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doi: 10.3934/ipi.2022001
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A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification

1. 

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

2. 

Laboratoire SIE, Université IBN ZOHR Agadir, Maroc

3. 

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

* Corresponding author: Amine Laghrib

Received  April 2021 Revised  October 2021 Early access January 2022

We propose a new variational framework to remove a mixture of Gaussian and impulse noise from images. This framework is based on a non-convex PDE-constrained with a fractional-order operator. The non-convex norm is applied to the impulse component controlled by a weighted parameter $ \gamma $, which depends on the level of the impulse noise and image feature. Furthermore, the fractional operator is used to preserve image texture and edges. In a first part, we study the theoretical properties of the proposed PDE-constrained, and we show some well-posdnees results. In a second part, after having demonstrated how to numerically find a minimizer, a proximal linearized algorithm combined with a Primal-Dual approach is introduced. Moreover, a bi-level optimization framework with a projected gradient algorithm is proposed in order to automatically select 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 other competitive denoising methods. Finally, we show extensive denoising experiments on various images and noise intensities and we report conventional numerical results which confirm the validity of the non-convex PDE-constrained, its analysis and also the proposed bi-level optimization with learning data.

Citation: Lekbir Afraites, Aissam Hadri, Amine Laghrib, Mourad Nachaoui. A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification. Inverse Problems and Imaging, doi: 10.3934/ipi.2022001
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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, Inverse Problems, 36 (2020), 025006, 40 pp. doi: 10.1088/1361-6420/ab5178.

[2]

H. Antil, Z. W. Di and R. Khatri, Bilevel optimization, deep learning and fractional laplacian regularization with applications in tomography, Inverse Problems, 36 (2020), 064001, 22 pp. doi: 10.1088/1361-6420/ab80d7.

[3]

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

[4]

J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications (Nonconvex Optimization and Its Applications), 30. Kluwer Academic Publishers, Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1.

[5]

E. M. BednarczukL. 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.

[6]

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.

[7]

K. BrediesK. Kunisch and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), 492-526.  doi: 10.1137/090769521.

[8]

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), IEEE, 2011, 1–8. doi: 10.1109/ICCPHOT.2011.5753128.

[9]

L. Calatroni, C. Cao, J. C. De los Reyes, C.-B. Schönlieb and T. Valkonen, Bilevel approaches for learning of variational imaging models, in Variational Methods, vol. 18 of Radon Ser. Comput. Appl. Math., De Gruyter, Berlin, 2017,252–290.

[10]

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

[11]

L. Calatroni and K. Papafitsoros, Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal, Inverse Problems, 35 (2019), 114001, 37 pp. doi: 10.1088/1361-6420/ab291a.

[12]

V. CasellesA. Chambolle and M. Novaga, Total variation in imaging, Handbook of Mathematical Methods in Imaging, 1, 2, 3, (2015), 1455-1499. 

[13]

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.

[14]

T. ChanA. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169.

[15]

T. F. Chan and S. Esedoglu, Aspects of total variation regularized $L^1$ function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.  doi: 10.1137/040604297.

[16]

F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, vol. 264 of Graduate Texts in Mathematics, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.

[17]

F. H. ClarkeR. 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.

[18]

C. Clason and B. Jin, A semismooth Newton method for nonlinear parameter identification problems with impulsive noise, SIAM J. Imaging Sci., 5 (2012), 505-536.  doi: 10.1137/110826187.

[19]

C. Clason and T. Valkonen, Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization, SIAM J. Optim., 27 (2017), 1314-1339.  doi: 10.1137/16M1080859.

[20]

K. DabovA. FoiV. Katkovnik and K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.

[21]

J. C. De los Reyes and C.-B. Schönlieb, Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization, Inverse Probl. Imaging, 7 (2013), 1183-1214.  doi: 10.3934/ipi.2013.7.1183.

[22]

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

[23]

F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, 2012. doi: 10.1007/978-1-4471-2807-6.

[24]

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

[25]

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

[26]

F. Dong and Q. Ma, Single image blind deblurring based on the fractional-order differential, Comput. Math. Appl., 78 (2019), 1960-1977.  doi: 10.1016/j.camwa.2019.03.033.

[27]

S. DurandJ. Fadili and M. Nikolova, Multiplicative noise removal using l1 fidelity on frame coefficients, Journal of Mathematical Imaging and Vision, 36 (2010), 201-226. 

[28]

S. Durand and M. Nikolova, Denoising of frame coefficients using $l^1$ data-fidelity term and edge-preserving regularization, Multiscale Model. Simul., 6 (2007), 547-576.  doi: 10.1137/06065828X.

[29]

I. El MourabitM. El RhabiA. HakimA. Laghrib and E. Moreau, A new denoising model for multi-frame super-resolution image reconstruction, Signal Processing, 132 (2017), 51-65.  doi: 10.1016/j.sigpro.2016.09.014.

[30]

Y.-R. FanA. BucciniM. Donatelli and T.-Z. Huang, A non-convex regularization approach for compressive sensing, Adv. Comput. Math., 45 (2019), 563-588.  doi: 10.1007/s10444-018-9627-3.

[31]

S. FarsiuM. D. RobinsonM. Elad and P. Milanfar, Fast and robust multiframe super resolution, IEEE Transactions on Image Processing, 13 (2004), 1327-1344.  doi: 10.1109/TIP.2004.834669.

[32]

P. Guidotti and K. Longo, Two enhanced fourth order diffusion models for image denoising, J. Math. Imaging Vision, 40 (2011), 188-198.  doi: 10.1007/s10851-010-0256-9.

[33]

P. Guidotti and K. Longo, Well-posedness for a class of fourth order diffusions for image processing, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 407-425.  doi: 10.1007/s00030-011-0101-x.

[34]

A. HadriL. AfraitesA. Laghrib and M. Nachaoui, A novel image denoising approach based on a non-convex constrained PDE: Application to ultrasound images, Signal, Image and Video Processing, 15 (2021), 1057-1064.  doi: 10.1007/s11760-020-01831-z.

[35]

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Figure 1.  TGV$ ^2 $ regularization with non-convex fidelity term results using different choices of the parameter $ \gamma $ for the Triangle image with the respective surfaces associated with the peak part. Note that the impulse noise is added with parameter $ r = 0.12 $ such as $ \mathfrak{L}(v) = \dfrac{e^{\frac{-|v|}{r}}}{2r} $, where $ \mathfrak{L} $ is the Laplace function
Figure 2.  The obtained solution when the noise is a mixture of impulse and Gaussian noise with a parameter $ r = 10^{-3} $ and $ \sigma^2 = 0.02 $. The first line presents the restored function with L$ ^1 $ and non-convex norm compared with the noisy and clean ones, while the second line presents the associated contours in the same order
Figure 3.  The obtained solution when the noise is a mixture of Gaussian and impulse noise with a variance $ \sigma^2 = 0.02 $ and parameter $ r = 5.10^{-2} $, respectively. The first line presents the restored function compared with the noisy and clean ones, while the second line presents the associated contours in the same order
Figure 4.  The obtained solution and impulse noise component with the associated contours compared to the original function when the noise is a mixture of impulse and Gaussian noise with a variance $ \sigma^2 = 0.02 $ and parameter $ r = 0.1 $
Figure 5.  The influence of the parameter $ \gamma $ on the obtained solution when the noise is a mixture of Gaussian and impulse noise with parameter $ \sigma^2 = 0.02 $ and $ r = 0.1 $, respectively
Figure 6.  The computed impulse noise component and the restored image compared to the L$ ^1 $ norm
Figure 7.  The computed impulse noise component and the restored image compared to the non-convex norm, when the impulse component is $ r = 10^{-3} $
Figure 8.  Comparisons with the CODE [1] method using four images and using different levels of noise. Note that for the first test, we consider a mixture of Gaussian noise with $ \sigma^2 = 0.03 $ and impulse noise with parameter $ r = 0.4 $, while for the second test $ \sigma^2 = 0.03 $ and $ r = 0.5 $. For the third test $ \sigma^2 = 0.04 $ and $ r = 0.5 $ where for the last test $ \sigma^2 = 0.04 $ and $ r = 0.6 $
Figure 9.  The efficiency of the fractional order operator in fixing the diffusion with respect to the parameter $ \alpha $ for the Castle image
Figure 10.  The efficiency of the fractional order operator in fixing the diffusion with respect to the parameter $ \alpha $ for the (Mountains image)
Figure 11.  Comparison between some denoising PDEs and the proposed fractional one for the (Brain MRI image). Note that the mixed noise is considered with $ r = 0.1 $ for the impulse parameter and $ \sigma^2 = 0.03 $ for the Gaussian variance noise
Figure 12.  Comparison between some denoising PDEs and the proposed fractional one for the (Knee MRI image). Note that the mixed noise is considered with $ r = 0.2 $ for the impulse parameter and $ \sigma^2 = 0.03 $ for the Gaussian variance noise)
Figure 13.  Comparison between some denoising PDEs and the proposed fractional one for the (Brain 2 MRI image). Note that the mixed noise is considered with $ r = 0.5 $ for the impulse parameter and $ \sigma^2 = 0.04 $ for the Gaussian variance noise
Figure 14.  The comparison with competitive denoising model for the (Bird image). Note that we consider a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.2 $ and $ r = 0.35 $
Figure 15.  The comparison with competitive denoising model for the (Baboon image). Note that we consider a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.3 $ and $ r = 0.5 $
Figure 16.  The denoising process using the proposed non-smooth Primal-Dual algorithm compared to Euler-Lagrange iterations as optimization approach for the PDE-constrained with final peak-signal-to-noise ratios for each restoration method. We use the (Tiger image) while the Gaussian noise is considered with $ \sigma^2 = 0.2 $ and impulse noise with parameter $ r = 0.3 $. We can see the robustness of the proposed Primal-Dual on both the quality of the restored image and the speed of convergence, compared to the Euler-Lagrange approach
Figure 17.  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 $
Figure 18.  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 consructed using a mixture of Gaussian and impule noise with paramaters $ \sigma^2 = 0.03 $ and $ r = 0.4 $
Table 1.  The PSNR and SSIM table for the four examples in Fig. 8
Image Method PSNR SSIM
Eyes CODE 27.95 0.7853
Our 28.41 0.8007
Penguin CODE 25.55 0.6653
Our 26.03 0.6880
Butterfly CODE 25.14 0.6222
Our 25.47 0.6429
goose CODE 24.05 0.5966
Our 24.88 0.6094
Image Method PSNR SSIM
Eyes CODE 27.95 0.7853
Our 28.41 0.8007
Penguin CODE 25.55 0.6653
Our 26.03 0.6880
Butterfly CODE 25.14 0.6222
Our 25.47 0.6429
goose CODE 24.05 0.5966
Our 24.88 0.6094
Table 2.  The set of parameters being used in denoising results presented in the two tests
Parameters Method
TV+TV2 TV$ ^{\alpha} $ TGV$ ^2 $ Nonconvex-TV Our Method
Iteration number $ N $ 2000 2000 2000 2000 2000
The parameter $ \alpha_0 $ 0.02 0.04
The parameter $ \alpha_1 $ 0.06 0.065
Spatially decaying effect $ (k_1,k_2) $ $ (35,35) $
The fractional order derivative $ \alpha $ 1.35 1.77
Regularization parameter $ \lambda $ 0.01 0.02
The concavity parameter $ \gamma $ 42 86
Parameters Method
TV+TV2 TV$ ^{\alpha} $ TGV$ ^2 $ Nonconvex-TV Our Method
Iteration number $ N $ 2000 2000 2000 2000 2000
The parameter $ \alpha_0 $ 0.02 0.04
The parameter $ \alpha_1 $ 0.06 0.065
Spatially decaying effect $ (k_1,k_2) $ $ (35,35) $
The fractional order derivative $ \alpha $ 1.35 1.77
Regularization parameter $ \lambda $ 0.01 0.02
The concavity parameter $ \gamma $ 42 86
Table 3.  PSNR and SSIM results obtained by applying different denoising methods with different levels of Gaussian and impulse noise to six selected images. In bold the best (highest) score of each line is shown
Image Method
$ \sigma $ noise Metric TV$ ^{\alpha} $ non-convex-TV TV+TV$ ^2 $ TGV TGV$ ^2 $ BM3D proposed
Baboon 0.2 PSNR 26.88 27.19 24.93 25.40 25.44 27.30 $ \mathbf{28.84} $
SSIM 0.781 0.788 0.714 0.737 0.748 0.789 $ \mathbf{0.811} $
0.5 PSNR 23.88 24.22 22.55 23.11 23.96 25.19 $ \mathbf{26.07} $
SSIM 0.641 0.685 0.598 0.612 0.631 0.688 $ \mathbf{0.722} $
Fly 0.2 PSNR 27.60 28.08 26.72 27.58 27.73 29.03 $ \mathbf{30.48} $
SSIM 0.808 0.802 0.767 0.748 0.741 0.794 $ \mathbf{0.834} $
0.4 PSNR 26.18 26.43 24.89 25.98 26.04 27.02 $ \mathbf{27.88} $
SSIM 0.720 0.733 0.668 0.696 0.695 0.742 $ \mathbf{0.747} $
Bird 0.1 PSNR 31.12 32.26 30.45 31.53 31.32 33.60 $ \mathbf{34.44} $
SSIM 0.886 0.880 0.788 0.805 0.822 0.868 $ \mathbf{0.909} $
0.3 PSNR 29.45 30.22 28.89 30.08 30.15 32.52 $ \mathbf{33.37} $
SSIM 0.826 0.838 0.738 0.794 0.810 0.850 $ \mathbf{0.876} $
Eyes 0.2 PSNR 29.87 29.97 28.01 29.68 29.62 30.30 $ \mathbf{31.77} $
SSIM 0.812 0.842 0.786 0.796 0.780 0.840 $ \mathbf{0.857} $
0.4 PSNR 27.44 28.15 26.93 27.94 27.55 28.55 $ \mathbf{29.03} $
SSIM 0.758 0.759 0.650 0.692 0.700 0.774 $ \mathbf{0.783} $
Gazelle 0.5 PSNR 25.19 25.36 23.55 24.02 24.12 25.04 $ \mathbf{26.49} $
SSIM 0.597 0.612 0.509 0.547 0.520 0.616 $ \mathbf{0.678} $
0.6 PSNR 23.45 23.50 22.11 22.44 22.22 23.29 $ \mathbf{23.66} $
SSIM 0.478 0.481 0.403 0.409 0.413 0.510 $ \mathbf{0.608} $
Goose 0.1 PSNR 33.49 33.70 30.88 31.01 31.44 34.17 $ \mathbf{34.65} $
SSIM 0.908 0.920 0.865 0.890 0.896 0.919 $ \mathbf{0.921} $
0.3 PSNR 31.08 31.13 28.91 29.17 29.34 31.01 $ \mathbf{32.36} $
SSIM 0.818 0.851 0.708 0.713 0.709 0.825 $ \mathbf{0.866} $
Image Method
$ \sigma $ noise Metric TV$ ^{\alpha} $ non-convex-TV TV+TV$ ^2 $ TGV TGV$ ^2 $ BM3D proposed
Baboon 0.2 PSNR 26.88 27.19 24.93 25.40 25.44 27.30 $ \mathbf{28.84} $
SSIM 0.781 0.788 0.714 0.737 0.748 0.789 $ \mathbf{0.811} $
0.5 PSNR 23.88 24.22 22.55 23.11 23.96 25.19 $ \mathbf{26.07} $
SSIM 0.641 0.685 0.598 0.612 0.631 0.688 $ \mathbf{0.722} $
Fly 0.2 PSNR 27.60 28.08 26.72 27.58 27.73 29.03 $ \mathbf{30.48} $
SSIM 0.808 0.802 0.767 0.748 0.741 0.794 $ \mathbf{0.834} $
0.4 PSNR 26.18 26.43 24.89 25.98 26.04 27.02 $ \mathbf{27.88} $
SSIM 0.720 0.733 0.668 0.696 0.695 0.742 $ \mathbf{0.747} $
Bird 0.1 PSNR 31.12 32.26 30.45 31.53 31.32 33.60 $ \mathbf{34.44} $
SSIM 0.886 0.880 0.788 0.805 0.822 0.868 $ \mathbf{0.909} $
0.3 PSNR 29.45 30.22 28.89 30.08 30.15 32.52 $ \mathbf{33.37} $
SSIM 0.826 0.838 0.738 0.794 0.810 0.850 $ \mathbf{0.876} $
Eyes 0.2 PSNR 29.87 29.97 28.01 29.68 29.62 30.30 $ \mathbf{31.77} $
SSIM 0.812 0.842 0.786 0.796 0.780 0.840 $ \mathbf{0.857} $
0.4 PSNR 27.44 28.15 26.93 27.94 27.55 28.55 $ \mathbf{29.03} $
SSIM 0.758 0.759 0.650 0.692 0.700 0.774 $ \mathbf{0.783} $
Gazelle 0.5 PSNR 25.19 25.36 23.55 24.02 24.12 25.04 $ \mathbf{26.49} $
SSIM 0.597 0.612 0.509 0.547 0.520 0.616 $ \mathbf{0.678} $
0.6 PSNR 23.45 23.50 22.11 22.44 22.22 23.29 $ \mathbf{23.66} $
SSIM 0.478 0.481 0.403 0.409 0.413 0.510 $ \mathbf{0.608} $
Goose 0.1 PSNR 33.49 33.70 30.88 31.01 31.44 34.17 $ \mathbf{34.65} $
SSIM 0.908 0.920 0.865 0.890 0.896 0.919 $ \mathbf{0.921} $
0.3 PSNR 31.08 31.13 28.91 29.17 29.34 31.01 $ \mathbf{32.36} $
SSIM 0.818 0.851 0.708 0.713 0.709 0.825 $ \mathbf{0.866} $
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