American Institute of Mathematical Sciences

Advanced
April  2021, 15(2): 339-366. doi: 10.3934/ipi.2020071

Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal

Jie Zhang 1,2, , Yuping Duan 3, , Yue Lu 1, , Michael K. Ng 2, and  Huibin Chang 1,,

1. 

School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China
2. 

Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong
3. 

Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

* Corresponding author: Huibin Chang

Received  January 2020 Revised  September 2020 Published  April 2021 Early access  November 2020

In this paper, we propose new operator-splitting algorithms for the total variation regularized infimal convolution (TV-IC) model [6] in order to remove mixed Poisson-Gaussian (MPG) noise. In the existing splitting algorithm for TV-IC, an inner loop by Newton method had to be adopted for one nonlinear optimization subproblem, which increased the computation cost per outer loop. By introducing a new bilinear constraint and applying the alternating direction method of multipliers (ADMM), all subproblems of the proposed algorithms named as BCA (short for Bilinear Constraint based ADMM algorithm) and BCA$ _{f} $ (short for a variant of BCA with $ {\bf f} $ully splitting form) can be very efficiently solved. Especially for the proposed BCA$ _{f} $, they can be calculated without any inner iterations. The convergence of the proposed algorithms are investigated, where particularly, a Huber type TV regularizer is adopted to guarantee the convergence of BCA$ _f $. Numerically, compared to existing primal-dual algorithms for the TV-IC model, the proposed algorithms, with fewer tunable parameters, converge much faster and produce comparable results meanwhile.

Keywords: Mixed Poisson-Gaussian noise, total variation, alternating direction method of multipliers, bilinear constraint, convergence.
Mathematics Subject Classification: Primary:65K10, 65F22;Secondary:94A08, 46N10.
Citation: Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems and Imaging, 2021, 15 (2) : 339-366. doi: 10.3934/ipi.2020071
References:
[1]

B. Begovic, V. Stankovic and L. Stankovic, Contrast enhancement and denoising of Poisson and Gaussian mixture noise for solar images, 18th IEEE International Conference on Image Processing (ICIP), Brussels, Belgium, 2011, 185-188. doi: 10.1109/ICIP.2011.6115829.

[2]

F. Benvenuto, A. L. Camera, C. Theys, A. Ferrari, H. Lantéri and M. Bertero, The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise, Inverse Problems, 24 (2008), 035016, 20pp. doi: 10.1088/0266-5611/24/3/035016.

[3]

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.  doi: 10.1561/9781601984616.

[4]

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

[5]

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, 37pp. doi: 10.1088/1361-6420/ab291a.

[6]

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

[7]

A. Chakrabarti and T. E. Zickler, Image restoration with signal-dependent Camera noise, preprint, arXiv: 1204.2994.

[8]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20 (2004), 89-97. 

[9]

H. Chang, P. Enfedaque and S. Marchesini, Blind ptychographic phase retrieval via convergent alternating direction method of multipliers, SIAM Journal on Imaging Sciences, 12 (2019), 153-185.  doi: 10.1137/18M1188446.

[10]

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

[11]

E. Chouzenoux, A. Jezierska, J. Pesquet and H. Talbot, A convex approach for image restoration with exact poisson-gaussian likelihood, SIAM J. Imaging Sciences, 8 (2015), 2662-2682.  doi: 10.1137/15M1014395.

[12]

W. Deng, M. Lai, Z. Peng and W. Yin, Parallel multi-block ADMM with O (1/k) convergence, Journal of Scientific Computing, 71 (2017), 712-736.  doi: 10.1007/s10915-016-0318-2.

[13]

Q. Ding and Y. Long and X. Zhang and J. A. Fessler, Statistical image reconstruction using mixed poisson-gaussian noise model for X-ray CT, preprint, arXiv: 1801.09533.

[14]

J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318.  doi: 10.1007/BF01581204.

[15]

A. Foi, M. Trimeche, V. Katkovnik and K. Egiazarian, Practical poissonian-gaussian noise modeling and fitting for single-image raw-data, IEEE Transactions on Image Processing, 17 (2008), 1737-1754.  doi: 10.1109/TIP.2008.2001399.

[16]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers & Mathematics with Applications, 2 (1976), 17-40.  doi: 10.1016/0898-1221(76)90003-1.

[17]

R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Revue Française D'automatique, Informatique, Recherche Opérationnelle. Analyse Numérique, 9 (1975), 41-76. doi: 10.1051/m2an/197509R200411.

[18]

D. Hajinezhad and Q. Shi, Alternating direction method of multipliers for a class of nonconvex bilinear optimization: convergence analysis and applications, Journal of Global Optimization, 70 (2018), 261-288.  doi: 10.1007/s10898-017-0594-x.

[19]

M. Hong, Z. Luo and M. Razaviyayn, Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems, SIAM Journal on Optimization, 26 (2016), 337-364.  doi: 10.1137/140990309.

[20]

P. J. Huber, Robust Estimation of a Location Parameter, The Annals of Mathematical Statistics, 35 (1964), 73-101.  doi: 10.1214/aoms/1177703732.

[21]

A. Jezierska, C. Chaux, J. Pesquet and H. Talbot, An EM approach for Poisson-Gaussian noise modeling, 19th European Signal Processing Conference (EUSIPCO), Barcelona, Spain, 2011, 2244-2248.

[22]

A. Lanza, S. Morigi, F. Sgallari and Y. Wen, Image restoration with Poisson-Gaussian mixed noise, Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2 (2014), 12-24.  doi: 10.1080/21681163.2013.811039.

[23]

T. Le, R. Chartrand and T. Asaki, A variational approach to reconstructing images corrupted by poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263.  doi: 10.1007/s10851-007-0652-y.

[24]

J. Li, Z. Shen, R. Yin and X. Zhang, A reweighted $L^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise, Inverse Problems & Imaging, 9 (2015), 875-894.  doi: 10.3934/ipi.2015.9.875.

[25]

T. Lin, S. Ma and S. Zhang, On the global linear convergence of the admm with multiblock variables, SIAM Journal on Optimization, 25 (2015), 1478-1497.  doi: 10.1137/140971178.

[26]

Y. Lou and M. Yan, Fast L1-L2 minimization via a proximal operator, Journal of Scientific Computing, 74 (2018), 767-785.  doi: 10.1007/s10915-017-0463-2.

[27]

M. Mäkitalo and A. Foi, Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise, IEEE Transactions on Image Processing, 22 (2013), 91-103.  doi: 10.1109/TIP.2012.2202675.

[28]

Y. Marnissi, Y. Zheng and J. Pesquet, Fast variational Bayesian signal recovery in the presence of Poisson-Gaussian noise, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Shanghai, China, 2016, 3964-3968. doi: 10.1109/ICASSP.2016.7472421.

[29]

J. Mei, Y. Dong, T. Huang and W. Yin, Cauchy noise removal by nonconvex ADMM with convergence guarantees, Journal of Scientific Computing, 74 (2018), 743-766.  doi: 10.1007/s10915-017-0460-5.

[30]

F. Murtagh, J.-L Starck and A. Bijaoui, Image restoration with noise suppression using a multiresolution support, Astronomy & Astrophysics, Suppl. Ser, 112 (1995), 179-189. 

[31]

B. O' Donoghue, G. Stathopoulos and S. Boyd, A splitting method for optimal control, IEEE Transactions on Control Systems Technology, 21 (2013), 2432-2442. 

[32]

C. T. Pham, G. Gamard, A. Kopylov and T. Tran, An algorithm for image restoration with mixed noise using total variation regularization, Turkish Journal of Electrical Engineering & Computer Sciences, 26 (2018), 2832-2846.  doi: 10.3906/elk-1803-100.

[33]

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

[34]

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.

[35] J.-L. Starck, F. Murtagh and A. Bijaoui, Image Processing and Data Analysis: The Multiscale Approach, Cambridge University Press, New York, USA, 1998.  doi: 10.1017/CBO9780511564352.
[36]

D. N. H. Thanh and S. D. Dvoenko, A method of total variation to remove the mixed Poisson-Gaussian noise, Pattern Recognition and Image Analysis, 26 (2016), 285-293.  doi: 10.1134/S1054661816020231.

[37]

Y. Wang, W. Yin and J. Zeng, Global convergence of ADMM in nonconvex nonsmooth optimization, Journal of Scientific Computing, 78 (2019), 29-63.  doi: 10.1007/s10915-018-0757-z.

[38]

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

[39]

C. Wu and X. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339.  doi: 10.1137/090767558.

[40]

C. Wu, J. Zhang and X. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems & Imaging, 5 (2011), 237-261.  doi: 10.3934/ipi.2011.5.237.

show all references

References:
[1]

B. Begovic, V. Stankovic and L. Stankovic, Contrast enhancement and denoising of Poisson and Gaussian mixture noise for solar images, 18th IEEE International Conference on Image Processing (ICIP), Brussels, Belgium, 2011, 185-188. doi: 10.1109/ICIP.2011.6115829.

[2]

F. Benvenuto, A. L. Camera, C. Theys, A. Ferrari, H. Lantéri and M. Bertero, The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise, Inverse Problems, 24 (2008), 035016, 20pp. doi: 10.1088/0266-5611/24/3/035016.

[3]

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.  doi: 10.1561/9781601984616.

[4]

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

[5]

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, 37pp. doi: 10.1088/1361-6420/ab291a.

[6]

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

[7]

A. Chakrabarti and T. E. Zickler, Image restoration with signal-dependent Camera noise, preprint, arXiv: 1204.2994.

[8]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20 (2004), 89-97. 

[9]

H. Chang, P. Enfedaque and S. Marchesini, Blind ptychographic phase retrieval via convergent alternating direction method of multipliers, SIAM Journal on Imaging Sciences, 12 (2019), 153-185.  doi: 10.1137/18M1188446.

[10]

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

[11]

E. Chouzenoux, A. Jezierska, J. Pesquet and H. Talbot, A convex approach for image restoration with exact poisson-gaussian likelihood, SIAM J. Imaging Sciences, 8 (2015), 2662-2682.  doi: 10.1137/15M1014395.

[12]

W. Deng, M. Lai, Z. Peng and W. Yin, Parallel multi-block ADMM with O (1/k) convergence, Journal of Scientific Computing, 71 (2017), 712-736.  doi: 10.1007/s10915-016-0318-2.

[13]

Q. Ding and Y. Long and X. Zhang and J. A. Fessler, Statistical image reconstruction using mixed poisson-gaussian noise model for X-ray CT, preprint, arXiv: 1801.09533.

[14]

J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318.  doi: 10.1007/BF01581204.

[15]

A. Foi, M. Trimeche, V. Katkovnik and K. Egiazarian, Practical poissonian-gaussian noise modeling and fitting for single-image raw-data, IEEE Transactions on Image Processing, 17 (2008), 1737-1754.  doi: 10.1109/TIP.2008.2001399.

[16]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers & Mathematics with Applications, 2 (1976), 17-40.  doi: 10.1016/0898-1221(76)90003-1.

[17]

R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Revue Française D'automatique, Informatique, Recherche Opérationnelle. Analyse Numérique, 9 (1975), 41-76. doi: 10.1051/m2an/197509R200411.

[18]

D. Hajinezhad and Q. Shi, Alternating direction method of multipliers for a class of nonconvex bilinear optimization: convergence analysis and applications, Journal of Global Optimization, 70 (2018), 261-288.  doi: 10.1007/s10898-017-0594-x.

[19]

M. Hong, Z. Luo and M. Razaviyayn, Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems, SIAM Journal on Optimization, 26 (2016), 337-364.  doi: 10.1137/140990309.

[20]

P. J. Huber, Robust Estimation of a Location Parameter, The Annals of Mathematical Statistics, 35 (1964), 73-101.  doi: 10.1214/aoms/1177703732.

[21]

A. Jezierska, C. Chaux, J. Pesquet and H. Talbot, An EM approach for Poisson-Gaussian noise modeling, 19th European Signal Processing Conference (EUSIPCO), Barcelona, Spain, 2011, 2244-2248.

[22]

A. Lanza, S. Morigi, F. Sgallari and Y. Wen, Image restoration with Poisson-Gaussian mixed noise, Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2 (2014), 12-24.  doi: 10.1080/21681163.2013.811039.

[23]

T. Le, R. Chartrand and T. Asaki, A variational approach to reconstructing images corrupted by poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263.  doi: 10.1007/s10851-007-0652-y.

[24]

J. Li, Z. Shen, R. Yin and X. Zhang, A reweighted $L^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise, Inverse Problems & Imaging, 9 (2015), 875-894.  doi: 10.3934/ipi.2015.9.875.

[25]

T. Lin, S. Ma and S. Zhang, On the global linear convergence of the admm with multiblock variables, SIAM Journal on Optimization, 25 (2015), 1478-1497.  doi: 10.1137/140971178.

[26]

Y. Lou and M. Yan, Fast L1-L2 minimization via a proximal operator, Journal of Scientific Computing, 74 (2018), 767-785.  doi: 10.1007/s10915-017-0463-2.

[27]

M. Mäkitalo and A. Foi, Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise, IEEE Transactions on Image Processing, 22 (2013), 91-103.  doi: 10.1109/TIP.2012.2202675.

[28]

Y. Marnissi, Y. Zheng and J. Pesquet, Fast variational Bayesian signal recovery in the presence of Poisson-Gaussian noise, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Shanghai, China, 2016, 3964-3968. doi: 10.1109/ICASSP.2016.7472421.

[29]

J. Mei, Y. Dong, T. Huang and W. Yin, Cauchy noise removal by nonconvex ADMM with convergence guarantees, Journal of Scientific Computing, 74 (2018), 743-766.  doi: 10.1007/s10915-017-0460-5.

[30]

F. Murtagh, J.-L Starck and A. Bijaoui, Image restoration with noise suppression using a multiresolution support, Astronomy & Astrophysics, Suppl. Ser, 112 (1995), 179-189. 

[31]

B. O' Donoghue, G. Stathopoulos and S. Boyd, A splitting method for optimal control, IEEE Transactions on Control Systems Technology, 21 (2013), 2432-2442. 

[32]

C. T. Pham, G. Gamard, A. Kopylov and T. Tran, An algorithm for image restoration with mixed noise using total variation regularization, Turkish Journal of Electrical Engineering & Computer Sciences, 26 (2018), 2832-2846.  doi: 10.3906/elk-1803-100.

[33]

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

[34]

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.

[35] J.-L. Starck, F. Murtagh and A. Bijaoui, Image Processing and Data Analysis: The Multiscale Approach, Cambridge University Press, New York, USA, 1998.  doi: 10.1017/CBO9780511564352.
[36]

D. N. H. Thanh and S. D. Dvoenko, A method of total variation to remove the mixed Poisson-Gaussian noise, Pattern Recognition and Image Analysis, 26 (2016), 285-293.  doi: 10.1134/S1054661816020231.

[37]

Y. Wang, W. Yin and J. Zeng, Global convergence of ADMM in nonconvex nonsmooth optimization, Journal of Scientific Computing, 78 (2019), 29-63.  doi: 10.1007/s10915-018-0757-z.

[38]

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

[39]

C. Wu and X. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339.  doi: 10.1137/090767558.

[40]

C. Wu, J. Zhang and X. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems & Imaging, 5 (2011), 237-261.  doi: 10.3934/ipi.2011.5.237.

Figure 1.  (A) Circles; (B) Fluorescent Cells; (C) Cameraman
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Recovery results by proposed algorithms and other compared algorithms (with SNRs(dB) below the figures) for the image "Circles" in the case of MPG noises which are generated with $ \eta = 4, \sigma = 10^{-4} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Recovery results by proposed algorithms and other compared algorithms (with SNRs(dB) below the figures) for the image "Fluorescent Cells" in the case of MPG noises which are generated with $ \eta=16, \sigma=10^{-4} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Recovery results by proposed algorithms and other compared algorithms (with SNRs(dB) below the figures) for the image "Cameraman" in the case of MPG noises which are generated with $ \eta = 64, \sigma = 10^{-1} $. Comparing with other results, there are less grey blocks in the restoration images of proposed algorithms on the blue circle, which can show the better performance of our proposed algorithms
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Line profiles of recovery results showed in Fig. 4
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Histories of SNR changes for proposed algorithms and TV+PD w.r.t. the elapsed CPU time (in log scale). Left: $ \eta = 4, \sigma = 10^{-1} $. Right: $ \eta = 16, \sigma = 10^{-1} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Rcovery results by SSN and BCA$ _f $-HTV. (A): Noisy image, SNR = 12.79. (B): Recovery result by SSN, $ SNR = 18.86, t = 6.763386s $. (C): Recovery result by BCA$ _f $-HTV, $ SNR = 18.88, t = 2.377923s $. (D): Poisson residuums of noisy images. (E): Poisson residuums by SSN. (F): Poisson residuums by BCA$ _f $-HTV
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  SE ($ \tfrac{\Vert u_{k+1}-u_{k}\Vert}{\Vert u_{k}\Vert} $) changes v.s. iteration number (both in log scale): Top: BCA; Medium:BCA$ _f $; Bottom: BCA$ _f $-HTV; Left: $ \eta = 1, \sigma = 10^{-1} $; Right: $ \eta = 1, \sigma = 10^{-4} $. The test image is Circles(Fig1(A))
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  The performance of BCA w.r.t. $ \alpha $ (in log scale), where $ \eta = 1, \sigma = 10^{-4} $, using test image Circles(Fig1(A))
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  The performance of BCA$ _f $ w.r.t. $ \alpha_w $ (in log scale), $ \alpha_p $ (in log scale), where $ \eta = 1, \sigma = 10^{-4} $, using test image Circles(Fig1(A))
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  The performance of BCA$ _f $-HTV w.r.t. $ \alpha_w $ (in log scale), $ \alpha_p $ (in log scale), where $ \eta = 1, \sigma = 10^{-4} $, using test image Circles(Fig1(A))
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  SNR changes w.r.t. the parameter $ \epsilon $ for BCA Algorithm, using test image Circles(Fig1(A))
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 13.  Minimum value curves of $ w $ for BCA Algorithm. The test image is Circles(Fig1(A))
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  SNR changes w.r.t. the number of inner iterations using gradient descent method [8] for proposed BCA Algorithm
$ \eta $ $ \sigma $ 1 2 5 10 20 100
1 $ 10^{-1} $ 15.20 18.08 17.57 18.16 18.17 18.17
1 $ 10^{-4} $ 16.20 18.39 16.35 18.40 18.40 18.40
4 $ 10^{-1} $ 19.92 21.92 21.90 21.98 21.97 21.98
4 $ 10^{-4} $ 20.05 22.37 22.36 22.47 22.48 22.49
16 $ 10^{-1} $ 22.50 25.17 25.27 25.28 25.26 25.28
16 $ 10^{-4} $ 23.59 26.17 26.40 26.37 26.38 26.40
Table Options

Download as excel

$ \eta $ $ \sigma $ 1 2 5 10 20 100
1 $ 10^{-1} $ 15.20 18.08 17.57 18.16 18.17 18.17
1 $ 10^{-4} $ 16.20 18.39 16.35 18.40 18.40 18.40
4 $ 10^{-1} $ 19.92 21.92 21.90 21.98 21.97 21.98
4 $ 10^{-4} $ 20.05 22.37 22.36 22.47 22.48 22.49
16 $ 10^{-1} $ 22.50 25.17 25.27 25.28 25.26 25.28
16 $ 10^{-4} $ 23.59 26.17 26.40 26.37 26.38 26.40
Table 2.  Denoising performance (First row: SNR in dB. Second row: SSIM.) with Poisson-Gaussian Noise
Image $ \eta $ $ \sigma $ Noisy TV+$ L^2 $ TV+KL TV+EPG TV+SP TV+KL+$ L^2 $ TV+PD BCA BCA$ _f $
Circle 1 $ 10^{-1} $ 2.50 17.21 16.68 17.07 16.76 17.28 17.44 17.84 17.97
0.0452 0.6147 0.4044 0.6580 0.4059 0.3975 0.7544 0.7125 0.9007
1 $ 10^{-4} $ 2.57 17.34 17.16 17.11 17.24 17.76 17.00 18.02 18.10
0.5494 0.6087 0.7997 0.7740 0.8678 0.8251 0.8711 0.8913 0.9029
4 $ 10^{-1} $ 5.92 21.48 20.44 20.74 20.77 21.29 21.64 22.01 21.78
0.0687 0.7149 0.4763 0.6260 0.5305 0.5259 0.9216 0.7153 0.9357
4 $ 10^{-4} $ 6.32 21.64 21.83 21.14 21.83 22.00 22.16 22.18 22.35
0.5793 0.7397 0.9385 0.9395 0.9385 0.9299 0.9466 0.9472 0.9180
16 $ 10^{-1} $ 9.88 24.64 22.33 22.54 22.62 23.89 23.54 25.28 25.04
0.0971 0.8411 0.5088 0.5315 0.5436 0.5438 0.8141 0.9697 0.8079
16 $ 10^{-4} $ 11.55 25.46 26.41 25.14 26.41 26.46 26.47 26.37 27.12
0.6149 0.8816 0.9500 0.9447 0.9500 0.9594 0.9651 0.9676 0.9168
Average 6.46 21.30 20.81 20.62 20.94 21.45 21.38 21.95 22.06
0.3258 0.7335 0.6796 0.745 0.7061 0.6969 0.8788 0.8673 0.8970
Fluorescent Cells 1 $ 10^{-1} $ 1.16 9.88 9.72 9.29 9.72 9.96 9.48 10.37 10.33
0.0402 0.4861 0.4508 0.3149 0.4532 0.4512 0.4572 0.5026 0.4971
1 $ 10^{-4} $ 1.22 9.97 9.83 9.46 9.83 9.98 9.79 10.43 10.41
0.0598 0.5058 0.4954 0.3289 0.4954 0.5003 0.4471 0.5108 0.5014
4 $ 10^{-1} $ 3.14 11.10 11.58 11.12 11.54 11.75 11.62 11.66 12.06
0.1181 0.5554 0.5369 0.5093 0.5239 0.5588 0.5674 0.5753 0.5801
4 $ 10^{-4} $ 3.59 11.25 11.88 11.32 11.88 12.17 11.88 11.96 12.38
0.1680 0.5765 0.6078 0.4593 0.6078 0.6133 0.5531 0.6160 0.6139
16 $ 10^{-1} $ 5.77 13.43 12.66 12.52 12.64 13.27 13.45 13.37 13.50
0.2282 0.6424 0.5998 0.5271 0.5989 0.6383 0.6669 0.6557 0.6685
16 $ 10^{-4} $ 7.87 14.17 14.16 14.30 14.16 14.59 14.47 14.42 14.62
0.4003 0.7360 0.7228 0.6895 0.7228 0.7388 0.7309 0.7379 0.7368
Average 3.79 11.63 11.64 11.34 11.63 11.95 11.78 12.04 12.22
0.1691 0.5837 0.5689 0.4710 0.5670 0.5835 0.5704 0.5997 0.5996
Cameraman 1 $ 10^{-1} $ 1.97 13.06 14.25 13.13 14.19 14.30 14.30 14.59 14.56
0.0496 0.4167 0.5498 0.3452 0.5322 0.5432 0.5524 0.5633 0.5644
1 $ 10^{-4} $ 2.00 13.09 14.36 13.04 14.36 14.40 14.33 14.57 14.52
0.0628 0.4238 0.4602 0.3342 0.4602 0.4760 0.4362 0.5854 0.5659
4 $ 10^{-1} $ 5.02 15.66 16.46 15.5 16.43 16.56 16.17 16.79 16.83
0.1178 0.5729 0.6552 0.4863 0.6540 0.6720 0.6422 0.6417 0.6655
4 $ 10^{-4} $ 5.28 15.81 16.27 15.64 16.27 16.47 16.14 16.99 17.00
0.1514 0.5879 0.6301 0.5027 0.6301 0.6160 0.6047 0.6697 0.6770
16 $ 10^{-1} $ 9.06 18.25 18.60 18.24 18.60 18.81 18.18 18.99 19.02
0.1992 0.6393 0.7126 0.6355 0.7181 0.7163 0.6527 0.7112 0.7214
16 $ 10^{-4} $ 10.24 18.88 19.44 18.83 19.44 19.64 19.59 19.81 19.53
0.2920 0.6980 0.7321 0.6616 0.7321 0.7503 0.7317 0.7614 0.7764
Average 5.60 15.79 16.56 15.73 16.55 16.70 16.45 16.96 16.91
0.1455 0.5564 0.6233 0.4940 0.6211 0.6290 0.6033 0.6555 0.6618
Table Options

Download as excel

Image $ \eta $ $ \sigma $ Noisy TV+$ L^2 $ TV+KL TV+EPG TV+SP TV+KL+$ L^2 $ TV+PD BCA BCA$ _f $
Circle 1 $ 10^{-1} $ 2.50 17.21 16.68 17.07 16.76 17.28 17.44 17.84 17.97
0.0452 0.6147 0.4044 0.6580 0.4059 0.3975 0.7544 0.7125 0.9007
1 $ 10^{-4} $ 2.57 17.34 17.16 17.11 17.24 17.76 17.00 18.02 18.10
0.5494 0.6087 0.7997 0.7740 0.8678 0.8251 0.8711 0.8913 0.9029
4 $ 10^{-1} $ 5.92 21.48 20.44 20.74 20.77 21.29 21.64 22.01 21.78
0.0687 0.7149 0.4763 0.6260 0.5305 0.5259 0.9216 0.7153 0.9357
4 $ 10^{-4} $ 6.32 21.64 21.83 21.14 21.83 22.00 22.16 22.18 22.35
0.5793 0.7397 0.9385 0.9395 0.9385 0.9299 0.9466 0.9472 0.9180
16 $ 10^{-1} $ 9.88 24.64 22.33 22.54 22.62 23.89 23.54 25.28 25.04
0.0971 0.8411 0.5088 0.5315 0.5436 0.5438 0.8141 0.9697 0.8079
16 $ 10^{-4} $ 11.55 25.46 26.41 25.14 26.41 26.46 26.47 26.37 27.12
0.6149 0.8816 0.9500 0.9447 0.9500 0.9594 0.9651 0.9676 0.9168
Average 6.46 21.30 20.81 20.62 20.94 21.45 21.38 21.95 22.06
0.3258 0.7335 0.6796 0.745 0.7061 0.6969 0.8788 0.8673 0.8970
Fluorescent Cells 1 $ 10^{-1} $ 1.16 9.88 9.72 9.29 9.72 9.96 9.48 10.37 10.33
0.0402 0.4861 0.4508 0.3149 0.4532 0.4512 0.4572 0.5026 0.4971
1 $ 10^{-4} $ 1.22 9.97 9.83 9.46 9.83 9.98 9.79 10.43 10.41
0.0598 0.5058 0.4954 0.3289 0.4954 0.5003 0.4471 0.5108 0.5014
4 $ 10^{-1} $ 3.14 11.10 11.58 11.12 11.54 11.75 11.62 11.66 12.06
0.1181 0.5554 0.5369 0.5093 0.5239 0.5588 0.5674 0.5753 0.5801
4 $ 10^{-4} $ 3.59 11.25 11.88 11.32 11.88 12.17 11.88 11.96 12.38
0.1680 0.5765 0.6078 0.4593 0.6078 0.6133 0.5531 0.6160 0.6139
16 $ 10^{-1} $ 5.77 13.43 12.66 12.52 12.64 13.27 13.45 13.37 13.50
0.2282 0.6424 0.5998 0.5271 0.5989 0.6383 0.6669 0.6557 0.6685
16 $ 10^{-4} $ 7.87 14.17 14.16 14.30 14.16 14.59 14.47 14.42 14.62
0.4003 0.7360 0.7228 0.6895 0.7228 0.7388 0.7309 0.7379 0.7368
Average 3.79 11.63 11.64 11.34 11.63 11.95 11.78 12.04 12.22
0.1691 0.5837 0.5689 0.4710 0.5670 0.5835 0.5704 0.5997 0.5996
Cameraman 1 $ 10^{-1} $ 1.97 13.06 14.25 13.13 14.19 14.30 14.30 14.59 14.56
0.0496 0.4167 0.5498 0.3452 0.5322 0.5432 0.5524 0.5633 0.5644
1 $ 10^{-4} $ 2.00 13.09 14.36 13.04 14.36 14.40 14.33 14.57 14.52
0.0628 0.4238 0.4602 0.3342 0.4602 0.4760 0.4362 0.5854 0.5659
4 $ 10^{-1} $ 5.02 15.66 16.46 15.5 16.43 16.56 16.17 16.79 16.83
0.1178 0.5729 0.6552 0.4863 0.6540 0.6720 0.6422 0.6417 0.6655
4 $ 10^{-4} $ 5.28 15.81 16.27 15.64 16.27 16.47 16.14 16.99 17.00
0.1514 0.5879 0.6301 0.5027 0.6301 0.6160 0.6047 0.6697 0.6770
16 $ 10^{-1} $ 9.06 18.25 18.60 18.24 18.60 18.81 18.18 18.99 19.02
0.1992 0.6393 0.7126 0.6355 0.7181 0.7163 0.6527 0.7112 0.7214
16 $ 10^{-4} $ 10.24 18.88 19.44 18.83 19.44 19.64 19.59 19.81 19.53
0.2920 0.6980 0.7321 0.6616 0.7321 0.7503 0.7317 0.7614 0.7764
Average 5.60 15.79 16.56 15.73 16.55 16.70 16.45 16.96 16.91
0.1455 0.5564 0.6233 0.4940 0.6211 0.6290 0.6033 0.6555 0.6618
Table 3.  Computational time of the proposed algorithms and TV+PD (in seconds)
Image $ \eta $ $ \sigma $ TV+PD BCA BCA$ _f $
Circle 1 $ 10^{-1} $ 40.4971 4.6052 1.0314
1 $ 10^{-4} $ 18.5340 4.6922 1.2650
4 $ 10^{-1} $ 7.3436 0.7641 0.7001
4 $ 10^{-4} $ 39.6196 1.1891 0.6030
16 $ 10^{-1} $ 36.3221 1.2652 0.4643
16 $ 10^{-4} $ 39.5773 0.6123 0.3070
Average 30.3156 2.1880 0.7285
Fluorescent Cells 1 $ 10^{-1} $ 21.7999 8.3016 2.1735
1 $ 10^{-4} $ 41.6892 8.9243 2.5687
4 $ 10^{-1} $ 40.1406 4.1263 2.4025
4 $ 10^{-4} $ 38.6224 3.8464 2.0889
16 $ 10^{-1} $ 37.7174 7.3265 6.1742
16 $ 10^{-4} $ 7.7788 7.6185 1.1258
Average 31.2914 6.6906 2.7556
Cameraman 1 $ 10^{-1} $ 9.3735 1.3062 1.2272
1 $ 10^{-4} $ 40.9848 3.2183 0.9053
4 $ 10^{-1} $ 36.8623 0.6872 1.5084
4 $ 10^{-4} $ 36.4358 0.6179 1.4209
16 $ 10^{-1} $ 3.3691 0.5151 0.9925
16 $ 10^{-4} $ 11.0407 0.3607 0.5260
Average 23.0140 1.1176 1.0967
Table Options

Download as excel

Image $ \eta $ $ \sigma $ TV+PD BCA BCA$ _f $
Circle 1 $ 10^{-1} $ 40.4971 4.6052 1.0314
1 $ 10^{-4} $ 18.5340 4.6922 1.2650
4 $ 10^{-1} $ 7.3436 0.7641 0.7001
4 $ 10^{-4} $ 39.6196 1.1891 0.6030
16 $ 10^{-1} $ 36.3221 1.2652 0.4643
16 $ 10^{-4} $ 39.5773 0.6123 0.3070
Average 30.3156 2.1880 0.7285
Fluorescent Cells 1 $ 10^{-1} $ 21.7999 8.3016 2.1735
1 $ 10^{-4} $ 41.6892 8.9243 2.5687
4 $ 10^{-1} $ 40.1406 4.1263 2.4025
4 $ 10^{-4} $ 38.6224 3.8464 2.0889
16 $ 10^{-1} $ 37.7174 7.3265 6.1742
16 $ 10^{-4} $ 7.7788 7.6185 1.1258
Average 31.2914 6.6906 2.7556
Cameraman 1 $ 10^{-1} $ 9.3735 1.3062 1.2272
1 $ 10^{-4} $ 40.9848 3.2183 0.9053
4 $ 10^{-1} $ 36.8623 0.6872 1.5084
4 $ 10^{-4} $ 36.4358 0.6179 1.4209
16 $ 10^{-1} $ 3.3691 0.5151 0.9925
16 $ 10^{-4} $ 11.0407 0.3607 0.5260
Average 23.0140 1.1176 1.0967
[1]

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 and Management Optimization, 2019, 15 (1) : 159-175. doi: 10.3934/jimo.2018037
[2]

Jia Li, Zuowei Shen, Rujie Yin, Xiaoqun Zhang. A reweighted $l^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise. Inverse Problems and Imaging, 2015, 9 (3) : 875-894. doi: 10.3934/ipi.2015.9.875
[3]

Bingsheng He, Xiaoming Yuan. Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 247-260. doi: 10.3934/naco.2013.3.247
[4]

Russell E. Warren, Stanley J. Osher. Hyperspectral unmixing by the alternating direction method of multipliers. Inverse Problems and Imaging, 2015, 9 (3) : 917-933. doi: 10.3934/ipi.2015.9.917
[5]

Sohana Jahan. Supervised distance preserving projection using alternating direction method of multipliers. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1783-1799. doi: 10.3934/jimo.2019029
[6]

Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030
[7]

Linghai Kong, Suhua Wei. A variational method for Abel inversion tomography with mixed Poisson-Laplace-Gaussian noise. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022007
[8]

Yan Gu, Nobuo Yamashita. Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 487-510. doi: 10.3934/naco.2020047
[9]

Zhongming Wu, Xingju Cai, Deren Han. Linearized block-wise alternating direction method of multipliers for multiple-block convex programming. Journal of Industrial and Management Optimization, 2018, 14 (3) : 833-855. doi: 10.3934/jimo.2017078
[10]

Johnathan M. Bardsley. An efficient computational method for total variation-penalized Poisson likelihood estimation. Inverse Problems and Imaging, 2008, 2 (2) : 167-185. doi: 10.3934/ipi.2008.2.167
[11]

Yunhai Xiao, Junfeng Yang, Xiaoming Yuan. Alternating algorithms for total variation image reconstruction from random projections. Inverse Problems and Imaging, 2012, 6 (3) : 547-563. doi: 10.3934/ipi.2012.6.547
[12]

Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial and Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067
[13]

Mujibur Rahman Chowdhury, Jun Zhang, Jing Qin, Yifei Lou. Poisson image denoising based on fractional-order total variation. Inverse Problems and Imaging, 2020, 14 (1) : 77-96. doi: 10.3934/ipi.2019064
[14]

Jianbin Yang, Cong Wang. A wavelet frame approach for removal of mixed Gaussian and impulse noise on surfaces. Inverse Problems and Imaging, 2017, 11 (5) : 783-798. doi: 10.3934/ipi.2017037
[15]

Feng Ma, Jiansheng Shu, Yaxiong Li, Jian Wu. The dual step size of the alternating direction method can be larger than 1.618 when one function is strongly convex. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1173-1185. doi: 10.3934/jimo.2020016
[16]

Yue Lu, Ying-En Ge, Li-Wei Zhang. An alternating direction method for solving a class of inverse semi-definite quadratic programming problems. Journal of Industrial and Management Optimization, 2016, 12 (1) : 317-336. doi: 10.3934/jimo.2016.12.317
[17]

Liyan Ma, Lionel Moisan, Jian Yu, Tieyong Zeng. A stable method solving the total variation dictionary model with $L^\infty$ constraints. Inverse Problems and Imaging, 2014, 8 (2) : 507-535. doi: 10.3934/ipi.2014.8.507
[18]

Chunlin Wu, Juyong Zhang, Xue-Cheng Tai. Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Problems and Imaging, 2011, 5 (1) : 237-261. doi: 10.3934/ipi.2011.5.237
[19]

Yunhai Xiao, Soon-Yi Wu, Bing-Sheng He. A proximal alternating direction method for $\ell_{2,1}$-norm least squares problem in multi-task feature learning. Journal of Industrial and Management Optimization, 2012, 8 (4) : 1057-1069. doi: 10.3934/jimo.2012.8.1057
[20]

Frédéric Bernicot, Vjekoslav Kovač. Sobolev norm estimates for a class of bilinear multipliers. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1305-1315. doi: 10.3934/cpaa.2014.13.1305

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (311)
  • HTML views (200)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy