doi: 10.3934/ipi.2022041
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Nonconvex model for mixing noise with fractional-order regularization

1. 

Institute of Microelectronics of the Chinese Academy of Sciences, Beijing, China

2. 

Institute of Applied Physics and Computational Mathematics, Beijing, China

3. 

Department of Mathematics, The Chinese University of Hong Kong, Hong Kong

*Corresponding author: Guoxi Ni

Received  September 2021 Revised  June 2022 Early access August 2022

In this paper, we propose a restoration model for image degraded by different kinds of blur and mixing Gaussian-impulse noise. Our model consist of a nonconvex Exponential-Type (ET) function, a $ L_{2} $-norm data-fitting term and a fractional-order Besov norm as the regularization term. We employ the proximal linearized minimization (PLM) algorithm and alternating direction method of multipliers (ADMM) algorithm to solve our proposed minimization model, convergence analysis is carried out. The experimental outcomes demonstrate that the restored images by our proposed method are better than those existing relative methods in terms of PSNR, SSIM values and visual quality.

Citation: Shaowen Yan, Guoxi Ni, Tieyong Zeng. Nonconvex model for mixing noise with fractional-order regularization. Inverse Problems and Imaging, doi: 10.3934/ipi.2022041
References:
[1]

H. AttouchJ. Bolte and B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Math. Program., 137 (2013), 91-129.  doi: 10.1007/s10107-011-0484-9.

[2]

J. Bai and X.-C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Tran. Image Proc., 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.

[3]

J. Bolte, A. Daniilidis, A. Lewis, et al., Clarke subgradients of stratifiable functions, SIAM J. Optim., 18 (2007), 556-572. doi: 10.1137/060670080.

[4]

J. BolteS. Sabach and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Math. Program., 146 (2014), 459-494.  doi: 10.1007/s10107-013-0701-9.

[5]

J.-F. CaiR. H. Chan and M. Nikolova, Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise, Inverse Probl. Imaging, 2 (2008), 187-204.  doi: 10.3934/ipi.2008.2.187.

[6]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20 (2004), 89-97. 

[7]

A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows, Int. J. Comput. Vis., 84 (2009), 288-307.  doi: 10.1007/s11263-009-0238-9.

[8]

T. F. Chan and S. Esedoglu, Aspects of total variation regularized L1 function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.  doi: 10.1137/040604297.

[9]

D. ChenY. Chen and D. Xue, Fractional-order total variation image restoration based on primal-dual algorithm, Abstract and Applied Analysis, 2013 (2013), 717-718.  doi: 10.1155/2013/585310.

[10]

D. Chen, S. Sun, C. Zhang, et al., Fractional-order TV-L2 model for image denoising, Cent. Eur. J. Phys., 11 (2013), 1414-1422. doi: 10.2478/s11534-013-0241-1.

[11]

X. Chen, X. Liu, J. Zheng, et al., Nonlocal low-rank regularized two-phase approach for mixed noise removal, Inverse Probl., 37 (2021), 085001. doi: 10.1088/1361-6420/ac0c21.

[12]

P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200.  doi: 10.1137/050626090.

[13]

I. DaubechiesM. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.

[14]

I. Daubechies and G. Teschke, Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising, Appl. Comput. Harmon. Anal., 19 (2005), 1-16.  doi: 10.1016/j.acha.2004.12.004.

[15]

J. Delon and A. Desolneux, A patch-based approach for removing impulse or mixed Gaussian-impulse noise, SIAM J. Imaging Sci., 6 (2013), 1140-1174.  doi: 10.1137/120885000.

[16]

L.-J. DengR. Glowinski and X.-C. Tai, A new operator splitting method for the Euler elastica model for image smoothing, SIAM J. Imaging Sci., 12 (2019), 1190-1230.  doi: 10.1137/18M1226361.

[17]

L. Ding and W. Han, A projected gradient method for $\alpha l_1-\beta l_2$ sparsity regularization, Inverse Probl., 36 (2020), 125012, 30 pp. doi: 10.1088/1361-6420/abc857.

[18]

B. Dong, H. Ji, J. Li, et al., Wavelet frame based blind image inpainting, Appl. Comput. Harmon. Anal., 32 (2012), 268-279. doi: 10.1016/j. acha. 2011.06.001.

[19]

C. X. Gao, N. Y. Wang, Q. Yu, et al., A feasible nonconvex relaxation approach to feature selection, Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence, San Francisco, California, 2011, 356–361.

[20]

R. Garnett, T. Huegerich, C. Chui, et al., A universal noise removal algorithm with an impulse detector, IEEE Trans. Image Process., 14 (2005), 1747-1754. doi: 10.1109/TIP. 2005.857261.

[21]

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3$^rd$ edition, Prentice Hall, New Jersey, 2007.

[22]

Y. HeS. H. Kang and H. Liu, Curvature regularized surface reconstruction from point clouds, SIAM J. Imaging Sci., 13 (2020), 1834-1859.  doi: 10.1137/20M1314525.

[23]

M. Hintermuller and A. Langer, Subspace correction methods for a class of nonsmooth and nonadditive convex variational problems with mixed $L^1/L^2$ data-fidelity in image processing, SIAM J. Imaging Sci., 6 (2013), 2134-2173.  doi: 10.1137/120894130.

[24]

A. Langer, Automated parameter selection for total variation minimization in image restoration, J. Math. Imaging Vis., 57 (2017), 239-268.  doi: 10.1007/s10851-016-0676-2.

[25]

B. Li, Q. Liu, J. Xu, et al., A new method for removing mixed noises, Sci. China Inf. Sci., 54 (2011), 51-59. doi: 10.1007/s11432-010-4128-0.

[26]

J. LiuA. Ni and G. Ni, A nonconvex l1(l1-l2) model for image restoration with impulse noise, J. Comput. Appl. Math., 378 (2020), 112934.  doi: 10.1016/j.cam.2020.112934.

[27]

J. Liu, X. -C. Tai, H. Huang, et al., A weighted dictionary learning model for denoising images corrupted by mixed noise, IEEE Trans. Image Process., 22 (2012), 1108-1120. doi: 10.1109/TIP. 2012.2227766.

[28]

E. López-Rubio, Restoration of images corrupted by Gaussian and uniform impulsive noise, Pattern Recognit., 43 (2010), 1835-1846.  doi: 10.1016/j.patcog.2009.11.017.

[29]

L. MaL. Xu and T. Zeng, Low rank prior and total variation regularization for image deblurring, J. Sci. Comput., 70 (2017), 1336-1357.  doi: 10.1007/s10915-016-0282-x.

[30]

J. -J. Mei, Y. Dong, T. -Z. Huang, et al., Cauchy noise removal by nonconvex ADMM with convergence guarantees, J. Sci. Comput., 74 (2018), 743-766. doi: 10.1007/s10915-017-0460-5.

[31]

Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program., 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.

[32]

D. Qiu, M. Bai, M. K. Ng, et al., Nonlocal robust tensor recovery with nonconvex regularization, Inverse Probl., 37 (2021), 035001. doi: 10.1088/1361-6420/abd85b.

[33]

Y. ShenB. Han and E. Braverman, Removal of mixed Gaussian and impulse noise using directional tensor product complex tight framelets, J. Math. Imaging Vis., 54 (2016), 64-77.  doi: 10.1007/s10851-015-0589-5.

[34]

P. WeissL. Blanc-Féraud and G. Aubert, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. Sci. Comput., 31 (2009), 2047-2080.  doi: 10.1137/070696143.

[35]

M. Yan, Restoration of images corrupted by impulse noise and mixed Gaussian impulse noise using blind inpainting, SIAM J. Imaging Sci., 6 (2013), 1227-1245.  doi: 10.1137/12087178X.

[36]

J. Yang, W. Yin, Y. Zhang, et al., A fast algorithm for edge-preserving variational multichannel image restoration, SIAM J. Imaging Sci., 2 (2009), 569-592. doi: 10.1137/080730421.

[37]

J. YangY. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), 2842-2865.  doi: 10.1137/080732894.

[38]

J. YangY. Zhang and W. Yin, A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data, IEEE J. Sel. Top. Signal Process., 4 (2010), 288-297.  doi: 10.1109/JSTSP.2010.2042333.

[39]

K. ŽeljkoJ. Maly and V. Naumova, Computational approaches to nonconvex sparsity-inducing multi-penalty regularization, Inverse Probl., 37 (2021), 055008.  doi: 10.1088/1361-6420/abdd46.

[40]

J. Zhang and K. Chen, A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution, SIAM J. Imaging Sci., 8 (2015), 2487-2518.  doi: 10.1137/14097121X.

[41]

J. ZhangZ. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vis., 43 (2012), 39-49.  doi: 10.1007/s10851-011-0285-z.

[42]

X. ZhangM. Bai and M. K. Ng, Nonconvex-TV based image restoration with impulse noise removal, SIAM J. Imaging Sci., 10 (2017), 1627-1667.  doi: 10.1137/16M1076034.

show all references

References:
[1]

H. AttouchJ. Bolte and B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Math. Program., 137 (2013), 91-129.  doi: 10.1007/s10107-011-0484-9.

[2]

J. Bai and X.-C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Tran. Image Proc., 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.

[3]

J. Bolte, A. Daniilidis, A. Lewis, et al., Clarke subgradients of stratifiable functions, SIAM J. Optim., 18 (2007), 556-572. doi: 10.1137/060670080.

[4]

J. BolteS. Sabach and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Math. Program., 146 (2014), 459-494.  doi: 10.1007/s10107-013-0701-9.

[5]

J.-F. CaiR. H. Chan and M. Nikolova, Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise, Inverse Probl. Imaging, 2 (2008), 187-204.  doi: 10.3934/ipi.2008.2.187.

[6]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20 (2004), 89-97. 

[7]

A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows, Int. J. Comput. Vis., 84 (2009), 288-307.  doi: 10.1007/s11263-009-0238-9.

[8]

T. F. Chan and S. Esedoglu, Aspects of total variation regularized L1 function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.  doi: 10.1137/040604297.

[9]

D. ChenY. Chen and D. Xue, Fractional-order total variation image restoration based on primal-dual algorithm, Abstract and Applied Analysis, 2013 (2013), 717-718.  doi: 10.1155/2013/585310.

[10]

D. Chen, S. Sun, C. Zhang, et al., Fractional-order TV-L2 model for image denoising, Cent. Eur. J. Phys., 11 (2013), 1414-1422. doi: 10.2478/s11534-013-0241-1.

[11]

X. Chen, X. Liu, J. Zheng, et al., Nonlocal low-rank regularized two-phase approach for mixed noise removal, Inverse Probl., 37 (2021), 085001. doi: 10.1088/1361-6420/ac0c21.

[12]

P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200.  doi: 10.1137/050626090.

[13]

I. DaubechiesM. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.

[14]

I. Daubechies and G. Teschke, Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising, Appl. Comput. Harmon. Anal., 19 (2005), 1-16.  doi: 10.1016/j.acha.2004.12.004.

[15]

J. Delon and A. Desolneux, A patch-based approach for removing impulse or mixed Gaussian-impulse noise, SIAM J. Imaging Sci., 6 (2013), 1140-1174.  doi: 10.1137/120885000.

[16]

L.-J. DengR. Glowinski and X.-C. Tai, A new operator splitting method for the Euler elastica model for image smoothing, SIAM J. Imaging Sci., 12 (2019), 1190-1230.  doi: 10.1137/18M1226361.

[17]

L. Ding and W. Han, A projected gradient method for $\alpha l_1-\beta l_2$ sparsity regularization, Inverse Probl., 36 (2020), 125012, 30 pp. doi: 10.1088/1361-6420/abc857.

[18]

B. Dong, H. Ji, J. Li, et al., Wavelet frame based blind image inpainting, Appl. Comput. Harmon. Anal., 32 (2012), 268-279. doi: 10.1016/j. acha. 2011.06.001.

[19]

C. X. Gao, N. Y. Wang, Q. Yu, et al., A feasible nonconvex relaxation approach to feature selection, Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence, San Francisco, California, 2011, 356–361.

[20]

R. Garnett, T. Huegerich, C. Chui, et al., A universal noise removal algorithm with an impulse detector, IEEE Trans. Image Process., 14 (2005), 1747-1754. doi: 10.1109/TIP. 2005.857261.

[21]

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3$^rd$ edition, Prentice Hall, New Jersey, 2007.

[22]

Y. HeS. H. Kang and H. Liu, Curvature regularized surface reconstruction from point clouds, SIAM J. Imaging Sci., 13 (2020), 1834-1859.  doi: 10.1137/20M1314525.

[23]

M. Hintermuller and A. Langer, Subspace correction methods for a class of nonsmooth and nonadditive convex variational problems with mixed $L^1/L^2$ data-fidelity in image processing, SIAM J. Imaging Sci., 6 (2013), 2134-2173.  doi: 10.1137/120894130.

[24]

A. Langer, Automated parameter selection for total variation minimization in image restoration, J. Math. Imaging Vis., 57 (2017), 239-268.  doi: 10.1007/s10851-016-0676-2.

[25]

B. Li, Q. Liu, J. Xu, et al., A new method for removing mixed noises, Sci. China Inf. Sci., 54 (2011), 51-59. doi: 10.1007/s11432-010-4128-0.

[26]

J. LiuA. Ni and G. Ni, A nonconvex l1(l1-l2) model for image restoration with impulse noise, J. Comput. Appl. Math., 378 (2020), 112934.  doi: 10.1016/j.cam.2020.112934.

[27]

J. Liu, X. -C. Tai, H. Huang, et al., A weighted dictionary learning model for denoising images corrupted by mixed noise, IEEE Trans. Image Process., 22 (2012), 1108-1120. doi: 10.1109/TIP. 2012.2227766.

[28]

E. López-Rubio, Restoration of images corrupted by Gaussian and uniform impulsive noise, Pattern Recognit., 43 (2010), 1835-1846.  doi: 10.1016/j.patcog.2009.11.017.

[29]

L. MaL. Xu and T. Zeng, Low rank prior and total variation regularization for image deblurring, J. Sci. Comput., 70 (2017), 1336-1357.  doi: 10.1007/s10915-016-0282-x.

[30]

J. -J. Mei, Y. Dong, T. -Z. Huang, et al., Cauchy noise removal by nonconvex ADMM with convergence guarantees, J. Sci. Comput., 74 (2018), 743-766. doi: 10.1007/s10915-017-0460-5.

[31]

Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program., 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.

[32]

D. Qiu, M. Bai, M. K. Ng, et al., Nonlocal robust tensor recovery with nonconvex regularization, Inverse Probl., 37 (2021), 035001. doi: 10.1088/1361-6420/abd85b.

[33]

Y. ShenB. Han and E. Braverman, Removal of mixed Gaussian and impulse noise using directional tensor product complex tight framelets, J. Math. Imaging Vis., 54 (2016), 64-77.  doi: 10.1007/s10851-015-0589-5.

[34]

P. WeissL. Blanc-Féraud and G. Aubert, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. Sci. Comput., 31 (2009), 2047-2080.  doi: 10.1137/070696143.

[35]

M. Yan, Restoration of images corrupted by impulse noise and mixed Gaussian impulse noise using blind inpainting, SIAM J. Imaging Sci., 6 (2013), 1227-1245.  doi: 10.1137/12087178X.

[36]

J. Yang, W. Yin, Y. Zhang, et al., A fast algorithm for edge-preserving variational multichannel image restoration, SIAM J. Imaging Sci., 2 (2009), 569-592. doi: 10.1137/080730421.

[37]

J. YangY. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), 2842-2865.  doi: 10.1137/080732894.

[38]

J. YangY. Zhang and W. Yin, A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data, IEEE J. Sel. Top. Signal Process., 4 (2010), 288-297.  doi: 10.1109/JSTSP.2010.2042333.

[39]

K. ŽeljkoJ. Maly and V. Naumova, Computational approaches to nonconvex sparsity-inducing multi-penalty regularization, Inverse Probl., 37 (2021), 055008.  doi: 10.1088/1361-6420/abdd46.

[40]

J. Zhang and K. Chen, A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution, SIAM J. Imaging Sci., 8 (2015), 2487-2518.  doi: 10.1137/14097121X.

[41]

J. ZhangZ. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vis., 43 (2012), 39-49.  doi: 10.1007/s10851-011-0285-z.

[42]

X. ZhangM. Bai and M. K. Ng, Nonconvex-TV based image restoration with impulse noise removal, SIAM J. Imaging Sci., 10 (2017), 1627-1667.  doi: 10.1137/16M1076034.

Figure 1.  Original images. (a): 'Boat', (b): 'House', (c): 'Cameraman', (d): 'Goldhill', (e): 'Plate', (f): 'Bridge', (g): 'Barbara', (h): 'Mandrill'
Figure 2.  Comparison of results by different methods applied on images degraded by Average blur and mixed Gaussian, salt-and-pepper noise. First column (a, e, i, m): the degraded images; Second to fourth columns: the restoration results by AOP, L12TV and our proposed algorithm respectively
Figure 3.  Comparison of results by different methods applied on images degraded by Gaussian blur and mixed Gaussian, salt-and-pepper noise. First column (a, e, i, m): the degraded images; Second to fourth columns: the restoration results by AOP, L12TV and our proposed algorithm respectively
Figure 4.  Comparison of results by different methods applied on images degraded by Average blur and mixed Gaussian, random-valued noise. First column (a, e, i, m): the degraded images; Second to fourth columns: the restoration results by AOP, L12TV and our proposed algorithm respectively
Figure 5.  Comparison of results by different methods applied on images degraded by Gaussian blur and mixed Gaussian, random-valued noise. First column (a, e, i, m): the degraded images; Second to fourth columns: the restoration results by AOP, L12TV and our proposed algorithm respectively
Table 1.  PSNR(dB) and SSIM results for images corrupted by Average blur and mixing noise (Gaussian salt-and-pepper impulse)
Image Blur/BSNR/SP Evaluation AOP L12TV Ours
Boat AB(5, 5)/40/30% PSNR 26.54 29.56 32.77
OPSNR=10.09 SSIM 0.8481 0.8901 0.9274
AB(7, 7)/30/50% PSNR 24.15 25.84 27.22
OPSNR=7.88 SSIM 0.7430 0.7796 0.8189
House AB(5, 5)/40/30% PSNR 30.18 33.84 36.45
OPSNR=10.67 SSIM 0.8511 0.8879 0.9131
AB(7, 7)/30/50% PSNR 27.86 30.50 31.97
OPSNR=8.43 SSIM 0.8105 0.8353 0.8554
Cameraman AB(5, 5)/40/30% PSNR 26.96 25.98 30.22
OPSNR=10.18 SSIM 0.8512 0.8324 0.8904
AB(7, 7)/30/50% PSNR 24.71 24.34 25.57
OPSNR=8.02 SSIM 0.7889 0.7671 0.7794
Goldhill AB(5, 5)/40/30% PSNR 25.94 27.25 30.74
OPSNR=10.51 SSIM 0.7032 0.7335 0.8663
AB(7, 7)/30/50% PSNR 24.07 25.73 27.02
OPSNR=8.32 SSIM 0.5954 0.6374 0.7074
Plate AB(5, 5)/40/30% PSNR 24.74 29.08 32.26
OPSNR=10.05 SSIM 0.9067 0.9334 0.9479
AB(7, 7)/30/50% PSNR 22.29 24.34 26.29
OPSNR=7.89 SSIM 0.8227 0.8352 0.8591
Bridge AB(5, 5)/40/30% PSNR 24.85 25.10 28.41
OPSNR=10.34 SSIM 0.7398 0.7031 0.8611
AB(7, 7)/30/50% PSNR 23.33 23.84 24.86
OPSNR=8.17 SSIM 0.6098 0.6002 0.6682
Barbara AB(5, 5)/40/30% PSNR 25.33 24.67 27.96
OPSNR=10.49 SSIM 0.7546 0.7204 0.8355
AB(7, 7)/30/50% PSNR 23.83 24.01 24.78
OPSNR=8.36 SSIM 0.6660 0.6628 0.6936
Mandrill AB(5, 5)/40/30% PSNR 23.34 22.27 26.59
OPSNR=10.53 SSIM 0.6872 0.5949 0.8469
AB(7, 7)/30/50% PSNR 21.73 21.63 23.05
OPSNR=8.44 SSIM 0.5249 0.5195 0.6368
Image Blur/BSNR/SP Evaluation AOP L12TV Ours
Boat AB(5, 5)/40/30% PSNR 26.54 29.56 32.77
OPSNR=10.09 SSIM 0.8481 0.8901 0.9274
AB(7, 7)/30/50% PSNR 24.15 25.84 27.22
OPSNR=7.88 SSIM 0.7430 0.7796 0.8189
House AB(5, 5)/40/30% PSNR 30.18 33.84 36.45
OPSNR=10.67 SSIM 0.8511 0.8879 0.9131
AB(7, 7)/30/50% PSNR 27.86 30.50 31.97
OPSNR=8.43 SSIM 0.8105 0.8353 0.8554
Cameraman AB(5, 5)/40/30% PSNR 26.96 25.98 30.22
OPSNR=10.18 SSIM 0.8512 0.8324 0.8904
AB(7, 7)/30/50% PSNR 24.71 24.34 25.57
OPSNR=8.02 SSIM 0.7889 0.7671 0.7794
Goldhill AB(5, 5)/40/30% PSNR 25.94 27.25 30.74
OPSNR=10.51 SSIM 0.7032 0.7335 0.8663
AB(7, 7)/30/50% PSNR 24.07 25.73 27.02
OPSNR=8.32 SSIM 0.5954 0.6374 0.7074
Plate AB(5, 5)/40/30% PSNR 24.74 29.08 32.26
OPSNR=10.05 SSIM 0.9067 0.9334 0.9479
AB(7, 7)/30/50% PSNR 22.29 24.34 26.29
OPSNR=7.89 SSIM 0.8227 0.8352 0.8591
Bridge AB(5, 5)/40/30% PSNR 24.85 25.10 28.41
OPSNR=10.34 SSIM 0.7398 0.7031 0.8611
AB(7, 7)/30/50% PSNR 23.33 23.84 24.86
OPSNR=8.17 SSIM 0.6098 0.6002 0.6682
Barbara AB(5, 5)/40/30% PSNR 25.33 24.67 27.96
OPSNR=10.49 SSIM 0.7546 0.7204 0.8355
AB(7, 7)/30/50% PSNR 23.83 24.01 24.78
OPSNR=8.36 SSIM 0.6660 0.6628 0.6936
Mandrill AB(5, 5)/40/30% PSNR 23.34 22.27 26.59
OPSNR=10.53 SSIM 0.6872 0.5949 0.8469
AB(7, 7)/30/50% PSNR 21.73 21.63 23.05
OPSNR=8.44 SSIM 0.5249 0.5195 0.6368
Table 2.  PSNR(dB) and SSIM results for images corrupted by Gaussian blur and mixing noise (Gaussian and salt-and-pepper impulse)
Image Blur/BSNR/SP Evaluation AOP L12TV Ours
Boat GB(5, 2)/40/30% PSNR 27.26 29.73 32.38
OPSNR=10.11 SSIM 0.8689 0.8964 0.9255
GB(7, 3)/30/50% PSNR 24.82 26.01 27.36
OPSNR=7.89 SSIM 0.7734 0.7842 0.8195
House GB(5, 2)/40/30% PSNR 30.67 33.91 36.29
OPSNR=10.68 SSIM 0.8578 0.8910 0.9122
GB(7, 3)/30/50% PSNR 28.40 30.38 31.55
OPSNR=8.44 SSIM 0.8169 0.8329 0.8518
Cameraman GB(5, 2)/40/30% PSNR 26.23 27.39 29.65
OPSNR=10.21 SSIM 0.8460 0.8659 0.8937
GB(7, 3)/30/50% PSNR 24.47 24.41 25.29
OPSNR=8.03 SSIM 0.7837 0.7692 0.7735
Goldhill GB(5, 2)/40/30% PSNR 26.13 28.43 30.38
OPSNR=10.53 SSIM 0.7132 0.7940 0.8551
GB(7, 3)/30/50% PSNR 24.73 25.87 26.81
OPSNR=8.33 SSIM 0.6120 0.6504 0.6986
Plate GB(5, 2)/40/30% PSNR 24.99 29.62 31.90
OPSNR=10.09 SSIM 0.9129 0.9372 0.9493
GB(7, 3)/30/50% PSNR 22.41 24.74 26.38
OPSNR=7.91 SSIM 0.8483 0.8529 0.8781
Bridge GB(5, 2)/40/30% PSNR 25.09 25.84 28.34
OPSNR=10.36 SSIM 0.7444 0.7570 0.8580
GB(7, 3)/30/50% PSNR 23.51 23.91 24.82
OPSNR=8.18 SSIM 0.6143 0.6118 0.6686
Barbara GB(5, 2)/40/30% PSNR 24.52 24.87 27.76
OPSNR=10.50 SSIM 0.7236 0.7399 0.8261
GB(7, 3)/30/50% PSNR 23.92 24.08 24.68
OPSNR=8.37 SSIM 0.6705 0.6669 0.6931
Mandrill GB(5, 2)/40/30% PSNR 22.91 23.09 26.52
OPSNR=10.57 SSIM 0.6512 0.6783 0.8413
GB(7, 3)/30/50% PSNR 21.45 21.65 22.83
OPSNR=8.45 SSIM 0.4988 0.5265 0.6160
Image Blur/BSNR/SP Evaluation AOP L12TV Ours
Boat GB(5, 2)/40/30% PSNR 27.26 29.73 32.38
OPSNR=10.11 SSIM 0.8689 0.8964 0.9255
GB(7, 3)/30/50% PSNR 24.82 26.01 27.36
OPSNR=7.89 SSIM 0.7734 0.7842 0.8195
House GB(5, 2)/40/30% PSNR 30.67 33.91 36.29
OPSNR=10.68 SSIM 0.8578 0.8910 0.9122
GB(7, 3)/30/50% PSNR 28.40 30.38 31.55
OPSNR=8.44 SSIM 0.8169 0.8329 0.8518
Cameraman GB(5, 2)/40/30% PSNR 26.23 27.39 29.65
OPSNR=10.21 SSIM 0.8460 0.8659 0.8937
GB(7, 3)/30/50% PSNR 24.47 24.41 25.29
OPSNR=8.03 SSIM 0.7837 0.7692 0.7735
Goldhill GB(5, 2)/40/30% PSNR 26.13 28.43 30.38
OPSNR=10.53 SSIM 0.7132 0.7940 0.8551
GB(7, 3)/30/50% PSNR 24.73 25.87 26.81
OPSNR=8.33 SSIM 0.6120 0.6504 0.6986
Plate GB(5, 2)/40/30% PSNR 24.99 29.62 31.90
OPSNR=10.09 SSIM 0.9129 0.9372 0.9493
GB(7, 3)/30/50% PSNR 22.41 24.74 26.38
OPSNR=7.91 SSIM 0.8483 0.8529 0.8781
Bridge GB(5, 2)/40/30% PSNR 25.09 25.84 28.34
OPSNR=10.36 SSIM 0.7444 0.7570 0.8580
GB(7, 3)/30/50% PSNR 23.51 23.91 24.82
OPSNR=8.18 SSIM 0.6143 0.6118 0.6686
Barbara GB(5, 2)/40/30% PSNR 24.52 24.87 27.76
OPSNR=10.50 SSIM 0.7236 0.7399 0.8261
GB(7, 3)/30/50% PSNR 23.92 24.08 24.68
OPSNR=8.37 SSIM 0.6705 0.6669 0.6931
Mandrill GB(5, 2)/40/30% PSNR 22.91 23.09 26.52
OPSNR=10.57 SSIM 0.6512 0.6783 0.8413
GB(7, 3)/30/50% PSNR 21.45 21.65 22.83
OPSNR=8.45 SSIM 0.4988 0.5265 0.6160
Table 3.  PSNR(dB) and SSIM results for images corrupted by Average blur and mixing noise (Gaussian and random-valued impulse)
Image Blur/BSNR/RV Evaluation AOP L12TV Ours
Boat AB(5, 5)/40/30% PSNR 25.13 29.17 32.21
OPSNR=13.17 SSIM 0.7588 0.8852 0.9205
AB(7, 7)/30/50% PSNR 21.54 24.14 26.53
OPSNR=11.02 SSIM 0.5977 0.7095 0.7660
House AB(5, 5)/40/30% PSNR 29.00 33.97 36.18
OPSNR=14.40 SSIM 0.8149 0.8904 0.9111
AB(7, 7)/30/50% PSNR 24.80 29.21 31.22
OPSNR=12.20 SSIM 0.7270 0.8192 0.8476
Cameraman AB(5, 5)/40/30% PSNR 24.26 26.79 29.66
OPSNR=13.32 SSIM 0.7506 0.8518 0.8724
AB(7, 7)/30/50% PSNR 20.99 22.36 24.91
OPSNR=11.23 SSIM 0.6189 0.6823 0.7013
Goldhill AB(5, 5)/40/30% PSNR 25.11 28.39 30.26
OPSNR=14.07 SSIM 0.6238 0.7866 0.8547
AB(7, 7)/30/50% PSNR 22.44 24.87 26.45
OPSNR=11.91 SSIM 0.4618 0.6034 0.6782
Plate AB(5, 5)/40/30% PSNR 23.11 29.30 31.62
OPSNR=13.07 SSIM 0.8340 0.9315 0.9388
AB(7, 7)/30/50% PSNR 19.42 22.80 25.37
OPSNR=10.95 SSIM 0.6408 0.7765 0.8130
Bridge AB(5, 5)/40/30% PSNR 23.58 26.35 28.01
OPSNR=13.69 SSIM 0.6007 0.7798 0.8493
AB(7, 7)/30/50% PSNR 21.20 22.95 24.39
OPSNR=11.60 SSIM 0.4046 0.5466 0.6416
Barbara AB(5, 5)/40/30% PSNR 23.94 24.83 27.85
OPSNR=14.04 SSIM 0.6576 0.7294 0.8275
AB(7, 7)/30/50% PSNR 22.61 23.70 24.48
OPSNR=12.03 SSIM 0.5762 0.6418 0.6445
Mandrill AB(5, 5)/40/30% PSNR 21.81 22.82 25.70
OPSNR=14.13 SSIM 0.5348 0.6464 0.8178
AB(7, 7)/30/50% PSNR 20.14 21.22 22.48
OPSNR=12.22 SSIM 0.3396 0.4700 0.5959
Image Blur/BSNR/RV Evaluation AOP L12TV Ours
Boat AB(5, 5)/40/30% PSNR 25.13 29.17 32.21
OPSNR=13.17 SSIM 0.7588 0.8852 0.9205
AB(7, 7)/30/50% PSNR 21.54 24.14 26.53
OPSNR=11.02 SSIM 0.5977 0.7095 0.7660
House AB(5, 5)/40/30% PSNR 29.00 33.97 36.18
OPSNR=14.40 SSIM 0.8149 0.8904 0.9111
AB(7, 7)/30/50% PSNR 24.80 29.21 31.22
OPSNR=12.20 SSIM 0.7270 0.8192 0.8476
Cameraman AB(5, 5)/40/30% PSNR 24.26 26.79 29.66
OPSNR=13.32 SSIM 0.7506 0.8518 0.8724
AB(7, 7)/30/50% PSNR 20.99 22.36 24.91
OPSNR=11.23 SSIM 0.6189 0.6823 0.7013
Goldhill AB(5, 5)/40/30% PSNR 25.11 28.39 30.26
OPSNR=14.07 SSIM 0.6238 0.7866 0.8547
AB(7, 7)/30/50% PSNR 22.44 24.87 26.45
OPSNR=11.91 SSIM 0.4618 0.6034 0.6782
Plate AB(5, 5)/40/30% PSNR 23.11 29.30 31.62
OPSNR=13.07 SSIM 0.8340 0.9315 0.9388
AB(7, 7)/30/50% PSNR 19.42 22.80 25.37
OPSNR=10.95 SSIM 0.6408 0.7765 0.8130
Bridge AB(5, 5)/40/30% PSNR 23.58 26.35 28.01
OPSNR=13.69 SSIM 0.6007 0.7798 0.8493
AB(7, 7)/30/50% PSNR 21.20 22.95 24.39
OPSNR=11.60 SSIM 0.4046 0.5466 0.6416
Barbara AB(5, 5)/40/30% PSNR 23.94 24.83 27.85
OPSNR=14.04 SSIM 0.6576 0.7294 0.8275
AB(7, 7)/30/50% PSNR 22.61 23.70 24.48
OPSNR=12.03 SSIM 0.5762 0.6418 0.6445
Mandrill AB(5, 5)/40/30% PSNR 21.81 22.82 25.70
OPSNR=14.13 SSIM 0.5348 0.6464 0.8178
AB(7, 7)/30/50% PSNR 20.14 21.22 22.48
OPSNR=12.22 SSIM 0.3396 0.4700 0.5959
Table 4.  PSNR(dB) and SSIM results for images corrupted by Gaussian blur and mixing noise (Gaussian and random-valued impulse)
Image Blur/BSNR/RV Evaluation AOP L12TV Ours
Boat GB(5, 2)/40/30% PSNR 25.38 29.10 31.90
OPSNR=13.21 SSIM 0.7679 0.8853 0.9191
GB(7, 3)/30/50% PSNR 21.92 24.28 26.85
OPSNR=11.05 SSIM 0.6009 0.7141 0.8044
House GB(5, 2)/40/30% PSNR 29.39 33.52 35.83
OPSNR=14.43 SSIM 0.8212 0.8857 0.9081
GB(7, 3)/30/50% PSNR 25.17 29.00 30.91
OPSNR=12.21 SSIM 0.7330 0.8172 0.8399
Cameraman GB(5, 2)/40/30% PSNR 24.41 26.75 29.19
OPSNR=13.38 SSIM 0.7589 0.8524 0.8870
GB(7, 3)/30/50% PSNR 21.11 22.40 24.83
OPSNR=11.26 SSIM 0.6229 0.6819 0.7766
Goldhill GB(5, 2)/40/30% PSNR 25.30 28.31 29.93
OPSNR=14.11 SSIM 0.6374 0.7945 0.8441
GB(7, 3)/30/50% PSNR 22.68 24.84 26.52
OPSNR=11.92 SSIM 0.4746 0.6062 0.6875
Plate GB(5, 2)/40/30% PSNR 23.19 29.32 31.47
OPSNR=13.15 SSIM 0.8402 0.9323 0.9451
GB(7, 3)/30/50% PSNR 19.71 23.22 25.71
OPSNR=11.00 SSIM 0.6448 0.7939 0.8657
Bridge GB(5, 2)/40/30% PSNR 23.77 26.00 27.94
OPSNR=13.75 SSIM 0.6231 0.7668 0.8462
GB(7, 3)/30/50% PSNR 21.31 22.94 24.48
OPSNR=11.62 SSIM 0.4158 0.5495 0.6523
Barbara GB(5, 2)/40/30% PSNR 23.95 24.62 26.81
OPSNR=14.07 SSIM 0.6619 0.7206 0.8073
GB(7, 3)/30/50% PSNR 22.66 23.73 24.47
OPSNR=12.03 SSIM 0.5799 0.6450 0.6865
Mandrill GB(5, 2)/40/30% PSNR 21.92 23.54 25.68
OPSNR=14.21 SSIM 0.5644 0.7108 0.8138
GB(7, 3)/30/50% PSNR 20.20 21.36 22.34
OPSNR=12.24 SSIM 0.3498 0.4962 0.5787
Image Blur/BSNR/RV Evaluation AOP L12TV Ours
Boat GB(5, 2)/40/30% PSNR 25.38 29.10 31.90
OPSNR=13.21 SSIM 0.7679 0.8853 0.9191
GB(7, 3)/30/50% PSNR 21.92 24.28 26.85
OPSNR=11.05 SSIM 0.6009 0.7141 0.8044
House GB(5, 2)/40/30% PSNR 29.39 33.52 35.83
OPSNR=14.43 SSIM 0.8212 0.8857 0.9081
GB(7, 3)/30/50% PSNR 25.17 29.00 30.91
OPSNR=12.21 SSIM 0.7330 0.8172 0.8399
Cameraman GB(5, 2)/40/30% PSNR 24.41 26.75 29.19
OPSNR=13.38 SSIM 0.7589 0.8524 0.8870
GB(7, 3)/30/50% PSNR 21.11 22.40 24.83
OPSNR=11.26 SSIM 0.6229 0.6819 0.7766
Goldhill GB(5, 2)/40/30% PSNR 25.30 28.31 29.93
OPSNR=14.11 SSIM 0.6374 0.7945 0.8441
GB(7, 3)/30/50% PSNR 22.68 24.84 26.52
OPSNR=11.92 SSIM 0.4746 0.6062 0.6875
Plate GB(5, 2)/40/30% PSNR 23.19 29.32 31.47
OPSNR=13.15 SSIM 0.8402 0.9323 0.9451
GB(7, 3)/30/50% PSNR 19.71 23.22 25.71
OPSNR=11.00 SSIM 0.6448 0.7939 0.8657
Bridge GB(5, 2)/40/30% PSNR 23.77 26.00 27.94
OPSNR=13.75 SSIM 0.6231 0.7668 0.8462
GB(7, 3)/30/50% PSNR 21.31 22.94 24.48
OPSNR=11.62 SSIM 0.4158 0.5495 0.6523
Barbara GB(5, 2)/40/30% PSNR 23.95 24.62 26.81
OPSNR=14.07 SSIM 0.6619 0.7206 0.8073
GB(7, 3)/30/50% PSNR 22.66 23.73 24.47
OPSNR=12.03 SSIM 0.5799 0.6450 0.6865
Mandrill GB(5, 2)/40/30% PSNR 21.92 23.54 25.68
OPSNR=14.21 SSIM 0.5644 0.7108 0.8138
GB(7, 3)/30/50% PSNR 20.20 21.36 22.34
OPSNR=12.24 SSIM 0.3498 0.4962 0.5787
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