.14E8 | 11.32 | 139 | 1.38E8 | 19.35 |
2 | 18 | 8.57E7 | 5.87 | 36 | 1.11E8 | 10.32 | 65 | 1.29E8 | 16.43 | |
3 | 18 | 8.56E7 | 6.18 | 44 | 1.07E8 | 10.65 | 32 | 1.25E8 | 14.47 | |
4 | 18 | 8.56E7 | 6.79 | 31 | 1.07E8 | 9.31 | 32 | 1.24E8 | 11.28 | |
5 | 18 | 8.56E7 | 6.77 | 31 | 1.07E8 | 9.78 | 32 | 1.24E8 | 12.02 | |
6 | 18 | 8.56E7 | 7.25 | 31 | 1.07E8 | 10.17 | 32 | 1.24E8 | 12.56 | |
#5 | 1 | 49 | 3.54E8 | 27.32 | 71 | 4.72E8 | 42.35 | 198 | 5.63E8 | 79.56 |
2 | 31 | 3.54E8 | 31.47 | 53 | 4.60E8 | 50.61 | 70 | 5.34E8 | 71.35 | |
3 | 28 | 3.53E8 | 32.06 | 43 | 4.59E8 | 49.35 | 54 | 5.24E8 | 60.21 | |
4 | 21 | 3.53E8 | 33.37 | 36 | 4.59E8 | 48.35 | 52 | 5.22E8 | 52.13 | |
5 | 21 | 3.53E8 | 34.25 | 36 | 4.59E8 | 50.22 | 52 | 5.22E8 | 66.32 | |
6 | 21 | 3.53E8 | 38.24 | 36 | 4.59E8 | 51.78 | 52 | 5.22E8 | 75.21 | |
#7 | 1 | 57 | 1.38E9 | 121.32 | 120 | 1.82E9 | 237.21 | 189 | 2.21E9 | 369.75 |
2 | 31 | 1.38E9 | 117.24 | 61 | 1.76E9 | 225.76 | 102 | 2.05E9 | 366.14 | |
3 | 26 | 1.37E9 | 115.78 | 44 | 1.76E9 | 193.35 | 65 | 1.99E9 | 299.78 | |
4 | 26 | 1.37E9 | 122.73 | 39 | 1.75E9 | 192.49 | 56 | 1.94E9 | 280.32 | |
5 | 26 | 1.37E9 | 126.25 | 39 | 1.75E9 | 208.15 | 53 | 1.94E9 | 288.25 | |
6 | 26 | 1.37E9 | 139.35 | 39 | 1.75E9 | 229.29 | 53 | 1.94E9 | 282.12 | |
MMC | Iter | Energy | CPU(s) | Iter | Energy | CPU(s) | Iter | Energy | CPU(s) | |
#1 | 1 | 33 | 2.31E7 | 0.85 | 72 | 3.12E7 | 1.72 | 112 | 3.79E7 | 2.77 |
2 | 21 | 2.31E7 | 0.69 | 41 | 3.05E7 | 1.22 | 37 | 3.63E7 | 1.21 | |
3 | 23 | 2.31E7 | 0.83 | 33 | 3.03E7 | 1.15 | 46 | 3.56E7 | 1.54 | |
4 | 21 | 2.31E7 | 0.88 | 33 | 3.03E7 | 1.21 | 34 | 3.54E7 | 1.20 | |
5 | 21 | 2.31E7 | 0.83 | 29 | 3.03E7 | 1.31 | 33 | 3.54E7 | 1.28 | |
6 | 20 | 2.31E7 | 0.82 | 23 | 3.03E7 | 1.95 | 33 | 3.54E7 | 1.55 | |
#3 | 1 | 39 | 8.59E7 | 4.96 | 80 | 1.14E8 | 9.65 | 134 | 1.39E8 | 15.84 |
2 | 28 | 8.58E7 | 4.07 | 36 | 1.11E8 | 5.13 | 54 | 1.31E8 | 7.54 | |
3 | 27 | 8.58E7 | 4.25 | 42 | 1.07E8 | 6.41 | 46 | 1.26E8 | 6.99 | |
4 | 27 | 8.58E7 | 4.27 | 37 | 1.07E8 | 5.97 | 34 | 1.24E8 | 5.75 | |
5 | 27 | 8.58E7 | 4.73 | 33 | 1.07E8 | 5.68 | 38 | 1.24E8 | 6.56 | |
6 | 26 | 8.58E7 | 4.56 | 30 | 1.07E8 | 5.93 | 32 | 1.24E8 | 6.73 | |
#5 | 1 | 51 | 3.54E8 | 23.32 | 99 | 4.71E8 | 44.23 | 161 | 5.64E8 | 74.36 |
2 | 33 | 3.54E8 | 20.67 | 49 | 4.60E8 | 29.95 | 66 | 5.37E8 | 42.74 | |
3 | 31 | 3.53E8 | 23.07 | 46 | 4.59E8 | 31.26 | 47 | 5.26E8 | 34.33 | |
4 | 30 | 3.53E8 | 22.07 | 40 | 4.59E8 | 28.98 | 48 | 5.23E8 | 34.28 | |
5 | 28 | 3.53E8 | 25.88 | 42 | 4.59E8 | 31.20 | 43 | 5.23E8 | 37.25 | |
6 | 29 | 3.53E8 | 30.15 | 39 | 4.59E8 | 36.59 | 44 | 5.23E8 | 42.67 | |
#7 | 1 | 57 | 1.38E9 | 121.96 | 120 | 1.82E9 | 237.88 | 189 | 2.21E9 | 369.02 |
2 | 36 | 1.38E9 | 100.06 | 56 | 1.76E9 | 124.64 | 84 | 2.05E9 | 183.47 | |
3 | 34 | 1.38E9 | 86.18 | 44 | 1.75E9 | 110.30 | 62 | 1.99E9 | 150.35 | |
4 | 34 | 1.36E9 | 81.35 | 43 | 1.74E9 | 105.42 | 43 | 1.95E9 | 114.78 | |
5 | 37 | 1.36E9 | 90.06 | 43 | 1.74E9 | 109.73 | 39 | 1.95E9 | 117.75 | |
6 | 36 | 1.36E9 | 98.81 | 43 | 1.74E9 | 115.06 | 40 | 1.95E9 | 123.42 |
10 | 15 | 20 | ||||||||
MML | Iter | Energy | CPU(s) | Iter | Energy | CPU(s) | Iter | Energy | CPU(s) | |
#1 | 1 | 54 | 2.32E7 | 1.10 | 86 | 3.12E7 | 2.05 | 128 | 3.80E7 | 3.12 |
2 | 23 | 2.31E7 | 1.21 | 50 | 3.05E7 | 2.54 | 80 | 3.62E7 | 3.49 | |
3 | 24 | 2.29E7 | 1.48 | 40 | 3.03E7 | 2.46 | 31 | 3.53E7 | 3.67 | |
4 | 22 | 2.29E7 | 1.53 | 32 | 3.03E7 | 3.35 | 29 | 3.52E7 | 3.21 | |
5 | 24 | 2.29E7 | 1.85 | 32 | 3.03E7 | 4.88 | 29 | 3.52E7 | 4.51 | |
6 | 22 | 2.29E7 | 2.08 | 23 | 3.03E7 | 5.73 | 29 | 3.52E7 | 5.71 | |
#3 | 1 | 35 | 8.60E7 | 5.64 | 82 | 1.14E8 | 11.32 | 139 | 1.38E8 | 19.35 |
2 | 18 | 8.57E7 | 5.87 | 36 | 1.11E8 | 10.32 | 65 | 1.29E8 | 16.43 | |
3 | 18 | 8.56E7 | 6.18 | 44 | 1.07E8 | 10.65 | 32 | 1.25E8 | 14.47 | |
4 | 18 | 8.56E7 | 6.79 | 31 | 1.07E8 | 9.31 | 32 | 1.24E8 | 11.28 | |
5 | 18 | 8.56E7 | 6.77 | 31 | 1.07E8 | 9.78 | 32 | 1.24E8 | 12.02 | |
6 | 18 | 8.56E7 | 7.25 | 31 | 1.07E8 | 10.17 | 32 | 1.24E8 | 12.56 | |
#5 | 1 | 49 | 3.54E8 | 27.32 | 71 | 4.72E8 | 42.35 | 198 | 5.63E8 | 79.56 |
2 | 31 | 3.54E8 | 31.47 | 53 | 4.60E8 | 50.61 | 70 | 5.34E8 | 71.35 | |
3 | 28 | 3.53E8 | 32.06 | 43 | 4.59E8 | 49.35 | 54 | 5.24E8 | 60.21 | |
4 | 21 | 3.53E8 | 33.37 | 36 | 4.59E8 | 48.35 | 52 | 5.22E8 | 52.13 | |
5 | 21 | 3.53E8 | 34.25 | 36 | 4.59E8 | 50.22 | 52 | 5.22E8 | 66.32 | |
6 | 21 | 3.53E8 | 38.24 | 36 | 4.59E8 | 51.78 | 52 | 5.22E8 | 75.21 | |
#7 | 1 | 57 | 1.38E9 | 121.32 | 120 | 1.82E9 | 237.21 | 189 | 2.21E9 | 369.75 |
2 | 31 | 1.38E9 | 117.24 | 61 | 1.76E9 | 225.76 | 102 | 2.05E9 | 366.14 | |
3 | 26 | 1.37E9 | 115.78 | 44 | 1.76E9 | 193.35 | 65 | 1.99E9 | 299.78 | |
4 | 26 | 1.37E9 | 122.73 | 39 | 1.75E9 | 192.49 | 56 | 1.94E9 | 280.32 | |
5 | 26 | 1.37E9 | 126.25 | 39 | 1.75E9 | 208.15 | 53 | 1.94E9 | 288.25 | |
6 | 26 | 1.37E9 | 139.35 | 39 | 1.75E9 | 229.29 | 53 | 1.94E9 | 282.12 | |
MMC | Iter | Energy | CPU(s) | Iter | Energy | CPU(s) | Iter | Energy | CPU(s) | |
#1 | 1 | 33 | 2.31E7 | 0.85 | 72 | 3.12E7 | 1.72 | 112 | 3.79E7 | 2.77 |
2 | 21 | 2.31E7 | 0.69 | 41 | 3.05E7 | 1.22 | 37 | 3.63E7 | 1.21 | |
3 | 23 | 2.31E7 | 0.83 | 33 | 3.03E7 | 1.15 | 46 | 3.56E7 | 1.54 | |
4 | 21 | 2.31E7 | 0.88 | 33 | 3.03E7 | 1.21 | 34 | 3.54E7 | 1.20 | |
5 | 21 | 2.31E7 | 0.83 | 29 | 3.03E7 | 1.31 | 33 | 3.54E7 | 1.28 | |
6 | 20 | 2.31E7 | 0.82 | 23 | 3.03E7 | 1.95 | 33 | 3.54E7 | 1.55 | |
#3 | 1 | 39 | 8.59E7 | 4.96 | 80 | 1.14E8 | 9.65 | 134 | 1.39E8 | 15.84 |
2 | 28 | 8.58E7 | 4.07 | 36 | 1.11E8 | 5.13 | 54 | 1.31E8 | 7.54 | |
3 | 27 | 8.58E7 | 4.25 | 42 | 1.07E8 | 6.41 | 46 | 1.26E8 | 6.99 | |
4 | 27 | 8.58E7 | 4.27 | 37 | 1.07E8 | 5.97 | 34 | 1.24E8 | 5.75 | |
5 | 27 | 8.58E7 | 4.73 | 33 | 1.07E8 | 5.68 | 38 | 1.24E8 | 6.56 | |
6 | 26 | 8.58E7 | 4.56 | 30 | 1.07E8 | 5.93 | 32 | 1.24E8 | 6.73 | |
#5 | 1 | 51 | 3.54E8 | 23.32 | 99 | 4.71E8 | 44.23 | 161 | 5.64E8 | 74.36 |
2 | 33 | 3.54E8 | 20.67 | 49 | 4.60E8 | 29.95 | 66 | 5.37E8 | 42.74 | |
3 | 31 | 3.53E8 | 23.07 | 46 | 4.59E8 | 31.26 | 47 | 5.26E8 | 34.33 | |
4 | 30 | 3.53E8 | 22.07 | 40 | 4.59E8 | 28.98 | 48 | 5.23E8 | 34.28 | |
5 | 28 | 3.53E8 | 25.88 | 42 | 4.59E8 | 31.20 | 43 | 5.23E8 | 37.25 | |
6 | 29 | 3.53E8 | 30.15 | 39 | 4.59E8 | 36.59 | 44 | 5.23E8 | 42.67 | |
#7 | 1 | 57 | 1.38E9 | 121.96 | 120 | 1.82E9 | 237.88 | 189 | 2.21E9 | 369.02 |
2 | 36 | 1.38E9 | 100.06 | 56 | 1.76E9 | 124.64 | 84 | 2.05E9 | 183.47 | |
3 | 34 | 1.38E9 | 86.18 | 44 | 1.75E9 | 110.30 | 62 | 1.99E9 | 150.35 | |
4 | 34 | 1.36E9 | 81.35 | 43 | 1.74E9 | 105.42 | 43 | 1.95E9 | 114.78 | |
5 | 37 | 1.36E9 | 90.06 | 43 | 1.74E9 | 109.73 | 39 | 1.95E9 | 117.75 | |
6 | 36 | 1.36E9 | 98.81 | 43 | 1.74E9 | 115.06 | 40 | 1.95E9 | 123.42 |
MMC | MML | Dual | P-D | MMC | MML | Dual | P-D | MMC | MML | Dual | P-D | ||
PSNR | #1 | 28.77 | 28.77 | 28.71 | 28.70 | 29.26 | 29.27 | 29.23 | 29.27 | 29.14 | 29.14 | 29.13 | 29.17 |
#2 | 28.43 | 28.42 | 28.40 | 28.41 | 28.64 | 28.64 | 28.60 | 28.62 | 27.61 | 27.59 | 27.55 | 27.58 | |
#3 | 29.73 | 29.73 | 29.70 | 29.73 | 30.79 | 30.77 | 30.72 | 30.75 | 30.60 | 30.63 | 30.58 | 30.60 | |
#4 | 29.84 | 29.86 | 29.83 | 29.83 | 31.14 | 31.13 | 31.09 | 31.11 | 30.28 | 30.32 | 30.28 | 30.30 | |
#5 | 29.18 | 29.16 | 29.15 | 29.15 | 29.64 | 29.64 | 29.63 | 29.63 | 29.35 | 29.34 | 29.30 | 29.33 | |
#6 | 27.81 | 27.82 | 27.80 | 27.79 | 27.95 | 27.95 | 27.89 | 27.94 | 26.58 | 26.58 | 26.55 | 26.55 | |
#7 | 30.70 | 30.65 | 30.60 | 30.68 | 32.45 | 32.44 | 32.43 | 32.40 | 32.33 | 32.33 | 32.26 | 32.38 | |
#8 | 30.43 | 30.40 | 30.39 | 30.41 | 32.28 | 32.25 | 32.23 | 32.27 | 31.27 | 31.26 | 31.20 | 31.25 | |
SSIM | #1 | 0.7786 | 0.7771 | 0.7735 | 0.7757 | 0.8511 | 0.8521 | 0.8515 | 0.8523 | 0.8492 | 0.8499 | 0.8491 | 0.8501 |
#2 | 0.7473 | 0.7451 | 0.7376 | 0.7391 | 0.8303 | 0.8279 | 0.8299 | 0.8255 | 0.8243 | 0.8216 | 0.8255 | 0.8299 | |
#3 | 0.7342 | 0.7277 | 0.7225 | 0.7245 | 0.8256 | 0.8221 | 0.8242 | 0.8253 | 0.8195 | 0.8207 | 0.8217 | 0.8228 | |
#4 | 0.7544 | 0.7431 | 0.7419 | 0.7433 | 0.8548 | 0.8562 | 0.854 | 0.8537 | 0.8518 | 0.8548 | 0.8520 | 0.8521 | |
#5 | 0.7287 | 0.7298 | 0.7267 | 0.7294 | 0.7649 | 0.7631 | 0.7622 | 0.7641 | 0.7381 | 0.7394 | 0.7388 | 0.7395 | |
#6 | 0.7953 | 0.7909 | 0.7943 | 0.7952 | 0.8031 | 0.8131 | 0.8124 | 0.8122 | 0.7883 | 0.7894 | 0.7788 | 0.7793 | |
#7 | 0.7447 | 0.7412 | 0.7368 | 0.7342 | 0.8865 | 0.8877 | 0.8788 | 0.8791 | 0.8725 | 0.8717 | 0.8736 | 0.8744 | |
#8 | 0.7386 | 0.7252 | 0.7237 | 0.7241 | 0.9203 | 0.9107 | 0.9154 | 0.9165 | 0.9154 | 0.9199 | 0.9198 | 0.9192 | |
CPU(s) | #1 | 0.88 | 1.53 | 2.41 | 1.40 | 1.21 | 3.35 | 4.49 | 2.15 | 1.20 | 3.61 | 7.61 | 2.71 |
#2 | 1.00 | 1.64 | 1.56 | 1.72 | 1.17 | 2.01 | 2.81 | 1.97 | 1.18 | 3.11 | 9.01 | 2.21 | |
#3 | 4.27 | 6.94 | 8.26 | 5.62 | 5.97 | 9.83 | 14.42 | 8.60 | 5.81 | 11.28 | 20.67 | 13.94 | |
#4 | 4.22 | 6.81 | 8.56 | 6.13 | 5.13 | 9.13 | 15.01 | 8.11 | 5.22 | 12.10 | 21.35 | 11.65 | |
#5 | 22.68 | 33.19 | 30.12 | 19.73 | 28.68 | 48.35 | 54.08 | 29.93 | 34.28 | 52.13 | 63.59 | 41.57 | |
#6 | 18.33 | 23.19 | 31.01 | 20.21 | 26.78 | 50.14 | 51.35 | 28.86 | 28.48 | 56.13 | 66.01 | 39.73 | |
#7 | 88.80 | 122.73 | 184.93 | 113.37 | 106.43 | 192.49 | 463.12 | 161.99 | 115.83 | 280.32 | 560.32 | 188.35 | |
#8 | 116.35 | 133.84 | 190.21 | 121.61 | 123.28 | 183.17 | 454.35 | 159.73 | 134.78 | 244.35 | 555.38 | 198.46 | |
Energy | #1 | 2.31E7 | 2.29E7 | 2.31E7 | 2.31E7 | 3.03E7 | 3.01E7 | 3.03E7 | 3.03E7 | 3.52E7 | 3.52E7 | 3.53E7 | 3.53E7 |
#2 | 2.44E7 | 2.43E7 | 2.43E7 | 2.43E7 | 3.23E7 | 3.21E7 | 3.23E7 | 3.23E7 | 3.70E7 | 3.71E7 | 3.71E7 | 3.70E7 | |
#3 | 1.28E8 | 1.29E8 | 1.29E8 | 1.28E8 | 2.77E8 | 2.78E8 | 2.78E8 | 2.78E8 | 4.66E8 | 4.67E8 | 4.67E8 | 4.67E8 | |
#4 | 8.39E7 | 8.37E7 | 8.39E7 | 8.39E7 | 1.08E8 | 1.09E8 | 1.09E8 | 1.09E8 | 1.22E8 | 1.23E8 | 1.22E8 | 1.23E8 | |
#5 | 3.53E8 | 3.51E8 | 3.53E8 | 3.53E8 | 4.59E8 | 4.57E8 | 4.57E8 | 4.57E8 | 5.23E8 | 5.21E8 | 5.22E8 | 5.22E8 | |
#6 | 3.71E8 | 3.72E8 | 3.72E8 | 3.72E8 | 4.71E8 | 4.72E8 | 4.71E8 | 4.71E8 | 5.97E8 | 5.97E8 | 5.98E8 | 5.98E8 | |
#7 | 1.37E9 | 1.36E9 | 1.37E9 | 1.36E9 | 1.75E9 | 1.75E9 | 1.75E9 | 1.75E9 | 1.95E9 | 1.94E9 | 1.95E9 | 1.95E9 | |
#8 | 1.40E9 | 1.42E9 | 1.40E9 | 1.41E9 | 1.81E9 | 1.80E9 | 1.81E9 | 1.80E9 | 1.97E9 | 1.98E9 | 1.98E9 | 1.97E9 |
MMC | MML | Dual | P-D | MMC | MML | Dual | P-D | MMC | MML | Dual | P-D | ||
PSNR | #1 | 28.77 | 28.77 | 28.71 | 28.70 | 29.26 | 29.27 | 29.23 | 29.27 | 29.14 | 29.14 | 29.13 | 29.17 |
#2 | 28.43 | 28.42 | 28.40 | 28.41 | 28.64 | 28.64 | 28.60 | 28.62 | 27.61 | 27.59 | 27.55 | 27.58 | |
#3 | 29.73 | 29.73 | 29.70 | 29.73 | 30.79 | 30.77 | 30.72 | 30.75 | 30.60 | 30.63 | 30.58 | 30.60 | |
#4 | 29.84 | 29.86 | 29.83 | 29.83 | 31.14 | 31.13 | 31.09 | 31.11 | 30.28 | 30.32 | 30.28 | 30.30 | |
#5 | 29.18 | 29.16 | 29.15 | 29.15 | 29.64 | 29.64 | 29.63 | 29.63 | 29.35 | 29.34 | 29.30 | 29.33 | |
#6 | 27.81 | 27.82 | 27.80 | 27.79 | 27.95 | 27.95 | 27.89 | 27.94 | 26.58 | 26.58 | 26.55 | 26.55 | |
#7 | 30.70 | 30.65 | 30.60 | 30.68 | 32.45 | 32.44 | 32.43 | 32.40 | 32.33 | 32.33 | 32.26 | 32.38 | |
#8 | 30.43 | 30.40 | 30.39 | 30.41 | 32.28 | 32.25 | 32.23 | 32.27 | 31.27 | 31.26 | 31.20 | 31.25 | |
SSIM | #1 | 0.7786 | 0.7771 | 0.7735 | 0.7757 | 0.8511 | 0.8521 | 0.8515 | 0.8523 | 0.8492 | 0.8499 | 0.8491 | 0.8501 |
#2 | 0.7473 | 0.7451 | 0.7376 | 0.7391 | 0.8303 | 0.8279 | 0.8299 | 0.8255 | 0.8243 | 0.8216 | 0.8255 | 0.8299 | |
#3 | 0.7342 | 0.7277 | 0.7225 | 0.7245 | 0.8256 | 0.8221 | 0.8242 | 0.8253 | 0.8195 | 0.8207 | 0.8217 | 0.8228 | |
#4 | 0.7544 | 0.7431 | 0.7419 | 0.7433 | 0.8548 | 0.8562 | 0.854 | 0.8537 | 0.8518 | 0.8548 | 0.8520 | 0.8521 | |
#5 | 0.7287 | 0.7298 | 0.7267 | 0.7294 | 0.7649 | 0.7631 | 0.7622 | 0.7641 | 0.7381 | 0.7394 | 0.7388 | 0.7395 | |
#6 | 0.7953 | 0.7909 | 0.7943 | 0.7952 | 0.8031 | 0.8131 | 0.8124 | 0.8122 | 0.7883 | 0.7894 | 0.7788 | 0.7793 | |
#7 | 0.7447 | 0.7412 | 0.7368 | 0.7342 | 0.8865 | 0.8877 | 0.8788 | 0.8791 | 0.8725 | 0.8717 | 0.8736 | 0.8744 | |
#8 | 0.7386 | 0.7252 | 0.7237 | 0.7241 | 0.9203 | 0.9107 | 0.9154 | 0.9165 | 0.9154 | 0.9199 | 0.9198 | 0.9192 | |
CPU(s) | #1 | 0.88 | 1.53 | 2.41 | 1.40 | 1.21 | 3.35 | 4.49 | 2.15 | 1.20 | 3.61 | 7.61 | 2.71 |
#2 | 1.00 | 1.64 | 1.56 | 1.72 | 1.17 | 2.01 | 2.81 | 1.97 | 1.18 | 3.11 | 9.01 | 2.21 | |
#3 | 4.27 | 6.94 | 8.26 | 5.62 | 5.97 | 9.83 | 14.42 | 8.60 | 5.81 | 11.28 | 20.67 | 13.94 | |
#4 | 4.22 | 6.81 | 8.56 | 6.13 | 5.13 | 9.13 | 15.01 | 8.11 | 5.22 | 12.10 | 21.35 | 11.65 | |
#5 | 22.68 | 33.19 | 30.12 | 19.73 | 28.68 | 48.35 | 54.08 | 29.93 | 34.28 | 52.13 | 63.59 | 41.57 | |
#6 | 18.33 | 23.19 | 31.01 | 20.21 | 26.78 | 50.14 | 51.35 | 28.86 | 28.48 | 56.13 | 66.01 | 39.73 | |
#7 | 88.80 | 122.73 | 184.93 | 113.37 | 106.43 | 192.49 | 463.12 | 161.99 | 115.83 | 280.32 | 560.32 | 188.35 | |
#8 | 116.35 | 133.84 | 190.21 | 121.61 | 123.28 | 183.17 | 454.35 | 159.73 | 134.78 | 244.35 | 555.38 | 198.46 | |
Energy | #1 | 2.31E7 | 2.29E7 | 2.31E7 | 2.31E7 | 3.03E7 | 3.01E7 | 3.03E7 | 3.03E7 | 3.52E7 | 3.52E7 | 3.53E7 | 3.53E7 |
#2 | 2.44E7 | 2.43E7 | 2.43E7 | 2.43E7 | 3.23E7 | 3.21E7 | 3.23E7 | 3.23E7 | 3.70E7 | 3.71E7 | 3.71E7 | 3.70E7 | |
#3 | 1.28E8 | 1.29E8 | 1.29E8 | 1.28E8 | 2.77E8 | 2.78E8 | 2.78E8 | 2.78E8 | 4.66E8 | 4.67E8 | 4.67E8 | 4.67E8 | |
#4 | 8.39E7 | 8.37E7 | 8.39E7 | 8.39E7 | 1.08E8 | 1.09E8 | 1.09E8 | 1.09E8 | 1.22E8 | 1.23E8 | 1.22E8 | 1.23E8 | |
#5 | 3.53E8 | 3.51E8 | 3.53E8 | 3.53E8 | 4.59E8 | 4.57E8 | 4.57E8 | 4.57E8 | 5.23E8 | 5.21E8 | 5.22E8 | 5.22E8 | |
#6 | 3.71E8 | 3.72E8 | 3.72E8 | 3.72E8 | 4.71E8 | 4.72E8 | 4.71E8 | 4.71E8 | 5.97E8 | 5.97E8 | 5.98E8 | 5.98E8 | |
#7 | 1.37E9 | 1.36E9 | 1.37E9 | 1.36E9 | 1.75E9 | 1.75E9 | 1.75E9 | 1.75E9 | 1.95E9 | 1.94E9 | 1.95E9 | 1.95E9 | |
#8 | 1.40E9 | 1.42E9 | 1.40E9 | 1.41E9 | 1.81E9 | 1.80E9 | 1.81E9 | 1.80E9 | 1.97E9 | 1.98E9 | 1.98E9 | 1.97E9 |
MMC | MML | Dual | P-D | MMC | MML | Dual | P-D | MMC | MML | Dual | P-D | ||
PSNR | #1 | 29.26 | 29.27 | 29.23 | 29.27 | 26.73 | 26.73 | 26.73 | 26.73 | 25.04 | 25.04 | 25.04 | 25.04 |
#2 | 28.64 | 28.64 | 28.60 | 28.62 | 26.49 | 26.44 | 26.40 | 26.42 | 25.01 | 24.95 | 24.96 | 24.96 | |
#3 | 30.79 | 30.77 | 30.72 | 30.75 | 28.78 | 28.78 | 28.78 | 28.78 | 27.25 | 27.21 | 27.21 | 27.21 | |
#4 | 31.14 | 31.13 | 31.09 | 31.11 | 29.13 | 29.11 | 29.11 | 29.12 | 27.21 | 27.21 | 27.21 | 27.21 | |
#5 | 29.64 | 29.64 | 29.63 | 29.63 | 28.73 | 27.75 | 27.75 | 27.74 | 26.40 | 26.38 | 26.39 | 26.40 | |
#6 | 27.95 | 27.95 | 27.89 | 27.94 | 25.44 | 25.45 | 25.42 | 24.13 | 23.93 | 23.94 | 23.92 | 23.94 | |
#7 | 32.45 | 32.44 | 32.43 | 32.40 | 29.92 | 29.91 | 29.90 | 29.91 | 28.20 | 28.20 | 28.20 | 28.20 | |
#8 | 32.28 | 32.25 | 32.23 | 32.27 | 29.95 | 29.93 | 29.85 | 29.87 | 27.52 | 27.52 | 27.51 | 27.51 | |
SSIM | #1 | 0.8511 | 0.8521 | 0.8515 | 0.8523 | 0.8051 | 0.8048 | 0.8046 | 0.8084 | 0.7561 | 0.7563 | 0.7601 | 0.7623 |
#2 | 0.8303 | 0.8279 | 0.8299 | 0.8255 | 0.7782 | 0.7755 | 0.7749 | 0.7811 | 0.7499 | 0.7476 | 0.7377 | 0.7492 | |
#3 | 0.8256 | 0.8221 | 0.8242 | 0.8253 | 0.7842 | 0.7827 | 0.7841 | 0.7851 | 0.7555 | 0.7519 | 0.7549 | 0.7537 | |
#4 | 0.8548 | 0.8562 | 0.8542 | 0.8537 | 0.8046 | 0.8011 | 0.8044 | 0.8012 | 0.7717 | 0.7782 | 0.7739 | 0.7717 | |
#5 | 0.7649 | 0.7631 | 0.7622 | 0.7641 | 0.7162 | 0.7144 | 0.7088 | 0.7101 | 0.6772 | 0.6766 | 0.6688 | 0.6768 | |
#6 | 0.8031 | 0.8131 | 0.8124 | 0.8122 | 0.7311 | 0.7353 | 0.7344 | 0.7349 | 0.6601 | 0.6721 | 0.6687 | 0.6699 | |
#7 | 0.8865 | 0.8877 | 0.8788 | 0.8791 | 0.8478 | 0.8516 | 0.8496 | 0.8502 | 0.8224 | 0.8236 | 0.8235 | 0.8247 | |
#8 | 0.9203 | 0.9107 | 0.9154 | 0.9165 | 0.9002 | 0.8997 | 0.8981 | 0.8994 | 0.8811 | 0.8868 | 0.8714 | 0.8726 | |
CPU(s) | #1 | 1.21 | 3.35 | 4.49 | 2.15 | 1.62 | 3.88 | 4.35 | 3.99 | 2.55 | 5.95 | 5.69 | 5.81 |
#2 | 1.17 | 2.01 | 2.81 | 1.97 | 1.79 | 3.61 | 7.13 | 4.23 | 3.05 | 11.78 | 7.13 | 8.68 | |
#3 | 5.97 | 9.83 | 14.42 | 8.60 | 6.93 | 16.06 | 17.14 | 11.86 | 9.88 | 37.98 | 17.19 | 15.06 | |
#4 | 5.13 | 9.13 | 15.01 | 8.11 | 7.68 | 15.76 | 19.06 | 8.99 | 9.85 | 20.74 | 18.99 | 15.04 | |
#5 | 28.68 | 48.35 | 54.08 | 29.93 | 35.66 | 76.67 | 64.43 | 49.30 | 48.53 | 190.87 | 78.30 | 56.35 | |
#6 | 26.78 | 50.14 | 51.35 | 28.86 | 38.63 | 79.12 | 78.84 | 58.84 | 50.25 | 180.62 | 64.30 | 55.44 | |
#7 | 106.43 | 192.49 | 463.12 | 161.99 | 161.37 | 334.45 | 452.12 | 188.14 | 186.92 | 491.83 | 443.35 | 269.35 | |
#8 | 123.28 | 183.17 | 454.35 | 159.73 | 164.28 | 303.28 | 443.18 | 191.18 | 208.15 | 482.78 | 482.06 | 275.05 | |
Energy | #1 | 3.03E7 | 3.01E7 | 3.03E7 | 3.03E7 | 6.56E7 | 6.57E7 | 6.57E7 | 6.56E7 | 1.18E8 | 1.17E8 | 1.18E8 | 1.17E8 |
#2 | 3.23E7 | 3.21E7 | 3.23E7 | 3.23E7 | 6.84E7 | 6.85E7 | 6.85E7 | 6.84E7 | 1.15E8 | 1.16E8 | 1.16E8 | 1.15E8 | |
#3 | 2.77E8 | 2.78E8 | 2.78E8 | 2.78E8 | 2.87E8 | 2.87E8 | 2.86E8 | 2.86E8 | 4.35E8 | 4.36E8 | 4.36E8 | 4.35E8 | |
#4 | 1.08E8 | 1.09E8 | 1.09E8 | 1.09E8 | 2.42E8 | 2.43E8 | 2.43E8 | 2.43E8 | 4.30E8 | 4.29E8 | 4.31E8 | 4.31E8 | |
#5 | 4.59E8 | 4.57E8 | 4.57E8 | 4.57E8 | 1.01E9 | 1.02E9 | 1.02E9 | 1.01E9 | 1.77E9 | 1.78E9 | 1.78E9 | 1.77E9 | |
#6 | 4.72E8 | 4.71E8 | 4.71E8 | 4.71E8 | 1.08E9 | 1.09E9 | 1.09E9 | 1.08E9 | 1.87E9 | 1.88E9 | 1.88E9 | 1.87E9 | |
#7 | 1.75E9 | 1.75E9 | 1.75E9 | 1.75E9 | 3.94E9 | 3.93E9 | 3.93E9 | 3.93E9 | 6.95E9 | 6.94E9 | 6.95E9 | 6.94E9 | |
#8 | 1.81E9 | 1.80E9 | 1.81E9 | 1.80E9 | 3.98E9 | 3.99E9 | 3.99E9 | 3.99E9 | 7.31E9 | 7.32E9 | 7.31E9 | 7.31E9 |
MMC | MML | Dual | P-D | MMC | MML | Dual | P-D | MMC | MML | Dual | P-D | ||
PSNR | #1 | 29.26 | 29.27 | 29.23 | 29.27 | 26.73 | 26.73 | 26.73 | 26.73 | 25.04 | 25.04 | 25.04 | 25.04 |
#2 | 28.64 | 28.64 | 28.60 | 28.62 | 26.49 | 26.44 | 26.40 | 26.42 | 25.01 | 24.95 | 24.96 | 24.96 | |
#3 | 30.79 | 30.77 | 30.72 | 30.75 | 28.78 | 28.78 | 28.78 | 28.78 | 27.25 | 27.21 | 27.21 | 27.21 | |
#4 | 31.14 | 31.13 | 31.09 | 31.11 | 29.13 | 29.11 | 29.11 | 29.12 | 27.21 | 27.21 | 27.21 | 27.21 | |
#5 | 29.64 | 29.64 | 29.63 | 29.63 | 28.73 | 27.75 | 27.75 | 27.74 | 26.40 | 26.38 | 26.39 | 26.40 | |
#6 | 27.95 | 27.95 | 27.89 | 27.94 | 25.44 | 25.45 | 25.42 | 24.13 | 23.93 | 23.94 | 23.92 | 23.94 | |
#7 | 32.45 | 32.44 | 32.43 | 32.40 | 29.92 | 29.91 | 29.90 | 29.91 | 28.20 | 28.20 | 28.20 | 28.20 | |
#8 | 32.28 | 32.25 | 32.23 | 32.27 | 29.95 | 29.93 | 29.85 | 29.87 | 27.52 | 27.52 | 27.51 | 27.51 | |
SSIM | #1 | 0.8511 | 0.8521 | 0.8515 | 0.8523 | 0.8051 | 0.8048 | 0.8046 | 0.8084 | 0.7561 | 0.7563 | 0.7601 | 0.7623 |
#2 | 0.8303 | 0.8279 | 0.8299 | 0.8255 | 0.7782 | 0.7755 | 0.7749 | 0.7811 | 0.7499 | 0.7476 | 0.7377 | 0.7492 | |
#3 | 0.8256 | 0.8221 | 0.8242 | 0.8253 | 0.7842 | 0.7827 | 0.7841 | 0.7851 | 0.7555 | 0.7519 | 0.7549 | 0.7537 | |
#4 | 0.8548 | 0.8562 | 0.8542 | 0.8537 | 0.8046 | 0.8011 | 0.8044 | 0.8012 | 0.7717 | 0.7782 | 0.7739 | 0.7717 | |
#5 | 0.7649 | 0.7631 | 0.7622 | 0.7641 | 0.7162 | 0.7144 | 0.7088 | 0.7101 | 0.6772 | 0.6766 | 0.6688 | 0.6768 | |
#6 | 0.8031 | 0.8131 | 0.8124 | 0.8122 | 0.7311 | 0.7353 | 0.7344 | 0.7349 | 0.6601 | 0.6721 | 0.6687 | 0.6699 | |
#7 | 0.8865 | 0.8877 | 0.8788 | 0.8791 | 0.8478 | 0.8516 | 0.8496 | 0.8502 | 0.8224 | 0.8236 | 0.8235 | 0.8247 | |
#8 | 0.9203 | 0.9107 | 0.9154 | 0.9165 | 0.9002 | 0.8997 | 0.8981 | 0.8994 | 0.8811 | 0.8868 | 0.8714 | 0.8726 | |
CPU(s) | #1 | 1.21 | 3.35 | 4.49 | 2.15 | 1.62 | 3.88 | 4.35 | 3.99 | 2.55 | 5.95 | 5.69 | 5.81 |
#2 | 1.17 | 2.01 | 2.81 | 1.97 | 1.79 | 3.61 | 7.13 | 4.23 | 3.05 | 11.78 | 7.13 | 8.68 | |
#3 | 5.97 | 9.83 | 14.42 | 8.60 | 6.93 | 16.06 | 17.14 | 11.86 | 9.88 | 37.98 | 17.19 | 15.06 | |
#4 | 5.13 | 9.13 | 15.01 | 8.11 | 7.68 | 15.76 | 19.06 | 8.99 | 9.85 | 20.74 | 18.99 | 15.04 | |
#5 | 28.68 | 48.35 | 54.08 | 29.93 | 35.66 | 76.67 | 64.43 | 49.30 | 48.53 | 190.87 | 78.30 | 56.35 | |
#6 | 26.78 | 50.14 | 51.35 | 28.86 | 38.63 | 79.12 | 78.84 | 58.84 | 50.25 | 180.62 | 64.30 | 55.44 | |
#7 | 106.43 | 192.49 | 463.12 | 161.99 | 161.37 | 334.45 | 452.12 | 188.14 | 186.92 | 491.83 | 443.35 | 269.35 | |
#8 | 123.28 | 183.17 | 454.35 | 159.73 | 164.28 | 303.28 | 443.18 | 191.18 | 208.15 | 482.78 | 482.06 | 275.05 | |
Energy | #1 | 3.03E7 | 3.01E7 | 3.03E7 | 3.03E7 | 6.56E7 | 6.57E7 | 6.57E7 | 6.56E7 | 1.18E8 | 1.17E8 | 1.18E8 | 1.17E8 |
#2 | 3.23E7 | 3.21E7 | 3.23E7 | 3.23E7 | 6.84E7 | 6.85E7 | 6.85E7 | 6.84E7 | 1.15E8 | 1.16E8 | 1.16E8 | 1.15E8 | |
#3 | 2.77E8 | 2.78E8 | 2.78E8 | 2.78E8 | 2.87E8 | 2.87E8 | 2.86E8 | 2.86E8 | 4.35E8 | 4.36E8 | 4.36E8 | 4.35E8 | |
#4 | 1.08E8 | 1.09E8 | 1.09E8 | 1.09E8 | 2.42E8 | 2.43E8 | 2.43E8 | 2.43E8 | 4.30E8 | 4.29E8 | 4.31E8 | 4.31E8 | |
#5 | 4.59E8 | 4.57E8 | 4.57E8 | 4.57E8 | 1.01E9 | 1.02E9 | 1.02E9 | 1.01E9 | 1.77E9 | 1.78E9 | 1.78E9 | 1.77E9 | |
#6 | 4.72E8 | 4.71E8 | 4.71E8 | 4.71E8 | 1.08E9 | 1.09E9 | 1.09E9 | 1.08E9 | 1.87E9 | 1.88E9 | 1.88E9 | 1.87E9 | |
#7 | 1.75E9 | 1.75E9 | 1.75E9 | 1.75E9 | 3.94E9 | 3.93E9 | 3.93E9 | 3.93E9 | 6.95E9 | 6.94E9 | 6.95E9 | 6.94E9 | |
#8 | 1.81E9 | 1.80E9 | 1.81E9 | 1.80E9 | 3.98E9 | 3.99E9 | 3.99E9 | 3.99E9 | 7.31E9 | 7.32E9 | 7.31E9 | 7.31E9 |
Image | Gaussian Blur | Motion Blur | |||||||
CPU | PSNR | SSIM | Energy | CPU | PSNR | SSIM | Energy | ||
MMC | Lena | 8.63 | 34.85 | 0.9207 | 1.4840 | 14.34 | 35.41 | 0.9401 | 1.5673 |
PD | 16.22 | 34.82 | 0.9234 | 1.4713 | 19.41 | 35.36 | 0.9382 | 1.5610 | |
MMC | Boat | 16.69 | 33.83 | 0.9204 | 1.9612 | 18.98 | 33.43 | 0.9178 | 1.9549 |
PD | 21.22 | 33.86 | 0.9222 | 1.9623 | 29.41 | 33.26 | 0.9166 | 1.9568 |
Image | Gaussian Blur | Motion Blur | |||||||
CPU | PSNR | SSIM | Energy | CPU | PSNR | SSIM | Energy | ||
MMC | Lena | 8.63 | 34.85 | 0.9207 | 1.4840 | 14.34 | 35.41 | 0.9401 | 1.5673 |
PD | 16.22 | 34.82 | 0.9234 | 1.4713 | 19.41 | 35.36 | 0.9382 | 1.5610 | |
MMC | Boat | 16.69 | 33.83 | 0.9204 | 1.9612 | 18.98 | 33.43 | 0.9178 | 1.9549 |
PD | 21.22 | 33.86 | 0.9222 | 1.9623 | 29.41 | 33.26 | 0.9166 | 1.9568 |
Image | 18 | 36 | 72 | |||||||||||
Size | CPU(s) | PSNR | SSIM | Energy | CPU(s) | PSNR | SSIM | Energy | CPU(s) | PSNR | SSIM | Energy | ||
MMC | Shepp | 256 | 15.64 | 39.28 | 0.9908 | 0.0508 | 20.01 | 50.19 | 0.9995 | 0.0292 | 33.21 | 58.30 | 0.9998 | 0.0292 |
PD | 27.59 | 39.16 | 0.9901 | 0.0508 | 59.33 | 50.37 | 0.9995 | 0.0292 | 112.04 | 58.21 | 0.9998 | 0.0292 | ||
MMC | Fobild | 256 | 21.33 | 27.96 | 0.9508 | 0.0595 | 24.15 | 37.66 | 0.9838 | 0.0421 | 40.21 | 45.06 | 0.9918 | 0.0423 |
PD | 25.02 | 27.92 | 0.9500 | 0.0600 | 71.21 | 37.17 | 0.9828 | 0.0421 | 140.10 | 45.16 | 0.9917 | 0.0425 | ||
MMC | Shepp | 512 | 96.62 | 35.79 | 0.9881 | 0.0991 | 103.54 | 43.22 | 0.9911 | 0.0587 | 129.91 | 50.60 | 0.9995 | 0.0585 |
PD | 176.86 | 35.72 | 0.9879 | 0.0991 | 204.57 | 43.21 | 0.9911 | 0.0585 | 370.14 | 50.51 | 0.9995 | 0.0585 | ||
MMC | Fobild | 512 | 130.01 | 31.29 | 0.9726 | 0.1501 | 153.52 | 37.90 | 0.9899 | 0.1557 | 179.58 | 48.88 | 0.9925 | 0.0861 |
PD | 187.02 | 31.23 | 0.9726 | 0.1501 | 248.01 | 37.86 | 0.9896 | 0.1557 | 457.51 | 48.83 | 0.9925 | 0.0861 |
Image | 18 | 36 | 72 | |||||||||||
Size | CPU(s) | PSNR | SSIM | Energy | CPU(s) | PSNR | SSIM | Energy | CPU(s) | PSNR | SSIM | Energy | ||
MMC | Shepp | 256 | 15.64 | 39.28 | 0.9908 | 0.0508 | 20.01 | 50.19 | 0.9995 | 0.0292 | 33.21 | 58.30 | 0.9998 | 0.0292 |
PD | 27.59 | 39.16 | 0.9901 | 0.0508 | 59.33 | 50.37 | 0.9995 | 0.0292 | 112.04 | 58.21 | 0.9998 | 0.0292 | ||
MMC | Fobild | 256 | 21.33 | 27.96 | 0.9508 | 0.0595 | 24.15 | 37.66 | 0.9838 | 0.0421 | 40.21 | 45.06 | 0.9918 | 0.0423 |
PD | 25.02 | 27.92 | 0.9500 | 0.0600 | 71.21 | 37.17 | 0.9828 | 0.0421 | 140.10 | 45.16 | 0.9917 | 0.0425 | ||
MMC | Shepp | 512 | 96.62 | 35.79 | 0.9881 | 0.0991 | 103.54 | 43.22 | 0.9911 | 0.0587 | 129.91 | 50.60 | 0.9995 | 0.0585 |
PD | 176.86 | 35.72 | 0.9879 | 0.0991 | 204.57 | 43.21 | 0.9911 | 0.0585 | 370.14 | 50.51 | 0.9995 | 0.0585 | ||
MMC | Fobild | 512 | 130.01 | 31.29 | 0.9726 | 0.1501 | 153.52 | 37.90 | 0.9899 | 0.1557 | 179.58 | 48.88 | 0.9925 | 0.0861 |
PD | 187.02 | 31.23 | 0.9726 | 0.1501 | 248.01 | 37.86 | 0.9896 | 0.1557 | 457.51 | 48.83 | 0.9925 | 0.0861 |
30 | 50 | ||||||||
Pattern | CPU | PSNR | SSIM | Energy | CPU | PSNR | SSIM | Energy | |
MMC | Radial | 4.78 | 32.15 | 0.9711 | 1.8840 | 10.19 | 38.41 | 0.9847 | 1.1673 |
PD | 13.22 | 32.08 | 0.9701 | 1.8840 | 19.41 | 38.36 | 0.9837 | 1.1675 | |
MMC | Random | 5.84 | 27.03 | 0.9500 | 2.0915 | 11.98 | 35.73 | 0.9812 | 2.0070 |
PD | 15.34 | 27.01 | 0.9504 | 2.0913 | 25.65 | 35.72 | 0.9812 | 2.0070 |
30 | 50 | ||||||||
Pattern | CPU | PSNR | SSIM | Energy | CPU | PSNR | SSIM | Energy | |
MMC | Radial | 4.78 | 32.15 | 0.9711 | 1.8840 | 10.19 | 38.41 | 0.9847 | 1.1673 |
PD | 13.22 | 32.08 | 0.9701 | 1.8840 | 19.41 | 38.36 | 0.9837 | 1.1675 | |
MMC | Random | 5.84 | 27.03 | 0.9500 | 2.0915 | 11.98 | 35.73 | 0.9812 | 2.0070 |
PD | 15.34 | 27.01 | 0.9504 | 2.0913 | 25.65 | 35.72 | 0.9812 | 2.0070 |
[1] |
Haijuan Hu, Jacques Froment, Baoyan Wang, Xiequan Fan. Spatial-Frequency domain nonlocal total variation for image denoising. Inverse Problems and Imaging, 2020, 14 (6) : 1157-1184. doi: 10.3934/ipi.2020059 |
[2] |
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 |
[3] |
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 |
[4] |
Yunhai Xiao, Junfeng Yang, Xiaoming Yuan. Alternating algorithms for total variation image reconstruction from random projections. Inverse Problems and Imaging, 2012, 6 (3) : 547-563. doi: 10.3934/ipi.2012.6.547 |
[5] |
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 |
[6] |
Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4963-4998. doi: 10.3934/dcdsb.2020321 |
[7] |
Juan C. Moreno, V. B. Surya Prasath, João C. Neves. Color image processing by vectorial total variation with gradient channels coupling. Inverse Problems and Imaging, 2016, 10 (2) : 461-497. doi: 10.3934/ipi.2016008 |
[8] |
Zhengmeng Jin, Chen Zhou, Michael K. Ng. A coupled total variation model with curvature driven for image colorization. Inverse Problems and Imaging, 2016, 10 (4) : 1037-1055. doi: 10.3934/ipi.2016031 |
[9] |
Sudeb Majee, Subit K. Jain, Rajendra K. Ray, Ananta K. Majee. A fuzzy edge detector driven telegraph total variation model for image despeckling. Inverse Problems and Imaging, 2022, 16 (2) : 367-396. doi: 10.3934/ipi.2021054 |
[10] |
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 |
[11] |
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 |
[12] |
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 |
[13] |
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 |
[14] |
Yuan Shen, Lei Ji. Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step. Journal of Industrial and Management Optimization, 2019, 15 (1) : 159-175. doi: 10.3934/jimo.2018037 |
[15] |
Baoli Shi, Zhi-Feng Pang, Jing Xu. Image segmentation based on the hybrid total variation model and the K-means clustering strategy. Inverse Problems and Imaging, 2016, 10 (3) : 807-828. doi: 10.3934/ipi.2016022 |
[16] |
Xavier Bresson, Tony F. Chan. Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Problems and Imaging, 2008, 2 (4) : 455-484. doi: 10.3934/ipi.2008.2.455 |
[17] |
Wei Wang, Na Sun, Michael K. Ng. A variational gamma correction model for image contrast enhancement. Inverse Problems and Imaging, 2019, 13 (3) : 461-478. doi: 10.3934/ipi.2019023 |
[18] |
Dominique Zosso, Jing An, James Stevick, Nicholas Takaki, Morgan Weiss, Liane S. Slaughter, Huan H. Cao, Paul S. Weiss, Andrea L. Bertozzi. Image segmentation with dynamic artifacts detection and bias correction. Inverse Problems and Imaging, 2017, 11 (3) : 577-600. doi: 10.3934/ipi.2017027 |
[19] |
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 |
[20] |
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 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]