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January  2019, 15(1): 159-175. doi: 10.3934/jimo.2018037

Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step

Yuan Shen , and  Lei Ji

School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210023, China

* Corresponding author: Yuan Shen

Received  May 2017 Revised  September 02, 2017 Published  January 2019 Early access  April 2018

Fund Project: Research supported by National Natural Science Foundation of China under grant 11401295 and by National Natural Science Foundation Tianyuan Project under grant 11726618 and by Jiangsu Provincial Natural Science Foundation under grant BK20141007 and by Major Program of the National Social Science Foundation of China under Grant 12&ZD114 and by National Social Science Foundation of China under Grant 15BGL58 and by Jiangsu Provincial Social Science Foundation under Grant 14EUA001 and by Qinglan Project of Jiangsu Province.

In this paper, we propose a partial convolution model for image deblurring and denoising. We also devise a new linearized alternating direction method of multipliers (ADMM) with an extension step. As the computation of its subproblem is simple enough to have closed-form solutions, its per-iteration cost is low; however, the relaxed parameter condition together with the extra extension step inspired by Ye and Yuan's ADMM enables faster convergence than the original linearized ADMM. Preliminary experimental results show that our algorithm can produce better quality results than some existing efficient algorithms within a similar computation time. The performance advantage of our algorithm is particularly evident at high noise ratios.

Keywords: Convex optimization, proximal point algorithm, augmented Lagrangian, total variation, deblur and denoise, partial convolution.
Mathematics Subject Classification: Primary: 90C30, 65K05; Secondary: 94A08.
Citation: Yuan Shen, Lei Ji. Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step. Journal of Industrial & Management Optimization, 2019, 15 (1) : 159-175. doi: 10.3934/jimo.2018037
References:
[1]

R. Acar and C. R. Vogel, Analysis of total variation penalty methods, Inverse Probl., 10 (1994), 1217-1229.  doi: 10.1088/0266-5611/10/6/003.  Google Scholar

[2]

S. Alliney, Digital filters as absolute norm regularizers, IEEE Trans. Signal Proces., 40 (1992), 1548-1562.   Google Scholar

[3]

J. Barzilai and J.M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[4]

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

[5]

S. Becker, J. Bobin and E. Candes, 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

[6]

D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.  Google Scholar

[7]

J. Bioucas-Dias and M. Figueiredo, A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration, IEEE Trans. Image Proces., 16 (2007), 2992-3004.  doi: 10.1109/TIP.2007.909319.  Google Scholar

[8]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2011), 1-122.   Google Scholar

[9]

J. Cai, R.H. Chan and M. Nikolova, Two phase methods for deblurring images corrupted by impulse plus gaussian noise, Inverse Probl. Imag., 2 (2008), 187-204.  doi: 10.3934/ipi.2008.2.187.  Google Scholar

[10]

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

[11]

R.H. Chan, C.W. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detector and detail-preserving regularization, IEEE Trans. Imag. Process., 14 (2005), 1479-1485.   Google Scholar

[12]

R.H. Chan, M. Tao and X. Yuan, Linearized alternating direction method of multipliers for constrained linear least-squares problem, E. Asian J. Appl. Math., 2 (2012), 326-341.  doi: 10.4208/eajam.270812.161112a.  Google Scholar

[13]

T.F. Chan and S. Esedoglu, Aspects of total variation regularized l1 function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.  doi: 10.1137/040604297.  Google Scholar

[14]

T.F. Chan, G. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.  Google Scholar

[15]

C. Chen, B. He, Y. Ye and X. Yuan, The direct extension of admm for multi-block convex minimization problems is not necessarily convergent, Math. Program. Series A, 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.  Google Scholar

[16]

C. Chen, Y. Shen and Y. You, On the convergence analysis of the alternating direction method of multipliers with three blocks, Abstr. Appl. Anal., 2013 (2013), Art. ID 183961, 7 pp.  Google Scholar

[17]

S. W.J. Cho and S. Lee, Handling outliers in non-blind image deconvolution, International Conference on Computer Vision, (2011), 495-502.   Google Scholar

[18]

I. Daubechies, M. Defrise and C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.  Google Scholar

[19]

J. Douglas and H.H. Rachford, On the numerical solution of the heat conduction problem in 2 and 3 space variables, Trans. Amer. Math. Soc., 82 (1956), 421-439.  doi: 10.1090/S0002-9947-1956-0084194-4.  Google Scholar

[20]

E. Esser, Applications of lagrangian-based alternating direction methods and connections to split bregman, Manuscript, (2009), ftp://ftp.math.ucla.edu/pub/camreport/cam09-31.pdf. Google Scholar

[21]

M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland, Amsterdam, 1983.  Google Scholar

[22]

D. Gabay, Applications of the method of multipliers to variational inequalities, In Augmented Lagrangian Methods: Applications to the Solution of Boundary Value Problems, M. Fortin and R. Glowinski, Eds. Horth-Hollan, Amsterdam, 1983.  Google Scholar

[23]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Comput. Math. Appl., (1976), 17-40.   Google Scholar

[24] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984.   Google Scholar
[25]

R. Glowinski and A. Marrocco, Sur lapproximation par elements finis dordre un, et la resolution par penalisation-dualite dune classe de problemes de dirichlet nonlineaires, Rev. Francaise dAut. Inf. Rech. Oper., 9 (1975), 41-76.   Google Scholar

[26]

B. He, M. Tao and X. Yuan, Alternating direction method with gaussian back substitution for separable convex programming, SIAM J. Optim., 22 (2012), 313-340.  doi: 10.1137/110822347.  Google Scholar

[27]

B. He and X. Yuan, On the convergence rate of douglas-rachford operator splitting method, Math. Program., 153 (2015), 715-722.  doi: 10.1007/s10107-014-0805-x.  Google Scholar

[28]

M. Hong and Z. Luo, On the linear convergence of the alternating direction method of multipliers, arXiv preprint, arXiv: 1208.3922, (2013). Google Scholar

[29]

Y. Huang, M. K. Ng and Y.-W. Wen, A fast total variation minimization method for image restoration, Multiscale Model. Sim., 7 (2008), 774-795.  doi: 10.1137/070703533.  Google Scholar

[30]

Y. Jiao, Q. Jin, X. Lu and W. Wang, Alternating direction method of multipliers for linear inverse problems, SIAM J. Numer. Anal., 54 (2016), 2114-2137.  doi: 10.1137/15M1029308.  Google Scholar

[31]

Y. Jiao, Q. Jin, X. Lu and W. Wang, Preconditioned alternating direction method of multipliers for inverse problems with constraints, nverse Probl., 33 (2017), 025004(34pp).   Google Scholar

[32]

C. Li, W. Yin and Y. Zhang, Tv Minimization by Augmented Lagrangian and Alternating Direction Algorithms, Tech. rep., Department of CAAM, Rice University, Houston, Texas, 77005, 2009. http://www.caam.rice.edu/~optimization/L1/TVAL3/. Google Scholar

[33]

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

[34]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503.  Google Scholar

[35]

Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program., 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.  Google Scholar

[36]

Y. Ouyang, Y. Chen, G. Lan and E.J. Pasiliao, An accelerated linearized alternating direction method of multipliers, SIAM J Imaging Sci., 8 (2015), 644-681.  doi: 10.1137/14095697X.  Google Scholar

[37]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[38] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston, New York, 1977.   Google Scholar
[39]

C. Vogel and M. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Trans. Image Proces., 7 (1998), 813-824.  doi: 10.1109/83.679423.  Google Scholar

[40]

X. Wang and X. Yuan, The linearized alternating direction method of multipliers for dantzig selector, SIAM J. Sci. Comput., 34 (2012), 2792-2811.  doi: 10.1137/110833543.  Google Scholar

[41]

Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Image Sci., 1 (2008), 248-272.  doi: 10.1137/080724265.  Google Scholar

[42]

T. Wu, Variable splitting based method for image restoration with impulse plus gaussian noise, Math. Probl. Eng., 2016 (2016), Article ID 3151303, 16 pages. Google Scholar

[43]

J. Yang and Y. Zhang, Alternating direction algorithms for $\ell_1$-problems in compressive sensing, SIAM J. Sci. Comput., 33 (2011), 250-278.  doi: 10.1137/090777761.  Google Scholar

[44]

J. Yang, Y. Zhang and W. Yin, An efficient tvl1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), 2842-2865.  doi: 10.1137/080732894.  Google Scholar

[45]

C. Ye and X. Yuan, A descent method for structured monotone variational inequalities, Optim. Methods Softw., 22 (2007), 329-338.  doi: 10.1080/10556780600552693.  Google Scholar

[46]

W. Yin, D. Goldfarb and S. Osher, The total variation regularized $L^1$ multiscale decomposition, Multiscale Model. Sim., 6 (2007), 190-211.  doi: 10.1137/060663027.  Google Scholar

show all references

References:
[1]

R. Acar and C. R. Vogel, Analysis of total variation penalty methods, Inverse Probl., 10 (1994), 1217-1229.  doi: 10.1088/0266-5611/10/6/003.  Google Scholar

[2]

S. Alliney, Digital filters as absolute norm regularizers, IEEE Trans. Signal Proces., 40 (1992), 1548-1562.   Google Scholar

[3]

J. Barzilai and J.M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[4]

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

[5]

S. Becker, J. Bobin and E. Candes, 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

[6]

D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.  Google Scholar

[7]

J. Bioucas-Dias and M. Figueiredo, A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration, IEEE Trans. Image Proces., 16 (2007), 2992-3004.  doi: 10.1109/TIP.2007.909319.  Google Scholar

[8]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2011), 1-122.   Google Scholar

[9]

J. Cai, R.H. Chan and M. Nikolova, Two phase methods for deblurring images corrupted by impulse plus gaussian noise, Inverse Probl. Imag., 2 (2008), 187-204.  doi: 10.3934/ipi.2008.2.187.  Google Scholar

[10]

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

[11]

R.H. Chan, C.W. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detector and detail-preserving regularization, IEEE Trans. Imag. Process., 14 (2005), 1479-1485.   Google Scholar

[12]

R.H. Chan, M. Tao and X. Yuan, Linearized alternating direction method of multipliers for constrained linear least-squares problem, E. Asian J. Appl. Math., 2 (2012), 326-341.  doi: 10.4208/eajam.270812.161112a.  Google Scholar

[13]

T.F. Chan and S. Esedoglu, Aspects of total variation regularized l1 function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.  doi: 10.1137/040604297.  Google Scholar

[14]

T.F. Chan, G. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.  Google Scholar

[15]

C. Chen, B. He, Y. Ye and X. Yuan, The direct extension of admm for multi-block convex minimization problems is not necessarily convergent, Math. Program. Series A, 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.  Google Scholar

[16]

C. Chen, Y. Shen and Y. You, On the convergence analysis of the alternating direction method of multipliers with three blocks, Abstr. Appl. Anal., 2013 (2013), Art. ID 183961, 7 pp.  Google Scholar

[17]

S. W.J. Cho and S. Lee, Handling outliers in non-blind image deconvolution, International Conference on Computer Vision, (2011), 495-502.   Google Scholar

[18]

I. Daubechies, M. Defrise and C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.  Google Scholar

[19]

J. Douglas and H.H. Rachford, On the numerical solution of the heat conduction problem in 2 and 3 space variables, Trans. Amer. Math. Soc., 82 (1956), 421-439.  doi: 10.1090/S0002-9947-1956-0084194-4.  Google Scholar

[20]

E. Esser, Applications of lagrangian-based alternating direction methods and connections to split bregman, Manuscript, (2009), ftp://ftp.math.ucla.edu/pub/camreport/cam09-31.pdf. Google Scholar

[21]

M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland, Amsterdam, 1983.  Google Scholar

[22]

D. Gabay, Applications of the method of multipliers to variational inequalities, In Augmented Lagrangian Methods: Applications to the Solution of Boundary Value Problems, M. Fortin and R. Glowinski, Eds. Horth-Hollan, Amsterdam, 1983.  Google Scholar

[23]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Comput. Math. Appl., (1976), 17-40.   Google Scholar

[24] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984.   Google Scholar
[25]

R. Glowinski and A. Marrocco, Sur lapproximation par elements finis dordre un, et la resolution par penalisation-dualite dune classe de problemes de dirichlet nonlineaires, Rev. Francaise dAut. Inf. Rech. Oper., 9 (1975), 41-76.   Google Scholar

[26]

B. He, M. Tao and X. Yuan, Alternating direction method with gaussian back substitution for separable convex programming, SIAM J. Optim., 22 (2012), 313-340.  doi: 10.1137/110822347.  Google Scholar

[27]

B. He and X. Yuan, On the convergence rate of douglas-rachford operator splitting method, Math. Program., 153 (2015), 715-722.  doi: 10.1007/s10107-014-0805-x.  Google Scholar

[28]

M. Hong and Z. Luo, On the linear convergence of the alternating direction method of multipliers, arXiv preprint, arXiv: 1208.3922, (2013). Google Scholar

[29]

Y. Huang, M. K. Ng and Y.-W. Wen, A fast total variation minimization method for image restoration, Multiscale Model. Sim., 7 (2008), 774-795.  doi: 10.1137/070703533.  Google Scholar

[30]

Y. Jiao, Q. Jin, X. Lu and W. Wang, Alternating direction method of multipliers for linear inverse problems, SIAM J. Numer. Anal., 54 (2016), 2114-2137.  doi: 10.1137/15M1029308.  Google Scholar

[31]

Y. Jiao, Q. Jin, X. Lu and W. Wang, Preconditioned alternating direction method of multipliers for inverse problems with constraints, nverse Probl., 33 (2017), 025004(34pp).   Google Scholar

[32]

C. Li, W. Yin and Y. Zhang, Tv Minimization by Augmented Lagrangian and Alternating Direction Algorithms, Tech. rep., Department of CAAM, Rice University, Houston, Texas, 77005, 2009. http://www.caam.rice.edu/~optimization/L1/TVAL3/. Google Scholar

[33]

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

[34]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503.  Google Scholar

[35]

Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program., 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.  Google Scholar

[36]

Y. Ouyang, Y. Chen, G. Lan and E.J. Pasiliao, An accelerated linearized alternating direction method of multipliers, SIAM J Imaging Sci., 8 (2015), 644-681.  doi: 10.1137/14095697X.  Google Scholar

[37]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[38] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston, New York, 1977.   Google Scholar
[39]

C. Vogel and M. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Trans. Image Proces., 7 (1998), 813-824.  doi: 10.1109/83.679423.  Google Scholar

[40]

X. Wang and X. Yuan, The linearized alternating direction method of multipliers for dantzig selector, SIAM J. Sci. Comput., 34 (2012), 2792-2811.  doi: 10.1137/110833543.  Google Scholar

[41]

Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Image Sci., 1 (2008), 248-272.  doi: 10.1137/080724265.  Google Scholar

[42]

T. Wu, Variable splitting based method for image restoration with impulse plus gaussian noise, Math. Probl. Eng., 2016 (2016), Article ID 3151303, 16 pages. Google Scholar

[43]

J. Yang and Y. Zhang, Alternating direction algorithms for $\ell_1$-problems in compressive sensing, SIAM J. Sci. Comput., 33 (2011), 250-278.  doi: 10.1137/090777761.  Google Scholar

[44]

J. Yang, Y. Zhang and W. Yin, An efficient tvl1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), 2842-2865.  doi: 10.1137/080732894.  Google Scholar

[45]

C. Ye and X. Yuan, A descent method for structured monotone variational inequalities, Optim. Methods Softw., 22 (2007), 329-338.  doi: 10.1080/10556780600552693.  Google Scholar

[46]

W. Yin, D. Goldfarb and S. Osher, The total variation regularized $L^1$ multiscale decomposition, Multiscale Model. Sim., 6 (2007), 190-211.  doi: 10.1137/060663027.  Google Scholar

Figure 1.  From left to right are the results of FTVD, New algorithm, TVAL3, and SNR history plot. First row: $NR$ = 60%, $airplane$; second row: $NR$ = 80%, $lena$
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Figure 2.  Performance v.s. noise ratio. Left: time; right: SNR. First row: $RC = 0.1$, $coins$; second row: $RC = 0$, $westconcordorthophoto$
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Figure 3.  Performance v.s. indices matrix quality. Left: time, right: SNR. First row: $moon$, $NR = 60%$; second row: $tire$, $NR = 80%$
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Figure 4.  From left to right to below are the results of FTVD, New algorithm, and TVAL3, and SNR history plot. First row: "average" kernel with dirichlet boundary condition; second row: "Gaussian" kernel; third row: "Disk" kernel; fourth row: "motion" kernel
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Table 1.  Numerical results of basic test
FTVD New LADMM TVAL3
Noise Ratio time SNR time SNR time SNR
60% 3.806 20.894 20.155 23.594 14.212 11.806
80% 8.174 8.751 19.687 15.226 26.442 7.106
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FTVD New LADMM TVAL3
Noise Ratio time SNR time SNR time SNR
60% 3.806 20.894 20.155 23.594 14.212 11.806
80% 8.174 8.751 19.687 15.226 26.442 7.106
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