A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification

Afraites Lekbir; Hadri Aissam; Laghrib Amine; Nachaoui Mourad

doi:10.3934/ipi.2022001

Article Contents

2022, Volume 16, Issue 4: 827-870. Doi: 10.3934/ipi.2022001

This issue Previous Article Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas Next Article Direct regularized reconstruction for the three-dimensional Calderón problem

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
^* Corresponding author: Amine Laghrib

Received: March 31, 2021

Revised: September 30, 2021

Early access: January 2022

Published: August 2022

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 65K10, 90C26; Secondary: 35R11, 68U10.

Citation:

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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

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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

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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

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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 $

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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

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Figure 6. The computed impulse noise component and the restored image compared to the L$ ^1 $ norm

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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} $

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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 $

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Figure 9. The efficiency of the fractional order operator in fixing the diffusion with respect to the parameter $ \alpha $ for the Castle image

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Figure 10. The efficiency of the fractional order operator in fixing the diffusion with respect to the parameter $ \alpha $ for the (Mountains image)

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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

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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)

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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

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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 $

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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 $

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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

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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 $

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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 $

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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

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Table 2. The set of parameters being used in denoising results presented in the two tests

Parameters	Method
Parameters	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

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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
Image	$ \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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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. 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.
[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. Bredies, K. 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. Calatroni, J. 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. Caselles, A. 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. Chan, A. 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. 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.
[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. Dabov, A. Foi, V. 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 Reyes, C.-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. Dempe, F. Harder, P. 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. 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.
[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 Mourabit, M. El Rhabi, A. Hakim, A. 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. Fan, A. Buccini, M. 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. Farsiu, M. D. Robinson, M. 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. Hadri, L. Afraites, A. 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]	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.
[36]	M. Hintermüller 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.
[37]	M. Hintermüller, K. Papafitsoros, C. N. Rautenberg and H. Sun, Dualization and automatic distributed parameter selection of total generalized variation via bilevel optimization, 2020.
[38]	M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE constraints, vol. 23, Springer, New York, 2009.
[39]	G. Holler, K. Kunisch and R. C. Barnard, A bilevel approach for parameter learning in inverse problems, Inverse Problems, 34 (2018), 115012, 28 pp. doi: 10.1088/1361-6420/aade77.
[40]	M. Janev, S. Pilipović, T. Atanacković, R. Obradović and N. Ralević, Fully fractional anisotropic diffusion for image denoising, Math. Comput. Modelling, 54 (2011), 729-741. doi: 10.1016/j.mcm.2011.03.017.
[41]	F. Knoll, K. Bredies, T. Pock and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491. doi: 10.1002/mrm.22595.
[42]	K. Kunisch and T. Pock, A bilevel optimization approach for parameter learning in variational models, SIAM J. Imaging Sci., 6 (2013), 938-983. doi: 10.1137/120882706.
[43]	A. Laghrib, A. Ben-Loghfyry, A. Hadri and A. Hakim, A nonconvex fractional order variational model for multi-frame image super-resolution, Signal Processing: Image Communication, 67 (2018), 1-11. doi: 10.1016/j.image.2018.05.011.
[44]	A. Laghrib, M. Ezzaki, M. El Rhabi, A. Hakim, P. Monasse and S. Raghay, Simultaneous deconvolution and denoising using a second order variational approach applied to image super resolution, Computer Vision and Image Understanding, 168 (2018), 50-63. doi: 10.1016/j.cviu.2017.08.007.
[45]	A. Laghrib, A. Ghazdali, A. Hakim and S. Raghay, A multi-frame super-resolution using diffusion registration and a nonlocal variational image restoration, Comput. Math. Appl., 72 (2016), 2535-2548. doi: 10.1016/j.camwa.2016.09.013.
[46]	A. Langer, Automated parameter selection for total variation minimization in image restoration, J. Math. Imaging Vision, 57 (2017), 239-268. doi: 10.1007/s10851-016-0676-2.
[47]	J. Lellmann, K. Papafitsoros, C. Schönlieb and D. Spector, Analysis and application of a nonlocal Hessian, SIAM J. Imaging Sci., 8 (2015), 2161-2202. doi: 10.1137/140993818.
[48]	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.
[49]	Q. Ma, F. Dong and D. Kong, A fractional differential fidelity-based PDE model for image denoising, Machine Vision and Applications, 28 (2017), 635-647. doi: 10.1007/s00138-017-0857-z.
[50]	J. V. Manjón, J. Carbonell-Caballero, J. J. Lull, G. García-Martí, L. Martí-Bonmatí and M. Robles, Mri denoising using non-local means, Medical Image Analysis, 12 (2008), 514-523.
[51]	P. Mehlitz, Necessary optimality conditions for a special class of bilevel programming problems with unique lower level solution, Optimization, 66 (2017), 1533-1562. doi: 10.1080/02331934.2017.1349123.
[52]	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.
[53]	P. Mehlitz and G. Wachsmuth, Weak and strong stationarity in generalized bilevel programming and bilevel optimal control, Optimization, 65 (2016), 907-935. doi: 10.1080/02331934.2015.1122007.
[54]	A. Mitsos, P. Lemonidis and P. I. Barton, Global solution of bilevel programs with a nonconvex inner program, J. Global Optim., 42 (2008), 475-513. doi: 10.1007/s10898-007-9260-z.
[55]	S. Mohaoui, A. Hakim and S. Raghay, Tensor completion via bilevel minimization with fixed-point constraint to estimate missing elements in noisy data, Adv. Comput. Math., 47 (2021), Paper No. 10, 27 pp. doi: 10.1007/s10444-020-09841-8.
[56]	M. Nikolova, Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers, SIAM J. Numer. Anal., 40 (2002), 965-994. doi: 10.1137/S0036142901389165.
[57]	M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vision, 20 (2004), 99-120.
[58]	P. Ochs, R. Ranftl, T. Brox and T. Pock, Techniques for gradient based bilevel optimization with nonsmooth lower level problems, J. Math. Imaging Vision, 56 (2016), 175–194, URL http://lmb.informatik.uni-freiburg.de/Publications/2016/OB16. doi: 10.1007/s10851-016-0663-7.
[59]	K. Palagachev and M. Gerdts, 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.
[60]	K. Papafitsoros and C.-B. Schönlieb, A combined first and second order variational approach for image reconstruction, J. Math. Imaging Vision, 48 (2014), 308-338. doi: 10.1007/s10851-013-0445-4.
[61]	J. Park, An overlapping domain decomposition framework without dual formulation for variational imaging problems, Adv. Comput. Math., 46 (2020), Paper No. 57, 29 pp. doi: 10.1007/s10444-020-09799-7.
[62]	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.
[63]	T. Valkonen, K. Bredies and F. Knoll, Total generalized variation in diffusion tensor imaging, SIAM J. Imaging Sci., 6 (2013), 487-525. doi: 10.1137/120867172.
[64]	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.
[65]	Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730. doi: 10.1109/83.869184.
[66]	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.
[67]	J. Zhang, Z. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vision, 43 (2012), 39-49. doi: 10.1007/s10851-011-0285-z.
[68]	X. Zhang, M. 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.
[69]	X.-L. Zhao, F. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM J. Imaging Sci., 7 (2014), 456-475. doi: 10.1137/13092472X.
[70]	X. Zhu and P. Guo, Approaches to four types of bilevel programming problems with nonconvex nonsmooth lower level programs and their applications to newsvendor problems, Math. Methods Oper. Res., 86 (2017), 255-275. doi: 10.1007/s00186-017-0592-2.