doi: 10.3934/ipi.2022031
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A non-convex PDE-constrained denoising model for impulse and Gaussian noise mixture reduction

1. 

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

2. 

Laboratoire SIE, Université IBN ZOHR Agadir, Maroc

*Corresponding author: Amine Laghrib

Received  July 2021 Revised  December 2021 Early access June 2022

This paper introduces a novel optimization procedure to reduce a mixture of Gaussian and impulse noise from images. This framework is based on a non-convex PDE-constrained with two diffusion operators: a local weickert and a fractional-order ones. The non-convex norm is applied to remove the impulse component, while the local and fractional operators are introduced to preserve image texture and edges. In the first part, we study the theoretical properties of the proposed PDE-constrained, and we show some well-posedness 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 with local convergence results. Finally, we show extensive denoising experiments on various images and noise intensities which confirms the validity of the non-convex PDE-constrained, its analysis and also the proposed optimization procedure.

Citation: Amine Laghrib, Lekbir Afraites, Aissam Hadri, Mourad Nachaoui. A non-convex PDE-constrained denoising model for impulse and Gaussian noise mixture reduction. Inverse Problems and Imaging, doi: 10.3934/ipi.2022031
References:
[1]

M. Afonso and J. M. Sanches, Adaptive order non-convex lp-norm regularization in image restoration, Journal of Physics: Conference Series, 904 (2017), 012016.  doi: 10.1088/1742-6596/904/1/012016.

[2]

L. AfraitesA. Hadri and A. Laghrib, A denoising model adapted for impulse and gaussian noises using a constrained-pde, Inverse Problems, 36 (2020), 025006.  doi: 10.1088/1361-6420/ab5178.

[3]

L. AfraitesA. HadriA. Laghrib and M. Nachaoui, A non-convex denoising model for impulse and gaussian noise mixture removing using bi-level parameter identification, Inverse Problems & Imaging, (2022).  doi: 10.3934/ipi.2022001.

[4]

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

[5]

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.

[6]

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

[7]

M. BurgerK. PapafitsorosE. Papoutsellis and C.-B. Schönlieb, Infimal convolution regularisation functionals of bv and ${\mathrm {l}}^{p} $ spaces, Journal of Mathematical Imaging and Vision, 55 (2016), 343-369.  doi: 10.1007/s10851-015-0624-6.

[8]

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

[9]

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

[10]

Q. ChenP. MontesinosQ. S. SunP. A. Heng and D. S. Xia, Adaptive total variation denoising based on difference curvature, Image and Vision Computing, 28 (2010), 298-306.  doi: 10.1016/j.imavis.2009.04.012.

[11]

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

[12]

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

[13]

F. Dong and Q. Ma, Single image blind deblurring based on the fractional-order differential, Computers & Mathematics with Applications, 78 (2019), 1960-1977.  doi: 10.1016/j.camwa.2019.03.033.

[14]

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.  doi: 10.1007/s10851-009-0180-z.

[15]

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.

[16]

C. Elion and L. A. Vese, An image decomposition model using the total variation and the infinity laplacian, In Computational Imaging V, volume 6498, page 64980W. International Society for Optics and Photonics, 2007. doi: 10.1117/12.716079.

[17]

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

[18]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2008), 1005-1028.  doi: 10.1137/070698592.

[19]

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

[20]

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

[21]

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.

[22]

A. HadriH. KhalfiA. Laghrib and M. Nachaoui, An improved spatially controlled reaction–diffusion equation with a non-linear second order operator for image super-resolution, Nonlinear Analysis: Real World Applications, 62 (2021), 103352.  doi: 10.1016/j.nonrwa.2021.103352.

[23]

M. Hintermuller and A. Langer, Subspace correction methods for a class of nonsmooth and nonadditive convex variational problems with mixed ${\rm{l\hat 1/l\hat 2}}$ data-fidelity in image processing, SIAM Journal on Imaging Sciences, 6 (2013), 2134-2173.  doi: 10.1137/120894130.

[24]

M. JanevS. PilipovićT. AtanackovićR. Obradović and N. Ralević, Fully fractional anisotropic diffusion for image denoising, Mathematical and Computer Modelling, 54 (2011), 729-741.  doi: 10.1016/j.mcm.2011.03.017.

[25]

A. Kaban and R. J. Durrant, Learning with l q < 1 vs l 1-norm regularisation with exponentially many irrelevant features, In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, Springer, 2008, 580–596. doi: 10.1007/978-3-540-87479-9_56.

[26]

F. KnollK. BrediesT. 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.

[27]

A. Kuijper, p-laplacian driven image processing, In 2007 IEEE International Conference on Image Processing, 5 (2007), pages V–257. doi: 10.1109/ICIP.2007.4379814.

[28]

A. LaghribA. Ben-LoghfyryA. 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.

[29]

A. LaghribM. EzzakiM. El RhabiA. HakimP. 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.

[30]

A. LaghribA. GhazdaliA. Hakim and S. Raghay, A multi-frame super-resolution using diffusion registration and a nonlocal variational image restoration, Computers & Mathematics with Applications, 72 (2016), 2535-2548.  doi: 10.1016/j.camwa.2016.09.013.

[31]

J. LellmannK. PapafitsorosC. Schönlieb and D. Spector, Analysis and application of a nonlocal hessian, SIAM Journal on Imaging Sciences, 8 (2015), 2161-2202.  doi: 10.1137/140993818.

[32]

L. F. Y. Z. G. Lutai, Resizing the noisy digital image using tikhonov regularization, Journal of Computer Aided Design & Computer Graphics, 7 (2002). 

[33]

Q. MaF. 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.

[34]

H. NaM. KangM. Jung and M. Kang, Nonconvex tgv regularization model for multiplicative noise removal with spatially varying parameters, Inverse Problems & Imaging, 13 (2019), 117-147.  doi: 10.3934/ipi.2019007.

[35]

M. NachaouiL. AfraitesA. Hadri and A. Laghrib, A non-convex non-smooth bi-level parameter learning for impulse and gaussian noise mixture removing, Communications on Pure & Applied Analysis, 21 (2022), 1249-1291.  doi: 10.3934/cpaa.2022018.

[36]

M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120. 

[37]

K. Papafitsoros and C. B. Schönlieb, A combined first and second order variational approach for image reconstruction, Journal of Mathematical Imaging and Vision, 48 (2014), 308-338.  doi: 10.1007/s10851-013-0445-4.

[38]

L. I. RudinS. 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.

[39]

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

[40]

D. Tuia, R. Flamary and M. Barlaud, To be or not to be convex? a study on regularization in hyperspectral image classification, In 2015 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), IEEE, 2015, 4947–4950. doi: 10.1109/IGARSS.2015.7326942.

[41]

T. ValkonenK. Bredies and F. Knoll, Total generalized variation in diffusion tensor imaging, SIAM Journal on Imaging Sciences, 6 (2013), 487-525.  doi: 10.1137/120867172.

[42]

J. Weickert, Coherence-enhancing diffusion filtering, International Journal of Computer Vision, 31 (1999), 111-127.  doi: 10.1023/A:1008009714131.

[43]

J. Weickert, Coherence-enhancing diffusion of colour images, Image and Vision Computing, 17 (1999), 201-212.  doi: 10.1016/S0262-8856(98)00102-4.

[44]

Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Transactions on Image Processing, 9 (2000), 1723-1730.  doi: 10.1109/83.869184.

[45]

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

[46]

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

[47]

X.-L. ZhaoF. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM Journal on Imaging Sciences, 7 (2014), 456-475.  doi: 10.1137/13092472X.

[48]

W. Zhu, A numerical study of a mean curvature denoising model using a novel augmented lagrangian method, Inverse Problems & Imaging, 11 (2017), 975-996.  doi: 10.3934/ipi.2017045.

show all references

References:
[1]

M. Afonso and J. M. Sanches, Adaptive order non-convex lp-norm regularization in image restoration, Journal of Physics: Conference Series, 904 (2017), 012016.  doi: 10.1088/1742-6596/904/1/012016.

[2]

L. AfraitesA. Hadri and A. Laghrib, A denoising model adapted for impulse and gaussian noises using a constrained-pde, Inverse Problems, 36 (2020), 025006.  doi: 10.1088/1361-6420/ab5178.

[3]

L. AfraitesA. HadriA. Laghrib and M. Nachaoui, A non-convex denoising model for impulse and gaussian noise mixture removing using bi-level parameter identification, Inverse Problems & Imaging, (2022).  doi: 10.3934/ipi.2022001.

[4]

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

[5]

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.

[6]

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

[7]

M. BurgerK. PapafitsorosE. Papoutsellis and C.-B. Schönlieb, Infimal convolution regularisation functionals of bv and ${\mathrm {l}}^{p} $ spaces, Journal of Mathematical Imaging and Vision, 55 (2016), 343-369.  doi: 10.1007/s10851-015-0624-6.

[8]

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

[9]

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

[10]

Q. ChenP. MontesinosQ. S. SunP. A. Heng and D. S. Xia, Adaptive total variation denoising based on difference curvature, Image and Vision Computing, 28 (2010), 298-306.  doi: 10.1016/j.imavis.2009.04.012.

[11]

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

[12]

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

[13]

F. Dong and Q. Ma, Single image blind deblurring based on the fractional-order differential, Computers & Mathematics with Applications, 78 (2019), 1960-1977.  doi: 10.1016/j.camwa.2019.03.033.

[14]

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.  doi: 10.1007/s10851-009-0180-z.

[15]

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.

[16]

C. Elion and L. A. Vese, An image decomposition model using the total variation and the infinity laplacian, In Computational Imaging V, volume 6498, page 64980W. International Society for Optics and Photonics, 2007. doi: 10.1117/12.716079.

[17]

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

[18]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2008), 1005-1028.  doi: 10.1137/070698592.

[19]

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

[20]

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

[21]

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.

[22]

A. HadriH. KhalfiA. Laghrib and M. Nachaoui, An improved spatially controlled reaction–diffusion equation with a non-linear second order operator for image super-resolution, Nonlinear Analysis: Real World Applications, 62 (2021), 103352.  doi: 10.1016/j.nonrwa.2021.103352.

[23]

M. Hintermuller and A. Langer, Subspace correction methods for a class of nonsmooth and nonadditive convex variational problems with mixed ${\rm{l\hat 1/l\hat 2}}$ data-fidelity in image processing, SIAM Journal on Imaging Sciences, 6 (2013), 2134-2173.  doi: 10.1137/120894130.

[24]

M. JanevS. PilipovićT. AtanackovićR. Obradović and N. Ralević, Fully fractional anisotropic diffusion for image denoising, Mathematical and Computer Modelling, 54 (2011), 729-741.  doi: 10.1016/j.mcm.2011.03.017.

[25]

A. Kaban and R. J. Durrant, Learning with l q < 1 vs l 1-norm regularisation with exponentially many irrelevant features, In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, Springer, 2008, 580–596. doi: 10.1007/978-3-540-87479-9_56.

[26]

F. KnollK. BrediesT. 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.

[27]

A. Kuijper, p-laplacian driven image processing, In 2007 IEEE International Conference on Image Processing, 5 (2007), pages V–257. doi: 10.1109/ICIP.2007.4379814.

[28]

A. LaghribA. Ben-LoghfyryA. 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.

[29]

A. LaghribM. EzzakiM. El RhabiA. HakimP. 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.

[30]

A. LaghribA. GhazdaliA. Hakim and S. Raghay, A multi-frame super-resolution using diffusion registration and a nonlocal variational image restoration, Computers & Mathematics with Applications, 72 (2016), 2535-2548.  doi: 10.1016/j.camwa.2016.09.013.

[31]

J. LellmannK. PapafitsorosC. Schönlieb and D. Spector, Analysis and application of a nonlocal hessian, SIAM Journal on Imaging Sciences, 8 (2015), 2161-2202.  doi: 10.1137/140993818.

[32]

L. F. Y. Z. G. Lutai, Resizing the noisy digital image using tikhonov regularization, Journal of Computer Aided Design & Computer Graphics, 7 (2002). 

[33]

Q. MaF. 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.

[34]

H. NaM. KangM. Jung and M. Kang, Nonconvex tgv regularization model for multiplicative noise removal with spatially varying parameters, Inverse Problems & Imaging, 13 (2019), 117-147.  doi: 10.3934/ipi.2019007.

[35]

M. NachaouiL. AfraitesA. Hadri and A. Laghrib, A non-convex non-smooth bi-level parameter learning for impulse and gaussian noise mixture removing, Communications on Pure & Applied Analysis, 21 (2022), 1249-1291.  doi: 10.3934/cpaa.2022018.

[36]

M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120. 

[37]

K. Papafitsoros and C. B. Schönlieb, A combined first and second order variational approach for image reconstruction, Journal of Mathematical Imaging and Vision, 48 (2014), 308-338.  doi: 10.1007/s10851-013-0445-4.

[38]

L. I. RudinS. 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.

[39]

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

[40]

D. Tuia, R. Flamary and M. Barlaud, To be or not to be convex? a study on regularization in hyperspectral image classification, In 2015 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), IEEE, 2015, 4947–4950. doi: 10.1109/IGARSS.2015.7326942.

[41]

T. ValkonenK. Bredies and F. Knoll, Total generalized variation in diffusion tensor imaging, SIAM Journal on Imaging Sciences, 6 (2013), 487-525.  doi: 10.1137/120867172.

[42]

J. Weickert, Coherence-enhancing diffusion filtering, International Journal of Computer Vision, 31 (1999), 111-127.  doi: 10.1023/A:1008009714131.

[43]

J. Weickert, Coherence-enhancing diffusion of colour images, Image and Vision Computing, 17 (1999), 201-212.  doi: 10.1016/S0262-8856(98)00102-4.

[44]

Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Transactions on Image Processing, 9 (2000), 1723-1730.  doi: 10.1109/83.869184.

[45]

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

[46]

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

[47]

X.-L. ZhaoF. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM Journal on Imaging Sciences, 7 (2014), 456-475.  doi: 10.1137/13092472X.

[48]

W. Zhu, A numerical study of a mean curvature denoising model using a novel augmented lagrangian method, Inverse Problems & Imaging, 11 (2017), 975-996.  doi: 10.3934/ipi.2017045.

Figure 1.  Illustration of the $ L^p $ norms regularization with diiferents values of $ p $
Figure 2.  The behaviour of the function $ \Psi $ using low values of $ \gamma $ for the two choices: $ \frac{t}{t+\gamma} $ and $ \frac{t}{1+\frac{t}{\gamma}} $
Figure 3.  The behaviour of the function $ \Psi $ using high values of $ \gamma $ for the two choices: $ \Psi(t) = \frac{t}{t+\gamma} $ and $ \Psi(t) = \frac{t}{1+\frac{t}{\gamma}} $
Figure 4.  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 = 0.05 $ and $ r = 10^{-5} $, respectively
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 evolution of the restored image with respect to the parameter $ \gamma $. The first line presents the restored image compared with the noisy and clean ones, while the second line presents the associated 3D surfaces in the same order. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.02 $ and $ r = 0.5 $
Figure 7.  The evolution of the restored image with respect to the parameter $ \gamma $. The first line presents the restored image compared with the noisy and clean ones, while the second line presents the associated 3D surfaces in the same order. 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 $
Figure 8.  The computed impulse noise component and the restored image compared to the non-convex norm, when the impulse component is $ r = 10^{-3} $
Figure 9.  The computed impulse noise component and the restored "Star" image compared to the non-convex norm, when the impulse component is $ r = 10^{-1} $
Figure 10.  The first line presents the restored function compared with the L$ ^1 $ norm and clean image of (Phantom, while the second line presents the associated 3D surfaces in the same order. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.002 $ and $ r = 0.03 $
Figure 11.  The first line presents the restored function compared with the L$ ^1 $ norm and clean image of (Phantom, while the second line presents the associated 3D surfaces in the same order. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.02 $ and $ r = 0.2 $
Figure 12.  The first line presents the restored function compared with the L$ ^1 $ norm and clean image of (Phantom, while the second line presents the associated 3D surfaces in the same order. 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 13.  Comparisons with the CODE [2] 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 14.  Comparison between some denoising PDEs and the proposed fractional one for the (Dune image). Note that the mixed noise is considered with $ r = 0.3 $ for the impulse parameter and $ \sigma^2 = 0.01 $ for the Gaussian variance noise
Figure 15.  Comparison between some denoising PDEs and the proposed fractional one for the (Barbara image). Note that the mixed noise is considered with $ r = 0.6 $ for the impulse parameter and $ \sigma^2 = 0.06 $ for the Gaussian variance noise
Figure 16.  Comparison between some denoising PDEs and the proposed fractional one for the (Barbara 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 17.  Comparison between some denoising PDEs and the proposed fractional one for the (Zebra image). Note that the mixed noise is considered with $ r = 0.4 $ for the impulse parameter and $ \sigma^2 = 0.035 $ for the Gaussian variance noise
Figure 18.  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 19.  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 20.  The comparison with competitive denoising model for the (Pirate image). Note that we consider a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.02 $ and $ r = 0.35 $
Figure 21.  The comparison with competitive denoising model for the (Penguin image). Note that we consider a mixture of Gaussian and impulse noise with parameters $ \sigma^2 = 0.05 $ and $ r = 0.6 $
Figure 22.  The efficiency of the fractional order operator in fixing the diffusion with respect to the parameter $ \alpha $ for the (Cameraman image)
Figure 23.  The efficiency of the fractional order operator in restoring sharp edges with respect to the parameter $ \beta $ for the (Star image)
Figure 24.  The efficiency of the fractional order operator in restoring sharp edges with respect to the parameter $ \beta $ for the (Woman image)
Figure 25.  The denoising process using the proposed Primal-Dual compared to Euler-Lagrange iterations as optimization approach for the PDE-constrained with final peak-signal-to-noise ratios for each restoration method. 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 26.  The PSNR value with respect to the first $ 1000 $ iterations for the four denoising tests
Figure 27.  The logarithm of the relative error $ E_k = \log_{10}(RelErr) $ with respect to the first $ 1000 $ iterations for the four previous denoising tests using the primal-dual and Euler-Lagrange algorithms
Table 1.  The PSNR and SSIM table for the four examples in Fig. 13
Image Method PSNR SSIM
Tiger CODE 28.33 0.7922
Our 28.84 0.8027
Penguin CODE 25.61 0.6661
Our 26.03 0.6880
Butterfly CODE 25.22 0.6228
Our 25.43 0.6433
Squirrel CODE 25.02 0.6118
Our 25.69 0.6344
Image Method PSNR SSIM
Tiger CODE 28.33 0.7922
Our 28.84 0.8027
Penguin CODE 25.61 0.6661
Our 26.03 0.6880
Butterfly CODE 25.22 0.6228
Our 25.43 0.6433
Squirrel CODE 25.02 0.6118
Our 25.69 0.6344
Table 2.  The set of parameters being used in denoising results presented in the three 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 $ \beta $ 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 $ \beta $ 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 nine 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
Bridge
0.1 PSNR 31.08 32.33 31.03 32.54 32.72 $ {\bf{33.14}} $ 32.94
SSIM 0.883 0.901 0.878 0.899 0.893 0.922 $ {\bf{0.922}} $
0.2 PSNR 30.57 31.43 30.45 31.17 31.66 $ {\bf{32.09}} $ 32.05
SSIM 0.641 0.685 0.598 0.612 0.631 0.688 $ {\bf{0.722}} $
Castle
0.3 PSNR 29.44 29.57 29.12 30.05 $ {\bf{30.18}} $ 30.14 30.08
SSIM 0.784 0.791 0.778 0.800 0.804 0.819 $ {\bf{0.821}} $
0.4 PSNR 28.81 28.94 28.75 29.22 29.85 $ {\bf{30.04}} $ 30.00
SSIM 0.711 0.725 0.699 0.737 0.738 0.768 $ {\bf{0.772}} $
Baboon
0.2 PSNR 26.88 27.19 24.93 25.40 25.44 27.30 $ {\bf{28.84}} $
SSIM 0.781 0.788 0.714 0.737 0.748 0.789 $ {\bf{0.811}} $
0.5 PSNR 23.88 24.22 22.55 23.11 23.96 25.19 $ {\bf{26.07}} $
SSIM 0.641 0.685 0.598 0.612 0.631 0.688 $ {\bf{0.722}} $
Fly
0.2 PSNR 27.60 28.08 26.72 27.58 27.73 29.03 $ {\bf{30.48}} $
SSIM 0.808 0.802 0.767 0.748 0.741 0.794 $ {\bf{0.834}} $
0.4 PSNR 26.18 26.43 24.89 25.98 26.04 27.02 $ {\bf{27.88}} $
SSIM 0.720 0.733 0.668 0.696 0.695 0.742 $ {\bf{0.747}} $
Bird
0.1 PSNR 31.12 32.26 30.45 31.53 31.32 33.60 $ {\bf{34.44}} $
SSIM 0.886 0.880 0.788 0.805 0.822 0.868 $ {\bf{0.909}} $
0.3 PSNR 29.45 30.22 28.89 30.08 30.15 32.52 $ {\bf{33.37}} $
SSIM 0.826 0.838 0.738 0.794 0.810 0.850 $ {\bf{0.876}} $
Woman
0.1 PSNR 33.66 32.93 34.04 33.08 33.88 $ {\bf{34.39}} $ 34.27
SSIM 0.911 0.922 0.906 0.902 0.920 $ {\bf{0.942}} $ 0.937
0.4 PSNR 24.71 25.12 24.66 24.99 25.83 26.04 $ {\bf{26.43}} $
SSIM 0.598 0.604 0.620 0.672 0.677 $ {\bf{0.684}} $ 0.682
Eyes
0.2 PSNR 29.87 29.97 28.01 29.68 29.62 30.30 $ {\bf{31.77}} $
SSIM 0.812 0.842 0.786 0.796 0.780 0.840 $ {\bf{0.857}} $
0.4 PSNR 27.44 28.15 26.93 27.94 27.55 28.55 $ {\bf{29.03}} $
SSIM 0.758 0.759 0.650 0.692 0.700 0.774 $ {\bf{0.783}} $
Gazelle
0.5 PSNR 25.19 25.36 23.55 24.02 24.12 25.04 $ {\bf{26.49}} $
SSIM 0.597 0.612 0.509 0.547 0.520 0.616 $ {\bf{0.678}} $
0.6 PSNR 23.45 23.50 22.11 22.44 22.22 23.29 $ {\bf{23.66}} $
SSIM 0.478 0.481 0.403 0.409 0.413 0.510 $ {\bf{0.608}} $
Goose
0.1 PSNR 33.49 33.70 30.88 31.01 31.44 34.17 $ {\bf{34.65}} $
SSIM 0.908 0.920 0.865 0.890 0.896 0.919 $ {\bf{0.921}} $
0.3 PSNR 31.08 31.13 28.91 29.17 29.34 31.01 $ {\bf{32.36}} $
SSIM 0.818 0.851 0.708 0.713 0.709 0.825 $ {\bf{0.866}} $
Image Method
$ \sigma $ noise Metric TV$ ^{\alpha} $ non-convex-TV TV+TV$ ^2 $ TGV TGV$ ^2 $ BM3D proposed
Bridge
0.1 PSNR 31.08 32.33 31.03 32.54 32.72 $ {\bf{33.14}} $ 32.94
SSIM 0.883 0.901 0.878 0.899 0.893 0.922 $ {\bf{0.922}} $
0.2 PSNR 30.57 31.43 30.45 31.17 31.66 $ {\bf{32.09}} $ 32.05
SSIM 0.641 0.685 0.598 0.612 0.631 0.688 $ {\bf{0.722}} $
Castle
0.3 PSNR 29.44 29.57 29.12 30.05 $ {\bf{30.18}} $ 30.14 30.08
SSIM 0.784 0.791 0.778 0.800 0.804 0.819 $ {\bf{0.821}} $
0.4 PSNR 28.81 28.94 28.75 29.22 29.85 $ {\bf{30.04}} $ 30.00
SSIM 0.711 0.725 0.699 0.737 0.738 0.768 $ {\bf{0.772}} $
Baboon
0.2 PSNR 26.88 27.19 24.93 25.40 25.44 27.30 $ {\bf{28.84}} $
SSIM 0.781 0.788 0.714 0.737 0.748 0.789 $ {\bf{0.811}} $
0.5 PSNR 23.88 24.22 22.55 23.11 23.96 25.19 $ {\bf{26.07}} $
SSIM 0.641 0.685 0.598 0.612 0.631 0.688 $ {\bf{0.722}} $
Fly
0.2 PSNR 27.60 28.08 26.72 27.58 27.73 29.03 $ {\bf{30.48}} $
SSIM 0.808 0.802 0.767 0.748 0.741 0.794 $ {\bf{0.834}} $
0.4 PSNR 26.18 26.43 24.89 25.98 26.04 27.02 $ {\bf{27.88}} $
SSIM 0.720 0.733 0.668 0.696 0.695 0.742 $ {\bf{0.747}} $
Bird
0.1 PSNR 31.12 32.26 30.45 31.53 31.32 33.60 $ {\bf{34.44}} $
SSIM 0.886 0.880 0.788 0.805 0.822 0.868 $ {\bf{0.909}} $
0.3 PSNR 29.45 30.22 28.89 30.08 30.15 32.52 $ {\bf{33.37}} $
SSIM 0.826 0.838 0.738 0.794 0.810 0.850 $ {\bf{0.876}} $
Woman
0.1 PSNR 33.66 32.93 34.04 33.08 33.88 $ {\bf{34.39}} $ 34.27
SSIM 0.911 0.922 0.906 0.902 0.920 $ {\bf{0.942}} $ 0.937
0.4 PSNR 24.71 25.12 24.66 24.99 25.83 26.04 $ {\bf{26.43}} $
SSIM 0.598 0.604 0.620 0.672 0.677 $ {\bf{0.684}} $ 0.682
Eyes
0.2 PSNR 29.87 29.97 28.01 29.68 29.62 30.30 $ {\bf{31.77}} $
SSIM 0.812 0.842 0.786 0.796 0.780 0.840 $ {\bf{0.857}} $
0.4 PSNR 27.44 28.15 26.93 27.94 27.55 28.55 $ {\bf{29.03}} $
SSIM 0.758 0.759 0.650 0.692 0.700 0.774 $ {\bf{0.783}} $
Gazelle
0.5 PSNR 25.19 25.36 23.55 24.02 24.12 25.04 $ {\bf{26.49}} $
SSIM 0.597 0.612 0.509 0.547 0.520 0.616 $ {\bf{0.678}} $
0.6 PSNR 23.45 23.50 22.11 22.44 22.22 23.29 $ {\bf{23.66}} $
SSIM 0.478 0.481 0.403 0.409 0.413 0.510 $ {\bf{0.608}} $
Goose
0.1 PSNR 33.49 33.70 30.88 31.01 31.44 34.17 $ {\bf{34.65}} $
SSIM 0.908 0.920 0.865 0.890 0.896 0.919 $ {\bf{0.921}} $
0.3 PSNR 31.08 31.13 28.91 29.17 29.34 31.01 $ {\bf{32.36}} $
SSIM 0.818 0.851 0.708 0.713 0.709 0.825 $ {\bf{0.866}} $
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