American Institute of Mathematical Sciences

Advanced
doi: 10.3934/ipi.2021014

Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

Yixuan Yang 1, , Yuchao Tang 1,, , Meng Wen 2, and  Tieyong Zeng 3,

1. 

Department of Mathematics, Nanchang University, Nanchang 330031, China
2. 

School of Science, Xi'an Polytechnic University, Xi'an, 710048, China
3. 

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

* Corresponding author: Yuchao Tang

Received  January 2020 Revised  December 2020 Published  February 2021

Fund Project: The second author is supported by the National Natural Science Foundation of China (12061045, 11661056), the China Postdoctoral Science Foundation (2015M571989) and the Jiangxi Province Postdoctoral Science Foundation (2015KY51). The third author is supported by the National Natural Science Foundation of China (12001416)

This paper is concerned with the monotone inclusion involving the sum of a finite number of maximally monotone operators and the parallel sum of two maximally monotone operators with bounded linear operators. To solve this monotone inclusion, we first transform it into the formulation of the sum of three maximally monotone operators in a proper product space. Then we derive two efficient iterative algorithms, which combine the partial inverse method with the preconditioned Douglas-Rachford splitting algorithm and the preconditioned proximal point algorithm. Furthermore, we develop an iterative algorithm, which relies on the preconditioned Douglas-Rachford splitting algorithm without using the partial inverse method. We carefully analyze the theoretical convergence of the proposed algorithms. Finally, in order to demonstrate the effectiveness and efficiency of these algorithms, we conduct numerical experiments on a novel image denoising model for salt-and-pepper noise removal. Numerical results show the good performance of the proposed algorithms.

Keywords: Maximally monotone operators, partial inverse, resolvent, Douglas-Rachford splitting algorithm, proximal point algorithm.
Mathematics Subject Classification: Primary: 90C25, 90C46; Secondary: 47A52.
Citation: Yixuan Yang, Yuchao Tang, Meng Wen, Tieyong Zeng. Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications. Inverse Problems & Imaging, doi: 10.3934/ipi.2021014
References:
[1]

M. A. Alghamdi, A. Alotaibi, P. L. Combettes and N. Shahzad, A primal-dual method of partial inverses for composite inclusions, Optim. Lett., 8 (2014), 2271-2284.  doi: 10.1007/s11590-014-0734-x.  Google Scholar

[2]

A. Alliney, A property of the minimum vectors of a regularization functional defined by means of the absolute norm, IEEE Trans. Signal Process., 45 (1997), 913-917.  doi: 10.1109/78.564179.  Google Scholar

[3]

A. Alotaibi, P. L. Combettes and N. Shahzad, Solving coupled composite monotone inclusions by successive fejér approximations of their Kukn-Tucker set, SIAM J. Optim., 24 (2014), 2076-2095.  doi: 10.1137/130950616.  Google Scholar

[4]

H. Attouch and M. Soueycatt, Augmented lagrangian and proximal alternating direction methods of multipliers in hilbert spaces, applications to games, pde's and control, Pacific. J. Optim., 5 (2008), 17-37.   Google Scholar

[5]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2$^nd$ edition, Springer, Cham, 2017. doi: 10.1007/978-3-319-48311-5.  Google Scholar

[6]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[7]

A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process., 18 (2009), 2419-2434.  doi: 10.1109/TIP.2009.2028250.  Google Scholar

[8]

S. Becker, J. Bobin and E. J. Candès, NESTA: A fast and accurate first-order method for sparse recovery, SIAM J. Imaging Sci., 4 (2011), 1-39.  doi: 10.1137/090756855.  Google Scholar

[9]

S. R. Becker and P. L. Combettes, An algorithm for splitting parallel sums of linearly composed monotone operatos with applications to signal recovery, J. Nonlinear Convex Anal., 15 (2014), 137-159.   Google Scholar

[10]

R. I. Boţ, E. R. Csetnek and A. Heinrich, A primal-dual splitting for finding zeros of sums of maximal monotone operators, SIAM J. Optim., 23 (2013), 2011-2036.  doi: 10.1137/12088255X.  Google Scholar

[11]

R. I. Boţ and C. Hendrich, Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators, Inverse Probl. Imaging, 10 (2016), 617-640.  doi: 10.3934/ipi.2016014.  Google Scholar

[12]

R. I. Boţ and C. Hendrich, Convergence analysis for a primal-dual monotone+skew splitting algorithm with applications to total variation minimization, J. Math. Imaging Vis., 49 (2014), 551-568.  doi: 10.1007/s10851-013-0486-8.  Google Scholar

[13]

R. I. Boţ and C. Hendrich, A Dougla-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel -sum type monotone operators, SIAM J. Optim., 23 (2013), 2541-2565.  doi: 10.1137/120901106.  Google Scholar

[14]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distrituted optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2010), 1-122.   Google Scholar

[15]

L. M. Briceño-Arias and P. L. Combettes, A monotone+skew splitting model for composite monotone inclusions in duality, SIAM J. Optim., 21 (2011), 1230-1250.  doi: 10.1137/10081602X.  Google Scholar

[16]

L. M. Briceño-Arias, Forward-Douglas-Rachford splitting and forward-partial inverse method for solving monotone inclusions, Optim., 64 (2015), 1239-1261.  doi: 10.1080/02331934.2013.855210.  Google Scholar

[17]

L. M. Briceño-Arias, Foward-partial inverse forward splitting for solving monotone inclusions, J. Optim. Theory Appl., 166 (2015), 391-413.  doi: 10.1007/s10957-015-0703-2.  Google Scholar

[18]

A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Springer-Verlag, Berlin Heidelberg, 2012.  Google Scholar

[19]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[20]

A. Chambolle and T. Pock, An introduction to continuous optimization for imaging, Acta Numer., 25 (2016), 161-319.  doi: 10.1017/S096249291600009X.  Google Scholar

[21]

F. Chen, L. Shen, Y. Xu and X. Zeng, The moreau envelope approach for the l1/TV image denoising model, Inverse Probl. Imaging, 8 (2014), 53-77.  doi: 10.3934/ipi.2014.8.53.  Google Scholar

[22]

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

[23]

P. L. Combettes and J.-C. Pesquet, A Douglas-Rachford splitting approach to nonsmooth conve variational signal recovery, IEEE J. Sel. Top. Signal Process, 1 (2007), 564-574.  doi: 10.1109/JSTSP.2007.910264.  Google Scholar

[24]

P. L. Combettes and J.-C. Pesquet, Primal-dual splitting algorithm for solving inclusions with mixtures of composite, lipschitzian, and parallel-sum type monotone operators, Set-Valued Var. Anal., 20 (2012), 307-330.  doi: 10.1007/s11228-011-0191-y.  Google Scholar

[25]

P. L. Combettes and B. C. Vũ, Variable metric forward-backward splitting with applications to monotone inclusions in duality, Optim., 63 (2014), 1289-1318.  doi: 10.1080/02331934.2012.733883.  Google Scholar

[26]

P. L. Combettes and J. Eckstein, Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions, Math. Program., 168 (2018), 645-672.  doi: 10.1007/s10107-016-1044-0.  Google Scholar

[27]

P. L. Combettes, Systems of structured monotone inclusions: Duality, algorithms, and applications, SIAM J. Optim., 23 (2013), 2420-2447.  doi: 10.1137/130904160.  Google Scholar

[28]

P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727-748.   Google Scholar

[29]

L. Condat, A primal-dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl., 158 (2013), 460-479.  doi: 10.1007/s10957-012-0245-9.  Google Scholar

[30]

E. X. Fang, B. He, H. Liu and X. Yuan, Generalized alternating direction method of multipliers: new theoretical insights and applications, Math. Program. Comput., 7 (2015), 149-187.  doi: 10.1007/s12532-015-0078-2.  Google Scholar

[31]

M. L. N. Goncalves, M. M. Alves and J. G. Melo, Pointwise and ergodic convergence rates of variable metric proximal alternating direction method of multipliers, J. Optim. Theory Appl., 177 (2018), 448-478.  doi: 10.1007/s10957-018-1232-6.  Google Scholar

[32]

P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979.  doi: 10.1137/0716071.  Google Scholar

[33]

Y.-J. Liu, D. Sun and K.-C. Toh, An implementable proximal point algorithmic framework for nuclear norm minimization, Math. Program., 133 (2012), 399-436.  doi: 10.1007/s10107-010-0437-8.  Google Scholar

[34]

C. A. Micchelli, L. Shen, Y. Xu and X. Zeng, Proximity algorithms for the l1/TV image denoising model, Adv. Comput. Math., 38 (2013), 401-426.  doi: 10.1007/s10444-011-9243-y.  Google Scholar

[35]

C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denoising, Inverse Probl., 27 (2011), 045009, 30 pp. doi: 10.1088/0266-5611/27/4/045009.  Google Scholar

[36]

M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vis., 20 (2004), 99-120.   Google Scholar

[37]

H. Raguet, J. Fadili and G. Peyré, A generalized forward-backward splitting, SIAM J. Imaging Sci., 6 (2013), 1199-1226.  doi: 10.1137/120872802.  Google Scholar

[38]

H. Raguet and L. Landrieu, Preconditioning of a generalized forward-backward splitting and application to optimization on graphs, SIAM J. Imaging Sci., 8 (2015), 2706-2739.  doi: 10.1137/15M1018253.  Google Scholar

[39]

H. Raguet, A note on the forward-Douglas-Rachford splitting for monotone inclusion and convex optimization, Optim. Lett., 13 (2019), 717-740.  doi: 10.1007/s11590-018-1272-8.  Google Scholar

[40]

R. Shefi and M. Teboulle, Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization, SIAM J. Optim., 24 (2014), 269-297.  doi: 10.1137/130910774.  Google Scholar

[41]

E. Y. Sidky, J. H. Jørgensen and X. Pan, Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm, Phys. Med. Biol., 57 (2012), 3065-3091.  doi: 10.1088/0031-9155/57/10/3065.  Google Scholar

[42]

J. E. Spingarn, Partial inverse of a monotone operator, Appl. Math. Optim., 10 (1983), 247-265.  doi: 10.1007/BF01448388.  Google Scholar

[43]

J. E. Spingarn, Applications of the method of partial inverses to convex programming: Decomposition, Math. Programming, 32 (1985), 199-223.  doi: 10.1007/BF01586091.  Google Scholar

[44]

Y. C. Tang, G. R. Wu and C. X. Zhu, An inner-outer iteration method for solving convex optimization problems involving the sum of three convex functions (in chinese), Sci. Sic. Math., 49 (2019), 831-858.  doi: 10.1360/SCM-2017-0313.  Google Scholar

[45]

Y. C. Tang, C. X. Zhu, M. Wen and J. G. Peng, A splitting primal-dual proximity algorithm for solving composite optimization problems, Acta. Math. Sin.-English Ser., 33 (2017), 868-886.  doi: 10.1007/s10114-016-5625-x.  Google Scholar

[46]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806.  Google Scholar

[47]

B. C. Vũ, A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comput. Math., 38 (2013), 667-681.  doi: 10.1007/s10444-011-9254-8.  Google Scholar

[48]

Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar

[49]

C. Zong, Y. Tang and Y. J. Cho, Convergence analysis of an inexact three-operator splitting algorithm, Symmetry, 10 (2018), 563. doi: 10.3390/sym10110563.  Google Scholar

show all references

References:
[1]

M. A. Alghamdi, A. Alotaibi, P. L. Combettes and N. Shahzad, A primal-dual method of partial inverses for composite inclusions, Optim. Lett., 8 (2014), 2271-2284.  doi: 10.1007/s11590-014-0734-x.  Google Scholar

[2]

A. Alliney, A property of the minimum vectors of a regularization functional defined by means of the absolute norm, IEEE Trans. Signal Process., 45 (1997), 913-917.  doi: 10.1109/78.564179.  Google Scholar

[3]

A. Alotaibi, P. L. Combettes and N. Shahzad, Solving coupled composite monotone inclusions by successive fejér approximations of their Kukn-Tucker set, SIAM J. Optim., 24 (2014), 2076-2095.  doi: 10.1137/130950616.  Google Scholar

[4]

H. Attouch and M. Soueycatt, Augmented lagrangian and proximal alternating direction methods of multipliers in hilbert spaces, applications to games, pde's and control, Pacific. J. Optim., 5 (2008), 17-37.   Google Scholar

[5]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2$^nd$ edition, Springer, Cham, 2017. doi: 10.1007/978-3-319-48311-5.  Google Scholar

[6]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[7]

A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process., 18 (2009), 2419-2434.  doi: 10.1109/TIP.2009.2028250.  Google Scholar

[8]

S. Becker, J. Bobin and E. J. Candès, NESTA: A fast and accurate first-order method for sparse recovery, SIAM J. Imaging Sci., 4 (2011), 1-39.  doi: 10.1137/090756855.  Google Scholar

[9]

S. R. Becker and P. L. Combettes, An algorithm for splitting parallel sums of linearly composed monotone operatos with applications to signal recovery, J. Nonlinear Convex Anal., 15 (2014), 137-159.   Google Scholar

[10]

R. I. Boţ, E. R. Csetnek and A. Heinrich, A primal-dual splitting for finding zeros of sums of maximal monotone operators, SIAM J. Optim., 23 (2013), 2011-2036.  doi: 10.1137/12088255X.  Google Scholar

[11]

R. I. Boţ and C. Hendrich, Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators, Inverse Probl. Imaging, 10 (2016), 617-640.  doi: 10.3934/ipi.2016014.  Google Scholar

[12]

R. I. Boţ and C. Hendrich, Convergence analysis for a primal-dual monotone+skew splitting algorithm with applications to total variation minimization, J. Math. Imaging Vis., 49 (2014), 551-568.  doi: 10.1007/s10851-013-0486-8.  Google Scholar

[13]

R. I. Boţ and C. Hendrich, A Dougla-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel -sum type monotone operators, SIAM J. Optim., 23 (2013), 2541-2565.  doi: 10.1137/120901106.  Google Scholar

[14]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distrituted optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2010), 1-122.   Google Scholar

[15]

L. M. Briceño-Arias and P. L. Combettes, A monotone+skew splitting model for composite monotone inclusions in duality, SIAM J. Optim., 21 (2011), 1230-1250.  doi: 10.1137/10081602X.  Google Scholar

[16]

L. M. Briceño-Arias, Forward-Douglas-Rachford splitting and forward-partial inverse method for solving monotone inclusions, Optim., 64 (2015), 1239-1261.  doi: 10.1080/02331934.2013.855210.  Google Scholar

[17]

L. M. Briceño-Arias, Foward-partial inverse forward splitting for solving monotone inclusions, J. Optim. Theory Appl., 166 (2015), 391-413.  doi: 10.1007/s10957-015-0703-2.  Google Scholar

[18]

A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Springer-Verlag, Berlin Heidelberg, 2012.  Google Scholar

[19]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[20]

A. Chambolle and T. Pock, An introduction to continuous optimization for imaging, Acta Numer., 25 (2016), 161-319.  doi: 10.1017/S096249291600009X.  Google Scholar

[21]

F. Chen, L. Shen, Y. Xu and X. Zeng, The moreau envelope approach for the l1/TV image denoising model, Inverse Probl. Imaging, 8 (2014), 53-77.  doi: 10.3934/ipi.2014.8.53.  Google Scholar

[22]

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

[23]

P. L. Combettes and J.-C. Pesquet, A Douglas-Rachford splitting approach to nonsmooth conve variational signal recovery, IEEE J. Sel. Top. Signal Process, 1 (2007), 564-574.  doi: 10.1109/JSTSP.2007.910264.  Google Scholar

[24]

P. L. Combettes and J.-C. Pesquet, Primal-dual splitting algorithm for solving inclusions with mixtures of composite, lipschitzian, and parallel-sum type monotone operators, Set-Valued Var. Anal., 20 (2012), 307-330.  doi: 10.1007/s11228-011-0191-y.  Google Scholar

[25]

P. L. Combettes and B. C. Vũ, Variable metric forward-backward splitting with applications to monotone inclusions in duality, Optim., 63 (2014), 1289-1318.  doi: 10.1080/02331934.2012.733883.  Google Scholar

[26]

P. L. Combettes and J. Eckstein, Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions, Math. Program., 168 (2018), 645-672.  doi: 10.1007/s10107-016-1044-0.  Google Scholar

[27]

P. L. Combettes, Systems of structured monotone inclusions: Duality, algorithms, and applications, SIAM J. Optim., 23 (2013), 2420-2447.  doi: 10.1137/130904160.  Google Scholar

[28]

P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727-748.   Google Scholar

[29]

L. Condat, A primal-dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl., 158 (2013), 460-479.  doi: 10.1007/s10957-012-0245-9.  Google Scholar

[30]

E. X. Fang, B. He, H. Liu and X. Yuan, Generalized alternating direction method of multipliers: new theoretical insights and applications, Math. Program. Comput., 7 (2015), 149-187.  doi: 10.1007/s12532-015-0078-2.  Google Scholar

[31]

M. L. N. Goncalves, M. M. Alves and J. G. Melo, Pointwise and ergodic convergence rates of variable metric proximal alternating direction method of multipliers, J. Optim. Theory Appl., 177 (2018), 448-478.  doi: 10.1007/s10957-018-1232-6.  Google Scholar

[32]

P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979.  doi: 10.1137/0716071.  Google Scholar

[33]

Y.-J. Liu, D. Sun and K.-C. Toh, An implementable proximal point algorithmic framework for nuclear norm minimization, Math. Program., 133 (2012), 399-436.  doi: 10.1007/s10107-010-0437-8.  Google Scholar

[34]

C. A. Micchelli, L. Shen, Y. Xu and X. Zeng, Proximity algorithms for the l1/TV image denoising model, Adv. Comput. Math., 38 (2013), 401-426.  doi: 10.1007/s10444-011-9243-y.  Google Scholar

[35]

C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denoising, Inverse Probl., 27 (2011), 045009, 30 pp. doi: 10.1088/0266-5611/27/4/045009.  Google Scholar

[36]

M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vis., 20 (2004), 99-120.   Google Scholar

[37]

H. Raguet, J. Fadili and G. Peyré, A generalized forward-backward splitting, SIAM J. Imaging Sci., 6 (2013), 1199-1226.  doi: 10.1137/120872802.  Google Scholar

[38]

H. Raguet and L. Landrieu, Preconditioning of a generalized forward-backward splitting and application to optimization on graphs, SIAM J. Imaging Sci., 8 (2015), 2706-2739.  doi: 10.1137/15M1018253.  Google Scholar

[39]

H. Raguet, A note on the forward-Douglas-Rachford splitting for monotone inclusion and convex optimization, Optim. Lett., 13 (2019), 717-740.  doi: 10.1007/s11590-018-1272-8.  Google Scholar

[40]

R. Shefi and M. Teboulle, Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization, SIAM J. Optim., 24 (2014), 269-297.  doi: 10.1137/130910774.  Google Scholar

[41]

E. Y. Sidky, J. H. Jørgensen and X. Pan, Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm, Phys. Med. Biol., 57 (2012), 3065-3091.  doi: 10.1088/0031-9155/57/10/3065.  Google Scholar

[42]

J. E. Spingarn, Partial inverse of a monotone operator, Appl. Math. Optim., 10 (1983), 247-265.  doi: 10.1007/BF01448388.  Google Scholar

[43]

J. E. Spingarn, Applications of the method of partial inverses to convex programming: Decomposition, Math. Programming, 32 (1985), 199-223.  doi: 10.1007/BF01586091.  Google Scholar

[44]

Y. C. Tang, G. R. Wu and C. X. Zhu, An inner-outer iteration method for solving convex optimization problems involving the sum of three convex functions (in chinese), Sci. Sic. Math., 49 (2019), 831-858.  doi: 10.1360/SCM-2017-0313.  Google Scholar

[45]

Y. C. Tang, C. X. Zhu, M. Wen and J. G. Peng, A splitting primal-dual proximity algorithm for solving composite optimization problems, Acta. Math. Sin.-English Ser., 33 (2017), 868-886.  doi: 10.1007/s10114-016-5625-x.  Google Scholar

[46]

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446.  doi: 10.1137/S0363012998338806.  Google Scholar

[47]

B. C. Vũ, A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comput. Math., 38 (2013), 667-681.  doi: 10.1007/s10444-011-9254-8.  Google Scholar

[48]

Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar

[49]

C. Zong, Y. Tang and Y. J. Cho, Convergence analysis of an inexact three-operator splitting algorithm, Symmetry, 10 (2018), 563. doi: 10.3390/sym10110563.  Google Scholar

Figure 1.  Test image. (a) The original image; (b) The noise image
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Figure 2.  Original images. (a) "Barbara"; (b) "Building"
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$ 10 \% $ to $ 90 \% $; The second column is the results of the PADMM; The third column is the results of the PDCP; The forth column is the results of the proposed Algorithm 2">Figure 3.  Noisy and restored "Barbara" images. The first column is the noise images of "Barbara" with the noise level from $ 10 \% $ to $ 90 \% $; The second column is the results of the PADMM; The third column is the results of the PDCP; The forth column is the results of the proposed Algorithm 2
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$ 10 \% $ to $ 90 \% $; The second column is the results of the PADMM; The third column is the results of the PDCP; The forth column is the results of the proposed Algorithm 2">Figure 4.  Noisy and restored "Building" images. The first column is the noise images of "Building" with the noise level from $ 10 \% $ to $ 90 \% $; The second column is the results of the PADMM; The third column is the results of the PDCP; The forth column is the results of the proposed Algorithm 2
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Table 1.  Numerical results of Algorithm 1, Algorithm 2 and Algorithm 3 for different selection of regularization parameters in terms of SNR, SSIM and the number of iterations (Iter)
Methods $ \lambda_1 $ $ \lambda_2 $ $ \epsilon = 10^{-2} $ $ \epsilon = 10^{-4} $
SNR (dB) SSIM Iter SNR (dB) SSIM Iter
Algorithm 1 $ 0 $ 0.5 $ 4.4173 $ $ 0.0159 $ $ 149 $ $ 4.4648 $ $ 0.0162 $ $ - $
$ 1 $ $ 4.4182 $ $ 0.0159 $ $ 147 $ $ 4.4749 $ $ 0.0162 $ $ - $
$ 20 $ $ 53.0111 $ $ 0.9988 $ $ 738 $ $ 62.4224 $ $ 0.9999 $ $ - $
$ 0.5 $ $ 0 $ $ 23.8749 $ $ 0.8813 $ $ 29 $ $ 24.3263 $ $ 0.9195 $ $ 110 $
$ 1 $ $ 30.3788 $ $ 0.9340 $ $ 30 $ $ 34.3648 $ $ 0.9905 $ $ 151 $
$ 2 $ $ 31.2330 $ $ 0.9331 $ $ 30 $ $ 52.4987 $ $ 0.9996 $ $ 276 $
$ 0.5 $ $ 0.5 $ $ 27.2958 $ $ 0.9269 $ $ 24 $ $ 30.7593 $ $ 0.9837 $ $ 139 $
$ 1 $ $ 1 $ $ 29.2181 $ $ 0.9434 $ $ 24 $ $ 47.8240 $ $ 0.9988 $ $ 135 $
$ 2 $ $ 2 $ $ 29.3063 $ $ 0.9391 $ $ 25 $ $ 79.2541 $ $ 1 $ $ 210 $
Algorithm 2 $ 0 $ 0.5 $ 4.5444 $ $ 0.0164 $ $ 24 $ $ 4.4263 $ $ 0.0159 $ $ 45 $
$ 1 $ $ 4.5044 $ $ 0.0164 $ $ 25 $ $ 4.4260 $ $ 0.0159 $ $ 46 $
$ 20 $ $ 32.8316 $ $ 0.9507 $ $ 33 $ $ 71.4020 $ $ 1 $ $ 89 $
$ 0.5 $ $ 0 $ $ 21.2139 $ $ 0.0.7467 $ $ 42 $ $ 22.8320 $ $ 0.9143 $ $ 223 $
$ 1 $ $ 27.0660 $ $ 0.7962 $ $ 49 $ $ 34.2508 $ $ 0.9903 $ $ 236 $
$ 2 $ $ 26.9064 $ $ 0.7831 $ $ 50 $ $ 53.0912 $ $ 0.9996 $ $ 272 $
$ 0.5 $ $ 0.5 $ $ 23.7828 $ $ 0.8072 $ $ 36 $ $ 30.5978 $ $ 0.9808 $ $ 168 $
$ 1 $ $ 1 $ $ 28.3761 $ $ 0.8454 $ $ 35 $ $ 48.3065 $ $ 0.9990 $ $ 111 $
$ 2 $ $ 2 $ $ 26.4981 $ $ 0.9092 $ $ 33 $ $ 76.5604 $ $ 1 $ $ 195 $
Algorithm 3 $ 0 $ 0.5 $ 4.4169 $ $ 0.0159 $ $ 373 $ $ 4.4219 $ $ 0.0159 $ $ - $
$ 1 $ $ 4.4246 $ $ 0.0159 $ $ 299 $ $ 4.4235 $ $ 0.0159 $ $ - $
$ 20 $ $ 39.9637 $ $ 0.9762 $ $ - $ $ 39.9637 $ $ 0.9762 $ $ - $
$ 0.5 $ $ 0 $ $ 23.1899 $ $ 0.9011 $ $ 61 $ $ 23.3077 $ $ 0.9151 $ $ 448 $
$ 1 $ $ 32.4599 $ $ 0.9695 $ $ 49 $ $ 34.7962 $ $ 0.9913 $ $ 308 $
$ 2 $ $ 35.2484 $ $ 0.9751 $ $ 53 $ $ 55.4857 $ $ 0.9998 $ $ 389 $
$ 0.5 $ $ 0.5 $ $ 29.7816 $ $ 0.9611 $ $ 54 $ $ 30.7667 $ $ 0.9838 $ $ 512 $
$ 1 $ $ 1 $ $ 35.3009 $ $ 0.9731 $ $ 43 $ $ 48.0544 $ $ 0.9989 $ $ 321 $
$ 2 $ $ 2 $ $ 35.5435 $ $ 0.9767 $ $ 41 $ $ 85.6388 $ $ 1 $ $ 279 $
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Methods $ \lambda_1 $ $ \lambda_2 $ $ \epsilon = 10^{-2} $ $ \epsilon = 10^{-4} $
SNR (dB) SSIM Iter SNR (dB) SSIM Iter
Algorithm 1 $ 0 $ 0.5 $ 4.4173 $ $ 0.0159 $ $ 149 $ $ 4.4648 $ $ 0.0162 $ $ - $
$ 1 $ $ 4.4182 $ $ 0.0159 $ $ 147 $ $ 4.4749 $ $ 0.0162 $ $ - $
$ 20 $ $ 53.0111 $ $ 0.9988 $ $ 738 $ $ 62.4224 $ $ 0.9999 $ $ - $
$ 0.5 $ $ 0 $ $ 23.8749 $ $ 0.8813 $ $ 29 $ $ 24.3263 $ $ 0.9195 $ $ 110 $
$ 1 $ $ 30.3788 $ $ 0.9340 $ $ 30 $ $ 34.3648 $ $ 0.9905 $ $ 151 $
$ 2 $ $ 31.2330 $ $ 0.9331 $ $ 30 $ $ 52.4987 $ $ 0.9996 $ $ 276 $
$ 0.5 $ $ 0.5 $ $ 27.2958 $ $ 0.9269 $ $ 24 $ $ 30.7593 $ $ 0.9837 $ $ 139 $
$ 1 $ $ 1 $ $ 29.2181 $ $ 0.9434 $ $ 24 $ $ 47.8240 $ $ 0.9988 $ $ 135 $
$ 2 $ $ 2 $ $ 29.3063 $ $ 0.9391 $ $ 25 $ $ 79.2541 $ $ 1 $ $ 210 $
Algorithm 2 $ 0 $ 0.5 $ 4.5444 $ $ 0.0164 $ $ 24 $ $ 4.4263 $ $ 0.0159 $ $ 45 $
$ 1 $ $ 4.5044 $ $ 0.0164 $ $ 25 $ $ 4.4260 $ $ 0.0159 $ $ 46 $
$ 20 $ $ 32.8316 $ $ 0.9507 $ $ 33 $ $ 71.4020 $ $ 1 $ $ 89 $
$ 0.5 $ $ 0 $ $ 21.2139 $ $ 0.0.7467 $ $ 42 $ $ 22.8320 $ $ 0.9143 $ $ 223 $
$ 1 $ $ 27.0660 $ $ 0.7962 $ $ 49 $ $ 34.2508 $ $ 0.9903 $ $ 236 $
$ 2 $ $ 26.9064 $ $ 0.7831 $ $ 50 $ $ 53.0912 $ $ 0.9996 $ $ 272 $
$ 0.5 $ $ 0.5 $ $ 23.7828 $ $ 0.8072 $ $ 36 $ $ 30.5978 $ $ 0.9808 $ $ 168 $
$ 1 $ $ 1 $ $ 28.3761 $ $ 0.8454 $ $ 35 $ $ 48.3065 $ $ 0.9990 $ $ 111 $
$ 2 $ $ 2 $ $ 26.4981 $ $ 0.9092 $ $ 33 $ $ 76.5604 $ $ 1 $ $ 195 $
Algorithm 3 $ 0 $ 0.5 $ 4.4169 $ $ 0.0159 $ $ 373 $ $ 4.4219 $ $ 0.0159 $ $ - $
$ 1 $ $ 4.4246 $ $ 0.0159 $ $ 299 $ $ 4.4235 $ $ 0.0159 $ $ - $
$ 20 $ $ 39.9637 $ $ 0.9762 $ $ - $ $ 39.9637 $ $ 0.9762 $ $ - $
$ 0.5 $ $ 0 $ $ 23.1899 $ $ 0.9011 $ $ 61 $ $ 23.3077 $ $ 0.9151 $ $ 448 $
$ 1 $ $ 32.4599 $ $ 0.9695 $ $ 49 $ $ 34.7962 $ $ 0.9913 $ $ 308 $
$ 2 $ $ 35.2484 $ $ 0.9751 $ $ 53 $ $ 55.4857 $ $ 0.9998 $ $ 389 $
$ 0.5 $ $ 0.5 $ $ 29.7816 $ $ 0.9611 $ $ 54 $ $ 30.7667 $ $ 0.9838 $ $ 512 $
$ 1 $ $ 1 $ $ 35.3009 $ $ 0.9731 $ $ 43 $ $ 48.0544 $ $ 0.9989 $ $ 321 $
$ 2 $ $ 2 $ $ 35.5435 $ $ 0.9767 $ $ 41 $ $ 85.6388 $ $ 1 $ $ 279 $
Table 2.  Regularization parameters $ \lambda_1 $ and $ \lambda_2 $ for the PADMM (76) and Algorithm 2 under different noise levels
Images 10$ \% $ 30$ \% $ 50$ \% $ 70$ \% $ 90$ \% $
$ \lambda_{1} $ $ \lambda_{2} $ $ \lambda_{1} $ $ \lambda_{2} $ $ \lambda_{1} $ $ \lambda_{2} $ $ \lambda_{1} $ $ \lambda_{2} $ $ \lambda_{1} $ $ \lambda_{2} $
Barbara $ 0.3 $ $ 3.4 $ $ 0.4 $ $ 3.3 $ $ 0.5 $ $ 4.3 $ $ 0.6 $ $ 4.6 $ $ 0.7 $ $ 4.0 $
Building $ 0.3 $ $ 5.8 $ $ 0.4 $ $ 5.6 $ $ 0.4 $ $ 8.4 $ $ 0.4 $ $ 10.2 $ $ 0.4 $ $ 10.5 $
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Images 10$ \% $ 30$ \% $ 50$ \% $ 70$ \% $ 90$ \% $
$ \lambda_{1} $ $ \lambda_{2} $ $ \lambda_{1} $ $ \lambda_{2} $ $ \lambda_{1} $ $ \lambda_{2} $ $ \lambda_{1} $ $ \lambda_{2} $ $ \lambda_{1} $ $ \lambda_{2} $
Barbara $ 0.3 $ $ 3.4 $ $ 0.4 $ $ 3.3 $ $ 0.5 $ $ 4.3 $ $ 0.6 $ $ 4.6 $ $ 0.7 $ $ 4.0 $
Building $ 0.3 $ $ 5.8 $ $ 0.4 $ $ 5.6 $ $ 0.4 $ $ 8.4 $ $ 0.4 $ $ 10.2 $ $ 0.4 $ $ 10.5 $
Table 3.  Regularization parameters $ \lambda_1 $ for the PDCP (77) under different noise levels
Images 10$ \% $ 30$ \% $ 50$ \% $ 70$ \% $ 90$ \% $
$ \lambda_{1} $ $ \lambda_{1} $ $ \lambda_{1} $ $ \lambda_{1} $ $ \lambda_{1} $
Barbara $ 0.4 $ $ 0.55 $ $ 0.75 $ $ 0.9 $ $ 1.2 $
Building $ 0.4 $ $ 0.55 $ $ 0.7 $ $ 0.85 $ $ 1.15 $
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Images 10$ \% $ 30$ \% $ 50$ \% $ 70$ \% $ 90$ \% $
$ \lambda_{1} $ $ \lambda_{1} $ $ \lambda_{1} $ $ \lambda_{1} $ $ \lambda_{1} $
Barbara $ 0.4 $ $ 0.55 $ $ 0.75 $ $ 0.9 $ $ 1.2 $
Building $ 0.4 $ $ 0.55 $ $ 0.7 $ $ 0.85 $ $ 1.15 $
Table 4.  Numerical results of the PADMM (76), the PDCP (77) and the proposed Algorithm 2 for different noise levels in terms of SNR (dB), PSNR (dB), SSIM, number of iterations (Iter) and CPU time (in seconds)
Level Methods Barbara Building
SNR PSNR SSIM Iter CPU SNR PSNR SSIM Iter CPU
10$ \% $ PADMM $ 23.1733 $ $ 29.5804 $ $ 0.9210 $ $ 890 $ $ 186.60 $ $ 25.7280 $ $ 31.5560 $ $ 0.9418 $ $ 990 $ $ 200.47 $
PDCP $ 21.7728 $ $ 28.1799 $ $ 0.9015 $ $ 297 $ $ 21.47 $ $ 21.9205 $ $ 27.7484 $ $ 0.8984 $ $ 262 $ $ 18.86 $
Algorithm 2 $ 23.1722 $ $ 29.5792 $ $ 0.9210 $ $ 122 $ $ 25.61 $ $ 25.7268 $ $ 31.5547 $ 0.9418 $ 110 $ $ 23.91 $
30$ \% $ PADMM 18.9947 $ 25.4018 $ $ 0.7976 $ $ 1232 $ $ 241.68 $ $ 20.0999 $ $ 25.9278 $ $ 0.7996 $ $ 1356 $ $ 260.67 $
PDCP $ 18.0216 $ $ 24.4287 $ $ 0.7553 $ $ 686 $ $ 48.35 $ $ 18.1008 $ $ 23.9288 $ $ 0.7109 $ $ 688 $ $ 52.04 $
Algorithm 2 $ 18.9937 $ $ 25.4008 $ $ 0.7981 $ $ 180 $ $ 37.26 $ $ 20.0902 $ $ 25.9181 $ $ 0.7988 $ $ 159 $ $ 33.12 $
50$ \% $ PADMM $ 17.2073 $ $ 23.6144 $ $ 0.6889 $ $ 2804 $ $ 557.60 $ $ 17.6625 $ $ 23.4905 $ $ 0.6647 $ $ 2001 $ $ 397.35 $
PDCP $ 17.0137 $ $ 23.4205 $ $ 0.6700 $ $ 904 $ $ 64.98 $ $ 15.9925 $ $ 21.8204 $ $ 0.5546 $ $ 861 $ $ 59.20 $
Algorithm 2 $ 17.2135 $ $ 23.6206 $ $ 0.6905 $ $ 210 $ $ 43.40 $ $ 17.6619 $ $ 23.4898 $ $ 0.6646 $ $ 169 $ $ 34.49 $
70$ \% $ PADMM $ 15.8733 $ $ 22.2804 $ $ 0.5972 $ $ 2001 $ $ 789.97 $ $ 15.6335 $ $ 21.4615 $ $ 0.5067 $ $ 2001 $ $ 529.72 $
PDCP $ 15.6142 $ $ 22.0213 $ $ 0.5852 $ $ 1322 $ $ 93.27 $ $ 13.8911 $ $ 19.7190 $ $ 0.3736 $ $ 1360 $ $ 94.47 $
Algorithm 2 $ 15.8856 $ $ 22.2927 $ $ 0.6000 $ $ 234 $ $ 47.93 $ $ 15.6387 $ $ 21.4666 $ $ 0.5070 $ $ 196 $ $ 41.08 $
90$ \% $ PADMM $ 12.4587 $ $ 18.8658 $ $ 0.4401 $ $ 2001 $ $ 856.01 $ $ 12.4100 $ $ 18.2379 $ $ 0.2730 $ $ 2001 $ $ 569.67 $
PDCP $ 12.4733 $ $ 18.8804 $ $ 0.4617 $ $ 2278 $ $ 172.28 $ $ 11.7455 $ $ 17.5735 $ $ 0.2278 $ $ 1987 $ $ 134.76 $
Algorithm 2 $ 12.4789 $ $ 18.8860 $ $ 0.4441 $ $ 362 $ $ 79.30 $ $ 12.4224 $ $ 18.2503 $ $ 0.2735 $ $ 332 $ $ 72.85 $
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Level Methods Barbara Building
SNR PSNR SSIM Iter CPU SNR PSNR SSIM Iter CPU
10$ \% $ PADMM $ 23.1733 $ $ 29.5804 $ $ 0.9210 $ $ 890 $ $ 186.60 $ $ 25.7280 $ $ 31.5560 $ $ 0.9418 $ $ 990 $ $ 200.47 $
PDCP $ 21.7728 $ $ 28.1799 $ $ 0.9015 $ $ 297 $ $ 21.47 $ $ 21.9205 $ $ 27.7484 $ $ 0.8984 $ $ 262 $ $ 18.86 $
Algorithm 2 $ 23.1722 $ $ 29.5792 $ $ 0.9210 $ $ 122 $ $ 25.61 $ $ 25.7268 $ $ 31.5547 $ 0.9418 $ 110 $ $ 23.91 $
30$ \% $ PADMM 18.9947 $ 25.4018 $ $ 0.7976 $ $ 1232 $ $ 241.68 $ $ 20.0999 $ $ 25.9278 $ $ 0.7996 $ $ 1356 $ $ 260.67 $
PDCP $ 18.0216 $ $ 24.4287 $ $ 0.7553 $ $ 686 $ $ 48.35 $ $ 18.1008 $ $ 23.9288 $ $ 0.7109 $ $ 688 $ $ 52.04 $
Algorithm 2 $ 18.9937 $ $ 25.4008 $ $ 0.7981 $ $ 180 $ $ 37.26 $ $ 20.0902 $ $ 25.9181 $ $ 0.7988 $ $ 159 $ $ 33.12 $
50$ \% $ PADMM $ 17.2073 $ $ 23.6144 $ $ 0.6889 $ $ 2804 $ $ 557.60 $ $ 17.6625 $ $ 23.4905 $ $ 0.6647 $ $ 2001 $ $ 397.35 $
PDCP $ 17.0137 $ $ 23.4205 $ $ 0.6700 $ $ 904 $ $ 64.98 $ $ 15.9925 $ $ 21.8204 $ $ 0.5546 $ $ 861 $ $ 59.20 $
Algorithm 2 $ 17.2135 $ $ 23.6206 $ $ 0.6905 $ $ 210 $ $ 43.40 $ $ 17.6619 $ $ 23.4898 $ $ 0.6646 $ $ 169 $ $ 34.49 $
70$ \% $ PADMM $ 15.8733 $ $ 22.2804 $ $ 0.5972 $ $ 2001 $ $ 789.97 $ $ 15.6335 $ $ 21.4615 $ $ 0.5067 $ $ 2001 $ $ 529.72 $
PDCP $ 15.6142 $ $ 22.0213 $ $ 0.5852 $ $ 1322 $ $ 93.27 $ $ 13.8911 $ $ 19.7190 $ $ 0.3736 $ $ 1360 $ $ 94.47 $
Algorithm 2 $ 15.8856 $ $ 22.2927 $ $ 0.6000 $ $ 234 $ $ 47.93 $ $ 15.6387 $ $ 21.4666 $ $ 0.5070 $ $ 196 $ $ 41.08 $
90$ \% $ PADMM $ 12.4587 $ $ 18.8658 $ $ 0.4401 $ $ 2001 $ $ 856.01 $ $ 12.4100 $ $ 18.2379 $ $ 0.2730 $ $ 2001 $ $ 569.67 $
PDCP $ 12.4733 $ $ 18.8804 $ $ 0.4617 $ $ 2278 $ $ 172.28 $ $ 11.7455 $ $ 17.5735 $ $ 0.2278 $ $ 1987 $ $ 134.76 $
Algorithm 2 $ 12.4789 $ $ 18.8860 $ $ 0.4441 $ $ 362 $ $ 79.30 $ $ 12.4224 $ $ 18.2503 $ $ 0.2735 $ $ 332 $ $ 72.85 $
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