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April  2022, 21(4): 1249-1291. doi: 10.3934/cpaa.2022018

A non-convex non-smooth bi-level parameter learning for impulse and Gaussian noise mixture removing

Mourad Nachaoui 1, , Lekbir Afraites 1, , Aissam Hadri 2, and  Amine Laghrib 1,,

1. 

EMI FST Béni-Mellal, Université Sultan Moulay Slimane, Maroc
2. 

Laboratoire SIE, Université IBN ZOHR Agadir

*Corresponding author

Received  May 2021 Revised  October 2021 Published  April 2022 Early access  February 2022

This paper introduce a novel optimization procedure to reduce mixture of Gaussian and impulse noise from images. This technique exploits a non-convex PDE-constrained characterized by a fractional-order operator. The used non-convex term facilitated the impulse component approximation controlled by a spatial parameter $ \gamma $. A non-convex and non-smooth bi-level optimization framework with a modified projected gradient algorithm is then proposed in order to learn the parameter $ \gamma $. Denoising tests confirm that the non-convex term and learned parameter $ \gamma $ lead in general to an improved reconstruction when compared to results of convex norm and manual parameter $ \lambda $ choice.

Keywords: Non-convex function, mixture noise, learning parameter, bi-level optimization.
Mathematics Subject Classification: Primary: 65K10; 90C26; Secondary: 35R11; 68U10.
Citation: Mourad Nachaoui, Lekbir Afraites, Aissam Hadri, Amine Laghrib. A non-convex non-smooth bi-level parameter learning for impulse and Gaussian noise mixture removing. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1249-1291. doi: 10.3934/cpaa.2022018
References:
[1]

L. Afraites, A. Hadri and A. Laghrib, A denoising model adapted for impulse and gaussian noises using a constrained-PDE, Inver. Prob., 36 (2020), 40 pp. doi: 10.1088/1361-6420/ab5178.

[2]

J. P. Aubin, Un théorème de compacité, Acad. Sci. Paris, 256 (1963), 5042-5044. 

[3]

E. M. Bednarczuk, L. I. Minchenko and K. E. Rutkowski, On lipschitz-like continuity of a class of set-valued mappings, Optimization, 69 (2020), 2535-2549.  doi: 10.1080/02331934.2019.1696339.

[4]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.

[5]

H. C. Burger, B. Schölkopf and S. Harmeling, Removing noise from astronomical images using a pixel-specific noise model, in 2011 IEEE International Conference on Computational Photography (ICCP), (2011), 1–8.

[6]

L. Calatroni, J. C. De Los Reyes and C.-B. Schönlieb, Infimal convolution of data discrepancies for mixed noise removal, SIAM J. Imag. Sci., 10 (2017), 1196-1233.  doi: 10.1137/16M1101684.

[7]

L. Calatroni and K. Papafitsoros, Analysis and automatic parameter selection of a variational model for mixed gaussian and salt-and-pepper noise removal, Inver. Prob., 35 (2019), 37 pp. doi: 10.1088/1361-6420/ab291a.

[8]

A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.  doi: 10.1007/s002110050258.

[9]

T. F. Chan and S. Esedoglu, Aspects of total variation regularized $\ell_1$ function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.  doi: 10.1137/040604297.

[10]

F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.

[11]

F. Clarke, R. J. Stern and P. R. Wolenski, Subgradient criteria for monotonicity, the lipschitz condition, and convexity, Canad. J. Math., 45 (1993), 1167-1183.  doi: 10.4153/CJM-1993-065-x.

[12]

J. C. De los Reyes, C. B. Schönlieb and T. Valkonen, Bilevel parameter learning for higher-order total variation regularisation models, J. Math. Imag. Vis., 57 (2017), 1-25.  doi: 10.1007/s10851-016-0662-8.

[13]

S. Dempe, F. Harder, P. Mehlitz and G. Wachsmuth, Solving inverse optimal control problems via value functions to global optimality, J. Glob. Optim., 74 (2019), 297-325.  doi: 10.1007/s10898-019-00758-1.

[14]

S. Dempe, V. Kalashnikov, G. A. Pérez-Valdés and N. Kalashnykova, Bilevel Programming Problems, Energy Systems Theory, algorithms and applications to energy networks, Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-45827-3.

[15]

I. El Mourabit, M. El Rhabi, A. Hakim, A. Laghrib and E. Moreau, A new denoising model for multi-frame super-resolution image reconstruction, Sign. Process., 132 (2017), 51-65. 

[16]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Mult. Model. Simul., 7 (2009), 1005-1028.  doi: 10.1137/070698592.

[17]

F. Harder and G. Wachsmuth, Optimality conditions for a class of inverse optimal control problems with partial differential equations, Optimization, 68 (2019), 615-643.  doi: 10.1080/02331934.2018.1495205.

[18]

M. Hintermüller, K. Papafitsoros, C. N. Rautenberg and H. Sun, Dualization and automatic distributed parameter selection of total generalized variation via bilevel optimization, 2020.,

[19]

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

[20]

F. Knoll, K. Bredies, T. Pock and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magnet. Resonan. Med., 65 (2011), 480-491. 

[21]

P. Konstantin and G. Mattias, Necessary conditions for a class of bilevel optimal control problems exploiting the value function, Pure Appl. Funct. Anal., 1 (2016), 505-524.  doi: 10.1260/174830107783133851.

[22]

K. Kunisch and T. Pock, A bilevel optimization approach for parameter learning in variational models, SIAM J. Imag. Sci., 6 (2013), 938-983.  doi: 10.1137/120882706.

[23]

A. Laghrib, A. Ben-Loghfyry, A. Hadri and A. Hakim, A nonconvex fractional order variational model for multi-frame image super-resolution, Sign. Process., 67 (2018), 1-11. 

[24]

G. H. Lin, M. Xu and J. J. Ye, On solving simple bilevel programs with a nonconvex lower level program, Math. Program., 144 (2014), 277-305.  doi: 10.1007/s10107-013-0633-4.

[25]

J. V. Manjón, J. Carbonell-Caballero, J. J. Lull, G. García-Martí, L. Martí-Bonmatí and M. Robles, MRI denoising using non-local means, Med. Imag. Anal., 12 (2008), 514-523. 

[26]

P. Mehlitz, L. I. Minchenko and A. B. Zemkoho, A note on partial calmness for bilevel optimization problems with linear structures at the lower level, Optim. Lett., 15 (2021), 1277-1291.  doi: 10.1007/s11590-020-01636-6.

[27]

A. Mitsos, P. Lemonidis and P. I. Barton, Global solution of bilevel programs with a nonconvex inner program, J. Glob. Optim., 42 (2008), 475-513.  doi: 10.1007/s10898-007-9260-z.

[28]

M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imag. Vis., 20 (2004), 99-120.  doi: 10.1023/B:JMIV.0000011920.58935.9c.

[29]

P. Ochs, R. Ranftl, T. Brox and T. Pock, Techniques for gradient based bilevel optimization with nonsmooth lower level problems, J. Math. Imag. Vis., 56 (2016), 175-194.  doi: 10.1007/s10851-016-0663-7.

[30]

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.

[31]

J. Simon, Compact sets in the space $l^p (0, t; b)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[32]

T. Valkonen, K. Bredies and F. Knoll, Total generalized variation in diffusion tensor imaging, SIAM J. Imag. Sci., 6 (2013), 487-525.  doi: 10.1137/120867172.

[33]

J. J. Ye, D. L. Zhu and Q. J. Zhu, Exact penalization and necessary optimality conditions for generalized bilevel programming problems, SIAM J. Optim., 7 (1997), 481-507.  doi: 10.1137/S1052623493257344.

[34]

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

[35]

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

[36]

X. L. Zhao, F. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM J. Imag. Sci., 7 (2014), 456-475.  doi: 10.1137/13092472X.

show all references

References:
[1]

L. Afraites, A. Hadri and A. Laghrib, A denoising model adapted for impulse and gaussian noises using a constrained-PDE, Inver. Prob., 36 (2020), 40 pp. doi: 10.1088/1361-6420/ab5178.

[2]

J. P. Aubin, Un théorème de compacité, Acad. Sci. Paris, 256 (1963), 5042-5044. 

[3]

E. M. Bednarczuk, L. I. Minchenko and K. E. Rutkowski, On lipschitz-like continuity of a class of set-valued mappings, Optimization, 69 (2020), 2535-2549.  doi: 10.1080/02331934.2019.1696339.

[4]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.

[5]

H. C. Burger, B. Schölkopf and S. Harmeling, Removing noise from astronomical images using a pixel-specific noise model, in 2011 IEEE International Conference on Computational Photography (ICCP), (2011), 1–8.

[6]

L. Calatroni, J. C. De Los Reyes and C.-B. Schönlieb, Infimal convolution of data discrepancies for mixed noise removal, SIAM J. Imag. Sci., 10 (2017), 1196-1233.  doi: 10.1137/16M1101684.

[7]

L. Calatroni and K. Papafitsoros, Analysis and automatic parameter selection of a variational model for mixed gaussian and salt-and-pepper noise removal, Inver. Prob., 35 (2019), 37 pp. doi: 10.1088/1361-6420/ab291a.

[8]

A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.  doi: 10.1007/s002110050258.

[9]

T. F. Chan and S. Esedoglu, Aspects of total variation regularized $\ell_1$ function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.  doi: 10.1137/040604297.

[10]

F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.

[11]

F. Clarke, R. J. Stern and P. R. Wolenski, Subgradient criteria for monotonicity, the lipschitz condition, and convexity, Canad. J. Math., 45 (1993), 1167-1183.  doi: 10.4153/CJM-1993-065-x.

[12]

J. C. De los Reyes, C. B. Schönlieb and T. Valkonen, Bilevel parameter learning for higher-order total variation regularisation models, J. Math. Imag. Vis., 57 (2017), 1-25.  doi: 10.1007/s10851-016-0662-8.

[13]

S. Dempe, F. Harder, P. Mehlitz and G. Wachsmuth, Solving inverse optimal control problems via value functions to global optimality, J. Glob. Optim., 74 (2019), 297-325.  doi: 10.1007/s10898-019-00758-1.

[14]

S. Dempe, V. Kalashnikov, G. A. Pérez-Valdés and N. Kalashnykova, Bilevel Programming Problems, Energy Systems Theory, algorithms and applications to energy networks, Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-45827-3.

[15]

I. El Mourabit, M. El Rhabi, A. Hakim, A. Laghrib and E. Moreau, A new denoising model for multi-frame super-resolution image reconstruction, Sign. Process., 132 (2017), 51-65. 

[16]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Mult. Model. Simul., 7 (2009), 1005-1028.  doi: 10.1137/070698592.

[17]

F. Harder and G. Wachsmuth, Optimality conditions for a class of inverse optimal control problems with partial differential equations, Optimization, 68 (2019), 615-643.  doi: 10.1080/02331934.2018.1495205.

[18]

M. Hintermüller, K. Papafitsoros, C. N. Rautenberg and H. Sun, Dualization and automatic distributed parameter selection of total generalized variation via bilevel optimization, 2020.,

[19]

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

[20]

F. Knoll, K. Bredies, T. Pock and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magnet. Resonan. Med., 65 (2011), 480-491. 

[21]

P. Konstantin and G. Mattias, Necessary conditions for a class of bilevel optimal control problems exploiting the value function, Pure Appl. Funct. Anal., 1 (2016), 505-524.  doi: 10.1260/174830107783133851.

[22]

K. Kunisch and T. Pock, A bilevel optimization approach for parameter learning in variational models, SIAM J. Imag. Sci., 6 (2013), 938-983.  doi: 10.1137/120882706.

[23]

A. Laghrib, A. Ben-Loghfyry, A. Hadri and A. Hakim, A nonconvex fractional order variational model for multi-frame image super-resolution, Sign. Process., 67 (2018), 1-11. 

[24]

G. H. Lin, M. Xu and J. J. Ye, On solving simple bilevel programs with a nonconvex lower level program, Math. Program., 144 (2014), 277-305.  doi: 10.1007/s10107-013-0633-4.

[25]

J. V. Manjón, J. Carbonell-Caballero, J. J. Lull, G. García-Martí, L. Martí-Bonmatí and M. Robles, MRI denoising using non-local means, Med. Imag. Anal., 12 (2008), 514-523. 

[26]

P. Mehlitz, L. I. Minchenko and A. B. Zemkoho, A note on partial calmness for bilevel optimization problems with linear structures at the lower level, Optim. Lett., 15 (2021), 1277-1291.  doi: 10.1007/s11590-020-01636-6.

[27]

A. Mitsos, P. Lemonidis and P. I. Barton, Global solution of bilevel programs with a nonconvex inner program, J. Glob. Optim., 42 (2008), 475-513.  doi: 10.1007/s10898-007-9260-z.

[28]

M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imag. Vis., 20 (2004), 99-120.  doi: 10.1023/B:JMIV.0000011920.58935.9c.

[29]

P. Ochs, R. Ranftl, T. Brox and T. Pock, Techniques for gradient based bilevel optimization with nonsmooth lower level problems, J. Math. Imag. Vis., 56 (2016), 175-194.  doi: 10.1007/s10851-016-0663-7.

[30]

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.

[31]

J. Simon, Compact sets in the space $l^p (0, t; b)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[32]

T. Valkonen, K. Bredies and F. Knoll, Total generalized variation in diffusion tensor imaging, SIAM J. Imag. Sci., 6 (2013), 487-525.  doi: 10.1137/120867172.

[33]

J. J. Ye, D. L. Zhu and Q. J. Zhu, Exact penalization and necessary optimality conditions for generalized bilevel programming problems, SIAM J. Optim., 7 (1997), 481-507.  doi: 10.1137/S1052623493257344.

[34]

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

[35]

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

[36]

X. L. Zhao, F. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM J. Imag. Sci., 7 (2014), 456-475.  doi: 10.1137/13092472X.

Figure 1.  The evolution of the two functions: ET and Geman with respect to various values of the parameter $ \gamma $. Note that the chosen values for the ET case are very small compared to the ones chosen in the case of Geman function
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Figure 2.  TGV$ ^2 $ regularization denoising with non-convex Geman function using different choices of the parameter $ \gamma $ for the Parrot image with the respective surfaces associated to a part of its beak. Note that the impulse noise is added with parameter $ 0.12 $
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Figure 3.  TGV$ ^2 $ regularization denoising result using different choices of the parameter $ \gamma $ in the ET function for the Parrot image with the respective surfaces associated to a part of its beak. Note that the impulse noise is added with parameter $ 0.12 $
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Figure 4.  The evolution of the restored image with respect to the computed parameter $ \gamma $. The first line presents the restored image compared with the noisy and clean one with the associated 3D surfaces in the same order of $ u(20:60,120:160) $, while the second line presents the approximate $ \gamma $ by the proposed bi-level approach. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.03 $ and $ r = 0.5 $
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Figure 5.  The evolution of the restored image with respect to the computed parameter $ \gamma $. The first line presents the restored image compared with the noisy and clean one with the associated 3D surfaces in the same order of $ u(30:70,110:150) $, while the second line presents the approximate $ \gamma $ by the proposed bi-level approach. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.03 $ and $ r = 0.6 $
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Figure 6.  The evolution of the restored image with respect to the computed parameter $ \gamma $. The first line presents the restored image compared with the noisy and clean one with the associated 3D surfaces in the same order of $ u(20:60,120:160) $, while the second line presents the impulse component and the approximate $ \gamma $ by the proposed bi-level approach. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.03 $ and $ r = 0.6 $
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Figure 7.  We present in the first row the original image, the noisy one and the respective 3D surface of a part from the image. The second row represents the restored images and the associated surfaces with two different values of $ \gamma $. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.03 $ and $ r = 0.4 $
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Figure 8.  Ten training images used for the computation of the optimal solution $ (\gamma, v, u) $
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Figure 9.  The comparison between the obtained image by the proposed bi-level approach with and without learning step. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.02 $ and $ r = 0.3 $
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Figure 10.  Five sample images of the 3D time of flight MR angiography of the circle of Willis from the OASIS MRI brain database: original images (upper row), noisy images (second row), free impulse images $ f_0 $ (third row) and optimal denoised images (bottom row)
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Figure 11.  The denoising process for the (Cameraman image) with different denoising methods. Note that we use the 3D representation of the image pixels $ u(40:100, 40:100) $ to better see the evolution of the piecewise constant regions
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Figure 12.  The denoising process for the (Tiger image) with different denoising methods. Note that we use the 3D representation of the image pixels $ u(10:30, 10:30) $ to better see the evolution of the smooth regions
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Figure 13.  The denoising process for the (Hot air balloon image) with different denoising methods. Note that we use the 3D representation of the image pixels $ u(10:80, 10:80) $ to better see the evolution of the transition from edges to smooth regions
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Figure 14.  The denoising process for the (Bridge image) using different restoration methods. Note that we use the 3D representation of the image pixels $ u(10:80, 10:80) $ to better see the evolution of the recovered sharp edges and also smooth regions
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Table 1.  The PSNR and SSIM values of ten selected images using our approach and compared to other denoising methods
Image Criterion Noise denoising algorithms
Noisy CCPDE [1] Nonconvex-TV [35] TV-IC [6] Our Method
Cameraman PSNR $ \sigma^2=0.02 $ and $ r=0.3 $ 24.84 31.26 31.02 31.96 32.44
SSIM 0.608 0.907 0.911 0.922 0.931
Tiger PSNR $ \sigma^2=0.03 $ and $ r=0.3 $ 21.27 26.77 26.82 27.06 27.13
SSIM 0.733 0.788 0.796 0.802 0.800
Hot air balloon PSNR $ \sigma^2=0.05 $ and $ r=0.4 $ 15.80 24.86 24.91 25.06 25.06
SSIM 0.433 0.605 0.614 0.617 0.622
Bridge PSNR $ \sigma^2=0.06 $ and $ r=0.5 $ 14.80 23.46 23.92 23.96 24.26
SSIM 0.333 0.555 0.564 0.547 0.592
Bird PSNR $ \sigma^2=0.03 $ and $ r=0.4 $ 18.63 26.56 26.88 26.77 26.98
SSIM 0.543 0.745 0.766 0.777 0.782
Lena PSNR $ \sigma^2=0.06 $ and $ r=0.6 $ 10.78 20.66 21.18 21.08 21.52
SSIM 0.303 0.555 0.563 0.577 0.598
Penguin PSNR $ \sigma^2=0.04 $ and $ r=0.1 $ 19.77 26.66 26.58 26.88 27.02
SSIM 0.483 0.685 0.693 0.704 0.715
Lion PSNR $ \sigma^2=0.01 $ and $ r=0.4 $ 16.42 24.86 25.14 25.48 25.92
SSIM 0.489 0.695 0.702 0.704 0.733
Bear PSNR $ \sigma^2=0.04 $ and $ r=0.4 $ 14.66 24.68 25.12 25.22 25.42
SSIM 0.433 0.555 0.566 0.583 0.608
Woman PSNR $ \sigma^2=0.05 $ and $ r=0.5 $ 13.67 18.55 24.88 24.95 25.07
SSIM 0.417 0.536 0.552 0.577 0.582
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Image Criterion Noise denoising algorithms
Noisy CCPDE [1] Nonconvex-TV [35] TV-IC [6] Our Method
Cameraman PSNR $ \sigma^2=0.02 $ and $ r=0.3 $ 24.84 31.26 31.02 31.96 32.44
SSIM 0.608 0.907 0.911 0.922 0.931
Tiger PSNR $ \sigma^2=0.03 $ and $ r=0.3 $ 21.27 26.77 26.82 27.06 27.13
SSIM 0.733 0.788 0.796 0.802 0.800
Hot air balloon PSNR $ \sigma^2=0.05 $ and $ r=0.4 $ 15.80 24.86 24.91 25.06 25.06
SSIM 0.433 0.605 0.614 0.617 0.622
Bridge PSNR $ \sigma^2=0.06 $ and $ r=0.5 $ 14.80 23.46 23.92 23.96 24.26
SSIM 0.333 0.555 0.564 0.547 0.592
Bird PSNR $ \sigma^2=0.03 $ and $ r=0.4 $ 18.63 26.56 26.88 26.77 26.98
SSIM 0.543 0.745 0.766 0.777 0.782
Lena PSNR $ \sigma^2=0.06 $ and $ r=0.6 $ 10.78 20.66 21.18 21.08 21.52
SSIM 0.303 0.555 0.563 0.577 0.598
Penguin PSNR $ \sigma^2=0.04 $ and $ r=0.1 $ 19.77 26.66 26.58 26.88 27.02
SSIM 0.483 0.685 0.693 0.704 0.715
Lion PSNR $ \sigma^2=0.01 $ and $ r=0.4 $ 16.42 24.86 25.14 25.48 25.92
SSIM 0.489 0.695 0.702 0.704 0.733
Bear PSNR $ \sigma^2=0.04 $ and $ r=0.4 $ 14.66 24.68 25.12 25.22 25.42
SSIM 0.433 0.555 0.566 0.583 0.608
Woman PSNR $ \sigma^2=0.05 $ and $ r=0.5 $ 13.67 18.55 24.88 24.95 25.07
SSIM 0.417 0.536 0.552 0.577 0.582
Table 2.  The elapsed CPU time (in seconds) during the computation of the algorithm of ten selected images using our approach and compared to other denoising methods
Image denoising algorithms
CCPDE [1] Nonconvex-TV [35] TV-IC [6] Our Method
Cameraman 44.24 21.66 35.99 52.64
Tiger 61.77 26.82 46.11 77.78
Hot air balloon 55.90 24.61 45.16 65.56
Bridge 54.80 25.56 43.99 64.36
Bird 68.33 28.96 46.07 76.38
Lena 47.73 22.26 31.88 61.22
Penguin 69.70 26.56 53.48 77.22
Lion 86.44 30.06 55.66 95.93
Bear 64.36 28.18 55.72 75.88
Woman 63.66 30.15 44.25 75.67
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Image denoising algorithms
CCPDE [1] Nonconvex-TV [35] TV-IC [6] Our Method
Cameraman 44.24 21.66 35.99 52.64
Tiger 61.77 26.82 46.11 77.78
Hot air balloon 55.90 24.61 45.16 65.56
Bridge 54.80 25.56 43.99 64.36
Bird 68.33 28.96 46.07 76.38
Lena 47.73 22.26 31.88 61.22
Penguin 69.70 26.56 53.48 77.22
Lion 86.44 30.06 55.66 95.93
Bear 64.36 28.18 55.72 75.88
Woman 63.66 30.15 44.25 75.67
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