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Image retinex based on the nonconvex TV-type regularization
An adaptive total variational despeckling model based on gray level indicator frame
1. | School of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
2. | School of Management, Harbin Institute of Technology, Harbin 150001, China |
For the characteristics of the degraded images with multiplicative noise, the gray level indicators for constructing adaptive total variation are proposed. Based on the new regularization term, we propose the new convex adaptive variational model. Then, considering the existence, uniqueness and comparison principle of the minimizer of the functional. The finite difference method with rescaling technique and the primal-dual method with adaptive step size are used to solve the minimization problem. The paper ends with a report on numerical tests for the denoising of images subject to multiplicative noise, the comparison with other methods is provided as well.
References:
[1] |
G. Aubert and J.-F. Aujol,
A variational approach to remove multiplicative noise, SIAM Journal on Applied Mathematics, 68 (2008), 925-946.
doi: 10.1137/060671814. |
[2] |
A. Beck and M. Teboulle,
Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Transactions on Image Processing, 18 (2009), 2419-2434.
doi: 10.1109/TIP.2009.2028250. |
[3] |
A. C. Bovik, T. S. Huang and D. C. Munson, A generalization of median filtering using linear combinations of order statistics, IEEE Trans Acoustics Speech Signal Processing, 31 (1983), 1342-1350. Google Scholar |
[4] |
A. C. Bovik, T. S. Huang and D. C. Munson,
The effect of median filtering on edge estimation and detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, 9 (1987), 181-194.
doi: 10.1109/TPAMI.1987.4767894. |
[5] |
K. Bredies, K. Kunisch and T. Pock,
Total generalized variation, SIAM Journal on Imaging Sciences, 3 (2010), 492-526.
doi: 10.1137/090769521. |
[6] |
A. Chambolle,
An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.
|
[7] |
A. Chambolle,
An algorithm for mean curvature motion, Interfaces and Free Boundaries, 6 (2004), 195-218.
doi: 10.4171/IFB/97. |
[8] |
A. Chambolle and T. Pock,
A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[9] |
C. Chaux, J.-C. Pesquet and N. Pustelnik,
Nested iterative algorithms for convex constrained image recovery problems, SIAM Journal on Imaging Sciences, 2 (2009), 730-762.
doi: 10.1137/080727749. |
[10] |
Y. Chen and M. Rao,
Minimization problems and associated flows related to weighted p energy and total variation, SIAM Journal on mathematical Analysis, 34 (2003), 1084-1104.
doi: 10.1137/S0036141002404577. |
[11] |
Y. Chen and T. Wunderli,
Adaptive total variation for image restoration in BV space, Journal of Mathematical Analysis and Applications, 272 (2002), 117-137.
doi: 10.1016/S0022-247X(02)00141-5. |
[12] |
P. L. Combettes and J.-C. Pesquet,
A douglas-rachford splitting approach to nonsmooth convex variational signal recovery, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 564-574.
doi: 10.1109/JSTSP.2007.910264. |
[13] |
A. Dauwe, B. Goossens, H. Luong and W. Philips, A fast non-local image denoising algorithm, Electronic Imaging, 6812 (2008), 681210.
doi: 10.1117/12.765505. |
[14] |
C.-A. Deledalle, L. Denis, S. Tabti and F. Tupin, MuLoG, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction?, IEEE Transactions on Image Processing, 26 (2017), 4389-4303.
doi: 10.1109/TIP.2017.2713946. |
[15] |
G. Dong, Z. Guo and B. Wu,
A convex adaptive total variation model based on the gray level indicator for multiplicative noise removal, Abstract and Applied Analysis, 2013 (2013), 1-21.
doi: 10.1155/2013/912373. |
[16] |
Y. Dong and T. Zeng,
A convex variational model for restoring blurred images with multiplicative noise, SIAM Journal on Imaging Sciences, 6 (2013), 1598-1625.
doi: 10.1137/120870621. |
[17] |
S. Durand, J. Fadili and M. Nikolova,
Multiplicative noise removal using L1 fidelity on frame coefficients, Journal of Mathematical Imaging and Vision, 36 (2010), 201-226.
doi: 10.1007/s10851-009-0180-z. |
[18] |
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 Journal on Imaging Sciences, 3 (2010), 1015-1046.
doi: 10.1137/09076934X. |
[19] |
K. Florian, B. Kristian, P. Thomas and S. Rudolf,
Second order total generalized variation (TGV) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491.
doi: 10.1002/mrm.22595. |
[20] |
E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, Birkhäuser Verlag, Basel, 80 1984, 7-26.
doi: 10.1007/978-1-4684-9486-0. |
[21] |
T. Goldstein and S. Osher,
The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.
doi: 10.1137/080725891. |
[22] |
H. Na, M. Kang, M. Jung and M. Kang,
Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters, Inverse Problems and Imaging, 13 (2019), 117-147.
doi: 10.3934/ipi.2019007. |
[23] |
P. Kornprobst, R. Deriche and G. Aubert,
Image sequence analysis via partial differential equations, Journal of Mathematical Imaging and Vision, 11 (1999), 5-26.
doi: 10.1023/A:1008318126505. |
[24] |
K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian,
Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Transactions on Image Processing, 16 (2007), 2080-2095.
doi: 10.1109/TIP.2007.901238. |
[25] |
T. Le, R. Chartrand and T. J. Asaki,
A variational approach to reconstructing images corrupted by poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263.
doi: 10.1007/s10851-007-0652-y. |
[26] |
S. Parrilli, M. Poderico, C. V. Angelino and L. Verdoliva,
A nonlocal SAR image denoising algorithm based on LLMMSE wavelet shrinkage, IEEE Transactions on Geoscience and Remote Sensing, 50 (2012), 606-616.
doi: 10.1109/TGRS.2011.2161586. |
[27] |
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. |
[28] |
T. Pock, D. Cremers, H. Bischof and A. Chambolle, An algorithm for minimizing the mumford-Shah functional, IEEE International Conference on Computer Vision, (2009), 1133-1140.
doi: 10.1109/ICCV.2009.5459348. |
[29] |
L. I. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[30] |
C. Poynton, Digital Video and HD: Algorithms and Interfaces, 2$^nd$ edition, Elsevier, USA, 2012.
doi: 10.1016/C2010-0-68987-5. |
[31] |
X. Shan, J. Sun and Z. Guo,
Multiplicative noise removal based on the smooth diffusion equation, Journal of Mathematical Imaging and Vision, 61 (2019), 763-779.
doi: 10.1007/s10851-018-00870-z. |
[32] |
R. Soorajkumar, P. K. Kumar, D. Girish and J. Rajan, Fourth order PDE based ultrasound despeckling using ENI classification, IEEE International Conference on Signal Processing and Communications (SPCOM), (2016).
doi: 10.1109/SPCOM.2016.7746633. |
[33] |
D. M. Strong and T. F. Chan, Spatially and scale adaptive total variation based regularization and anisotropic diffusion in image processing, Journal of Mathematical Imaging and Vision, (1996). Google Scholar |
[34] |
M. Tur, K. C. Chin and J. W. Goodman,
When is speckle noise multiplicative?, Applied Optics, 21 (1982), 1157-1159.
doi: 10.1364/AO.21.001157. |
[35] |
L. Verdoliva, R. Gaetano, G. Ruello and G. Poggi, Optical-driven nonlocal SAR despeckling, IEEE Geoscience and Remote Sensing Letters, 12 (2015), 314-318.
doi: 10.1109/LGRS.2014.2337515. |
[36] |
Z. Wang, A. C. Bovik and H. R. Sheikh, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 53 (2015), 2765-2774. Google Scholar |
[37] |
J. Weickert, Anisotropic Diffusion in Image Processing, Teubner Stuttgart, ECMI Series, Teubner, Stuttgart, 1998. |
[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 Journal on Imaging Sciences, 3 (2010), 300-339.
doi: 10.1137/090767558. |
[39] |
J.-H. Yang, X.-L. Zhao, T.-H. Ma, Y. Chen, T.-Z. Huang and M. Ding,
Emote sensing images destriping using unidirectional hybrid total variation and nonconvex low-rank regularization, Journal of Computational and Applied Mathematics, 363 (2020), 124-144.
doi: 10.1016/j.cam.2019.06.004. |
[40] |
X.-L. Zhao, F. Wang and M. K. Ng,
A new convex optimization model for multiplicative noise and blur removal, SIAM Journal on Imagingences, 7 (2014), 456-475.
doi: 10.1137/13092472X. |
[41] |
Y. Zhao, J. G. Liu, B. Zhang, W. Hong and Y.-R. Wu,
Adaptive total variation regularization based SAR image despeckling and despeckling evaluation index, IEEE Transactions on Geoscience and Remote Sensing, 53 (2015), 2765-2774.
doi: 10.1109/TGRS.2014.2364525. |
[42] |
Z. Zhou, Z. Guo, G. Dong, J. Sun, D. Zhang and B. Wu,
A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal, IEEE Transactions on Image Processing, 24 (2015), 249-260.
doi: 10.1109/TIP.2014.2376185. |
show all references
References:
[1] |
G. Aubert and J.-F. Aujol,
A variational approach to remove multiplicative noise, SIAM Journal on Applied Mathematics, 68 (2008), 925-946.
doi: 10.1137/060671814. |
[2] |
A. Beck and M. Teboulle,
Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Transactions on Image Processing, 18 (2009), 2419-2434.
doi: 10.1109/TIP.2009.2028250. |
[3] |
A. C. Bovik, T. S. Huang and D. C. Munson, A generalization of median filtering using linear combinations of order statistics, IEEE Trans Acoustics Speech Signal Processing, 31 (1983), 1342-1350. Google Scholar |
[4] |
A. C. Bovik, T. S. Huang and D. C. Munson,
The effect of median filtering on edge estimation and detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, 9 (1987), 181-194.
doi: 10.1109/TPAMI.1987.4767894. |
[5] |
K. Bredies, K. Kunisch and T. Pock,
Total generalized variation, SIAM Journal on Imaging Sciences, 3 (2010), 492-526.
doi: 10.1137/090769521. |
[6] |
A. Chambolle,
An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.
|
[7] |
A. Chambolle,
An algorithm for mean curvature motion, Interfaces and Free Boundaries, 6 (2004), 195-218.
doi: 10.4171/IFB/97. |
[8] |
A. Chambolle and T. Pock,
A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[9] |
C. Chaux, J.-C. Pesquet and N. Pustelnik,
Nested iterative algorithms for convex constrained image recovery problems, SIAM Journal on Imaging Sciences, 2 (2009), 730-762.
doi: 10.1137/080727749. |
[10] |
Y. Chen and M. Rao,
Minimization problems and associated flows related to weighted p energy and total variation, SIAM Journal on mathematical Analysis, 34 (2003), 1084-1104.
doi: 10.1137/S0036141002404577. |
[11] |
Y. Chen and T. Wunderli,
Adaptive total variation for image restoration in BV space, Journal of Mathematical Analysis and Applications, 272 (2002), 117-137.
doi: 10.1016/S0022-247X(02)00141-5. |
[12] |
P. L. Combettes and J.-C. Pesquet,
A douglas-rachford splitting approach to nonsmooth convex variational signal recovery, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 564-574.
doi: 10.1109/JSTSP.2007.910264. |
[13] |
A. Dauwe, B. Goossens, H. Luong and W. Philips, A fast non-local image denoising algorithm, Electronic Imaging, 6812 (2008), 681210.
doi: 10.1117/12.765505. |
[14] |
C.-A. Deledalle, L. Denis, S. Tabti and F. Tupin, MuLoG, or how to apply Gaussian denoisers to multi-channel SAR speckle reduction?, IEEE Transactions on Image Processing, 26 (2017), 4389-4303.
doi: 10.1109/TIP.2017.2713946. |
[15] |
G. Dong, Z. Guo and B. Wu,
A convex adaptive total variation model based on the gray level indicator for multiplicative noise removal, Abstract and Applied Analysis, 2013 (2013), 1-21.
doi: 10.1155/2013/912373. |
[16] |
Y. Dong and T. Zeng,
A convex variational model for restoring blurred images with multiplicative noise, SIAM Journal on Imaging Sciences, 6 (2013), 1598-1625.
doi: 10.1137/120870621. |
[17] |
S. Durand, J. Fadili and M. Nikolova,
Multiplicative noise removal using L1 fidelity on frame coefficients, Journal of Mathematical Imaging and Vision, 36 (2010), 201-226.
doi: 10.1007/s10851-009-0180-z. |
[18] |
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 Journal on Imaging Sciences, 3 (2010), 1015-1046.
doi: 10.1137/09076934X. |
[19] |
K. Florian, B. Kristian, P. Thomas and S. Rudolf,
Second order total generalized variation (TGV) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491.
doi: 10.1002/mrm.22595. |
[20] |
E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, Birkhäuser Verlag, Basel, 80 1984, 7-26.
doi: 10.1007/978-1-4684-9486-0. |
[21] |
T. Goldstein and S. Osher,
The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.
doi: 10.1137/080725891. |
[22] |
H. Na, M. Kang, M. Jung and M. Kang,
Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters, Inverse Problems and Imaging, 13 (2019), 117-147.
doi: 10.3934/ipi.2019007. |
[23] |
P. Kornprobst, R. Deriche and G. Aubert,
Image sequence analysis via partial differential equations, Journal of Mathematical Imaging and Vision, 11 (1999), 5-26.
doi: 10.1023/A:1008318126505. |
[24] |
K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian,
Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Transactions on Image Processing, 16 (2007), 2080-2095.
doi: 10.1109/TIP.2007.901238. |
[25] |
T. Le, R. Chartrand and T. J. Asaki,
A variational approach to reconstructing images corrupted by poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263.
doi: 10.1007/s10851-007-0652-y. |
[26] |
S. Parrilli, M. Poderico, C. V. Angelino and L. Verdoliva,
A nonlocal SAR image denoising algorithm based on LLMMSE wavelet shrinkage, IEEE Transactions on Geoscience and Remote Sensing, 50 (2012), 606-616.
doi: 10.1109/TGRS.2011.2161586. |
[27] |
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. |
[28] |
T. Pock, D. Cremers, H. Bischof and A. Chambolle, An algorithm for minimizing the mumford-Shah functional, IEEE International Conference on Computer Vision, (2009), 1133-1140.
doi: 10.1109/ICCV.2009.5459348. |
[29] |
L. I. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[30] |
C. Poynton, Digital Video and HD: Algorithms and Interfaces, 2$^nd$ edition, Elsevier, USA, 2012.
doi: 10.1016/C2010-0-68987-5. |
[31] |
X. Shan, J. Sun and Z. Guo,
Multiplicative noise removal based on the smooth diffusion equation, Journal of Mathematical Imaging and Vision, 61 (2019), 763-779.
doi: 10.1007/s10851-018-00870-z. |
[32] |
R. Soorajkumar, P. K. Kumar, D. Girish and J. Rajan, Fourth order PDE based ultrasound despeckling using ENI classification, IEEE International Conference on Signal Processing and Communications (SPCOM), (2016).
doi: 10.1109/SPCOM.2016.7746633. |
[33] |
D. M. Strong and T. F. Chan, Spatially and scale adaptive total variation based regularization and anisotropic diffusion in image processing, Journal of Mathematical Imaging and Vision, (1996). Google Scholar |
[34] |
M. Tur, K. C. Chin and J. W. Goodman,
When is speckle noise multiplicative?, Applied Optics, 21 (1982), 1157-1159.
doi: 10.1364/AO.21.001157. |
[35] |
L. Verdoliva, R. Gaetano, G. Ruello and G. Poggi, Optical-driven nonlocal SAR despeckling, IEEE Geoscience and Remote Sensing Letters, 12 (2015), 314-318.
doi: 10.1109/LGRS.2014.2337515. |
[36] |
Z. Wang, A. C. Bovik and H. R. Sheikh, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 53 (2015), 2765-2774. Google Scholar |
[37] |
J. Weickert, Anisotropic Diffusion in Image Processing, Teubner Stuttgart, ECMI Series, Teubner, Stuttgart, 1998. |
[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 Journal on Imaging Sciences, 3 (2010), 300-339.
doi: 10.1137/090767558. |
[39] |
J.-H. Yang, X.-L. Zhao, T.-H. Ma, Y. Chen, T.-Z. Huang and M. Ding,
Emote sensing images destriping using unidirectional hybrid total variation and nonconvex low-rank regularization, Journal of Computational and Applied Mathematics, 363 (2020), 124-144.
doi: 10.1016/j.cam.2019.06.004. |
[40] |
X.-L. Zhao, F. Wang and M. K. Ng,
A new convex optimization model for multiplicative noise and blur removal, SIAM Journal on Imagingences, 7 (2014), 456-475.
doi: 10.1137/13092472X. |
[41] |
Y. Zhao, J. G. Liu, B. Zhang, W. Hong and Y.-R. Wu,
Adaptive total variation regularization based SAR image despeckling and despeckling evaluation index, IEEE Transactions on Geoscience and Remote Sensing, 53 (2015), 2765-2774.
doi: 10.1109/TGRS.2014.2364525. |
[42] |
Z. Zhou, Z. Guo, G. Dong, J. Sun, D. Zhang and B. Wu,
A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal, IEEE Transactions on Image Processing, 24 (2015), 249-260.
doi: 10.1109/TIP.2014.2376185. |






Parameters | |||
Algorithm | |||
The Pie image (214 |
|||
AA ( |
5e-4 | 0.2 | 22 |
DZ ( |
0.3, 0.2 | 0.5, 0.04 | 5, 0.03 |
3e-7, 3e-8, 1.3 | 2e-3, 1e-3, 1.3 | 2e-3, 1e-3, 1.3 | |
1e-7, 1e-8, 1.39, 0.74 | 2e-3, 1e-3, 1.39, 0.74 | 2e-3, 3e-3, 1.39, 0.74 | |
4e-3, 6e-3, 1.3 | 4e-3, 5e-3, 1.3 | 3e-3, 6e-3, 1.3 | |
4e-2, 6e-3, 1.39, 0.74 | 4e-3, 5e-3, 1.38, 0.74 | 4e-3, 4e-3, 1.34, 0.74 | |
The Cameraman image( |
|||
AA ( |
0.18 | 40 | 75 |
DZ ( |
0.21, 0.25 | 1.6, 0.11 | 2.5, 0.03 |
1e-7, 5e-7, 1.3 | 2e-3, 2e-3, 1.3 | 3e-2, 2e-2, 1.3 | |
3e-7, 2e-7, 1.3, 0.8 | 3e-3, 2e-3, 1.33, 0.76 | 3e-2, 2e-2, 1.4, 0.78 | |
6e-3, 8e-3, 1.3 | 8e-3, 6e-3, 1.3 | 8e-3, 3e-3, 1.3 | |
1e-2, 6e-3, 1.3, 0.79 | 8e-3, 4e-3, 1.3, 0.75 | 6e-3, 6e-3, 1.3, 0.75 | |
The Parrot image( |
|||
AA ( |
0.6 | 1.5 | 8 |
DZ ( |
0.11, 0.2 | 1, 0.2 | 15, 0.3 |
2e-6, 2e-6, 1.3 | 1e-4, 1e-4, 1.3 | 3e-3, 3e-3, 1.3 | |
1e-6, 1e-6, 1.36, 0.74 | 1e-4, 1e-4, 1.35, 0.74 | 1e-3, 1e-3, 1.32, 0.74 | |
6e-3, 6e-3, 1.3 | 6e-3, 6e-3, 1.3 | 6e-3, 6e-3, 1.3 | |
6e-3, 6e-3, 1.36, 0.74 | 6e-3, 6e-3, 1.34, 0.74 | 6e-3, 6e-3, 1.32, 0.74 | |
The Synthetic image( |
|||
AA ( |
7e-4 | 4e-3 | 1e-3 |
DZ ( |
0.117, 0.125 | 0.155, 0.1 | 0.11, 0.2 |
1e-7, 1e-7, 1.3 | 2e-4, 2e-4, 1.3 | 1e-3, 1e-3, 1.3 | |
1e-7, 1e-7, 1.35, 0.78 | 1e-4, 1e-4, 1.36, 0.74 | 1e-3, 1e-3, 1.35, 0.74 | |
4e-3, 5e-3, 1.3 | 4e-3, 4e-3, 1.3 | 4e-3, 4e-3, 1.3 | |
4e-3, 5e-3, 1.4, 0.75 | 4e-3, 5e-3, 1.4, 0.74 | 4e-3, 4e-3, 1.32, 0.74 | |
The Piglet image( |
|||
AA ( |
5e-3 | 80 | 190 |
DZ ( |
3e-3, 0.16 | 2e-3, 3e-2 | 0.1, 7e-3 |
1e-7, 9e-8, 1.3 | 1e-4, 1e-4, 1.3 | 1e-3, 1e-3, 1.3 | |
2e-7, 9e-8, 1.38, 0.74 | 1e-4, 1e-4, 1.37, 0.74 | 1e-3, 1e-3, 1.39, 0.74 | |
4e-3, 4e-3, 1.3 | 4e-3, 7e-3, 1.3 | 6e-3, 8e-3, 1.3 | |
4e-3, 7e-3, 1.35, 0.74 | 4e-3, 7e-3, 1.35, 0.74 | 6e-3, 8e-3, 1.32, 0.74 |
Parameters | |||
Algorithm | |||
The Pie image (214 |
|||
AA ( |
5e-4 | 0.2 | 22 |
DZ ( |
0.3, 0.2 | 0.5, 0.04 | 5, 0.03 |
3e-7, 3e-8, 1.3 | 2e-3, 1e-3, 1.3 | 2e-3, 1e-3, 1.3 | |
1e-7, 1e-8, 1.39, 0.74 | 2e-3, 1e-3, 1.39, 0.74 | 2e-3, 3e-3, 1.39, 0.74 | |
4e-3, 6e-3, 1.3 | 4e-3, 5e-3, 1.3 | 3e-3, 6e-3, 1.3 | |
4e-2, 6e-3, 1.39, 0.74 | 4e-3, 5e-3, 1.38, 0.74 | 4e-3, 4e-3, 1.34, 0.74 | |
The Cameraman image( |
|||
AA ( |
0.18 | 40 | 75 |
DZ ( |
0.21, 0.25 | 1.6, 0.11 | 2.5, 0.03 |
1e-7, 5e-7, 1.3 | 2e-3, 2e-3, 1.3 | 3e-2, 2e-2, 1.3 | |
3e-7, 2e-7, 1.3, 0.8 | 3e-3, 2e-3, 1.33, 0.76 | 3e-2, 2e-2, 1.4, 0.78 | |
6e-3, 8e-3, 1.3 | 8e-3, 6e-3, 1.3 | 8e-3, 3e-3, 1.3 | |
1e-2, 6e-3, 1.3, 0.79 | 8e-3, 4e-3, 1.3, 0.75 | 6e-3, 6e-3, 1.3, 0.75 | |
The Parrot image( |
|||
AA ( |
0.6 | 1.5 | 8 |
DZ ( |
0.11, 0.2 | 1, 0.2 | 15, 0.3 |
2e-6, 2e-6, 1.3 | 1e-4, 1e-4, 1.3 | 3e-3, 3e-3, 1.3 | |
1e-6, 1e-6, 1.36, 0.74 | 1e-4, 1e-4, 1.35, 0.74 | 1e-3, 1e-3, 1.32, 0.74 | |
6e-3, 6e-3, 1.3 | 6e-3, 6e-3, 1.3 | 6e-3, 6e-3, 1.3 | |
6e-3, 6e-3, 1.36, 0.74 | 6e-3, 6e-3, 1.34, 0.74 | 6e-3, 6e-3, 1.32, 0.74 | |
The Synthetic image( |
|||
AA ( |
7e-4 | 4e-3 | 1e-3 |
DZ ( |
0.117, 0.125 | 0.155, 0.1 | 0.11, 0.2 |
1e-7, 1e-7, 1.3 | 2e-4, 2e-4, 1.3 | 1e-3, 1e-3, 1.3 | |
1e-7, 1e-7, 1.35, 0.78 | 1e-4, 1e-4, 1.36, 0.74 | 1e-3, 1e-3, 1.35, 0.74 | |
4e-3, 5e-3, 1.3 | 4e-3, 4e-3, 1.3 | 4e-3, 4e-3, 1.3 | |
4e-3, 5e-3, 1.4, 0.75 | 4e-3, 5e-3, 1.4, 0.74 | 4e-3, 4e-3, 1.32, 0.74 | |
The Piglet image( |
|||
AA ( |
5e-3 | 80 | 190 |
DZ ( |
3e-3, 0.16 | 2e-3, 3e-2 | 0.1, 7e-3 |
1e-7, 9e-8, 1.3 | 1e-4, 1e-4, 1.3 | 1e-3, 1e-3, 1.3 | |
2e-7, 9e-8, 1.38, 0.74 | 1e-4, 1e-4, 1.37, 0.74 | 1e-3, 1e-3, 1.39, 0.74 | |
4e-3, 4e-3, 1.3 | 4e-3, 7e-3, 1.3 | 6e-3, 8e-3, 1.3 | |
4e-3, 7e-3, 1.35, 0.74 | 4e-3, 7e-3, 1.35, 0.74 | 6e-3, 8e-3, 1.32, 0.74 |
PSNR | MAE | SSIM | #iter | |||||||||
The Pie image( |
||||||||||||
AA | 0.57 | 0.77 | 0.88 | 1978 | 743 | 464 | ||||||
DZ | 0.63 | 0.86 | 0.92 | 637 | 1162 | 1027 | ||||||
10.15 | 0.67 | 0.84 | 0.90 | 481 | 275 | 273 | ||||||
24.08 | 10.10 | 0.69 | 0.84 | 0.90 | 362 | 256 | 211 | |||||
0.71 | 0.84 | 0.90 | 1973 | 609 | 312 | |||||||
0.71 | 0.83 | 0.90 | 1475 | 485 | 239 | |||||||
The cameraman image( |
||||||||||||
AA | 19.06 | 22.96 | 25.09 | 16.79 | 10.09 | 7.94 | 0.56 | 0.69 | 0.74 | 1800 | 672 | 351 |
DZ | 19.34 | 22.81 | 25.22 | 16.64 | 10.45 | 7.94 | 0.53 | 0.70 | 0.75 | 387 | 291 | 338 |
21.56 | 24.44 | 26.37 | 12.95 | 9.19 | 7.29 | 0.63 | 0.71 | 0.77 | 442 | 361 | 289 | |
21.56 | 24.50 | 26.42 | 12.98 | 9.18 | 7.27 | 0.63 | 0.71 | 0.76 | 395 | 315 | 238 | |
21.70 | 24.28 | 26.21 | 12.60 | 9.01 | 7.29 | 0.65 | 0.73 | 0.78 | 1469 | 455 | 243 | |
21.76 | 24.30 | 26.23 | 11.96 | 9.00 | 7.35 | 0.66 | 0.73 | 0.77 | 1167 | 382 | 203 | |
The Parrot image( |
||||||||||||
AA | 20.52 | 24.46 | 26.89 | 14.73 | 10.15 | 7.86 | 0.60 | 0.68 | 0.72 | 1833 | 666 | 359 |
DZ | 20.59 | 25.19 | 27.52 | 14.22 | 9.38 | 7.29 | 0.57 | 0.71 | 0.76 | 533 | 187 | 69 |
23.84 | 26.73 | 28.57 | 10.88 | 7.95 | 6.49 | 0.66 | 0.74 | 0.78 | 445 | 391 | 342 | |
23.84 | 26.75 | 28.61 | 10.96 | 8.00 | 6.48 | 0.65 | 0.73 | 0.78 | 372 | 309 | 227 | |
23.88 | 26.73 | 28.55 | 10.93 | 7.94 | 6.50 | 0.65 | 0.73 | 0.78 | 1907 | 629 | 323 | |
23.81 | 26.71 | 28.60 | 11.96 | 9.00 | 6.48 | 0.65 | 0.73 | 0.78 | 1572 | 523 | 240 | |
The Synthetic image( |
||||||||||||
AA | 0.81 | 0.88 | 0.92 | 2664 | 1010 | 643 | ||||||
DZ | 0.77 | 0.94 | 0.96 | 1195 | 570 | 291 | ||||||
3.30 | 0.84 | 0.93 | 0.95 | 507 | 442 | 371 | ||||||
31.71 | 3.41 | 0.83 | 0.92 | 0.95 | 457 | 337 | 282 | |||||
0.89 | 0.93 | 0.96 | 2506 | 787 | 425 | |||||||
0.89 | 0.92 | 0.96 | 1759 | 645 | 338 | |||||||
The Piglet image( |
||||||||||||
AA | 0.83 | 0.89 | 0.91 | 2363 | 1068 | 660 | ||||||
DZ | 0.73 | 0.89 | 0.91 | 739 | 1540 | 4440 | ||||||
4.53 | 0.82 | 0.89 | 0.92 | 499 | 390 | 345 | ||||||
30.75 | 4.58 | 0.78 | 0.88 | 0.91 | 432 | 321 | 297 | |||||
0.84 | 0.88 | 0.91 | 1904 | 666 | 335 | |||||||
0.80 | 0.88 | 0.92 | 1463 | 530 | 280 |
PSNR | MAE | SSIM | #iter | |||||||||
The Pie image( |
||||||||||||
AA | 0.57 | 0.77 | 0.88 | 1978 | 743 | 464 | ||||||
DZ | 0.63 | 0.86 | 0.92 | 637 | 1162 | 1027 | ||||||
10.15 | 0.67 | 0.84 | 0.90 | 481 | 275 | 273 | ||||||
24.08 | 10.10 | 0.69 | 0.84 | 0.90 | 362 | 256 | 211 | |||||
0.71 | 0.84 | 0.90 | 1973 | 609 | 312 | |||||||
0.71 | 0.83 | 0.90 | 1475 | 485 | 239 | |||||||
The cameraman image( |
||||||||||||
AA | 19.06 | 22.96 | 25.09 | 16.79 | 10.09 | 7.94 | 0.56 | 0.69 | 0.74 | 1800 | 672 | 351 |
DZ | 19.34 | 22.81 | 25.22 | 16.64 | 10.45 | 7.94 | 0.53 | 0.70 | 0.75 | 387 | 291 | 338 |
21.56 | 24.44 | 26.37 | 12.95 | 9.19 | 7.29 | 0.63 | 0.71 | 0.77 | 442 | 361 | 289 | |
21.56 | 24.50 | 26.42 | 12.98 | 9.18 | 7.27 | 0.63 | 0.71 | 0.76 | 395 | 315 | 238 | |
21.70 | 24.28 | 26.21 | 12.60 | 9.01 | 7.29 | 0.65 | 0.73 | 0.78 | 1469 | 455 | 243 | |
21.76 | 24.30 | 26.23 | 11.96 | 9.00 | 7.35 | 0.66 | 0.73 | 0.77 | 1167 | 382 | 203 | |
The Parrot image( |
||||||||||||
AA | 20.52 | 24.46 | 26.89 | 14.73 | 10.15 | 7.86 | 0.60 | 0.68 | 0.72 | 1833 | 666 | 359 |
DZ | 20.59 | 25.19 | 27.52 | 14.22 | 9.38 | 7.29 | 0.57 | 0.71 | 0.76 | 533 | 187 | 69 |
23.84 | 26.73 | 28.57 | 10.88 | 7.95 | 6.49 | 0.66 | 0.74 | 0.78 | 445 | 391 | 342 | |
23.84 | 26.75 | 28.61 | 10.96 | 8.00 | 6.48 | 0.65 | 0.73 | 0.78 | 372 | 309 | 227 | |
23.88 | 26.73 | 28.55 | 10.93 | 7.94 | 6.50 | 0.65 | 0.73 | 0.78 | 1907 | 629 | 323 | |
23.81 | 26.71 | 28.60 | 11.96 | 9.00 | 6.48 | 0.65 | 0.73 | 0.78 | 1572 | 523 | 240 | |
The Synthetic image( |
||||||||||||
AA | 0.81 | 0.88 | 0.92 | 2664 | 1010 | 643 | ||||||
DZ | 0.77 | 0.94 | 0.96 | 1195 | 570 | 291 | ||||||
3.30 | 0.84 | 0.93 | 0.95 | 507 | 442 | 371 | ||||||
31.71 | 3.41 | 0.83 | 0.92 | 0.95 | 457 | 337 | 282 | |||||
0.89 | 0.93 | 0.96 | 2506 | 787 | 425 | |||||||
0.89 | 0.92 | 0.96 | 1759 | 645 | 338 | |||||||
The Piglet image( |
||||||||||||
AA | 0.83 | 0.89 | 0.91 | 2363 | 1068 | 660 | ||||||
DZ | 0.73 | 0.89 | 0.91 | 739 | 1540 | 4440 | ||||||
4.53 | 0.82 | 0.89 | 0.92 | 499 | 390 | 345 | ||||||
30.75 | 4.58 | 0.78 | 0.88 | 0.91 | 432 | 321 | 297 | |||||
0.84 | 0.88 | 0.91 | 1904 | 666 | 335 | |||||||
0.80 | 0.88 | 0.92 | 1463 | 530 | 280 |
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