American Institute of Mathematical Sciences

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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  (A) Circles; (B) Fluorescent Cells; (C) Cameraman
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$ \eta = 4, \sigma = 10^{-4} $">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} $
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$ \eta=16, \sigma=10^{-4} $">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} $
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$ \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 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
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Figure 5.  Line profiles of recovery results showed in Fig. 4
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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} $
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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 $-HTV
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Figure 8.  SE ($ \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))
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Figure 9.  The performance of BCA w.r.t. $ \alpha $ (in log scale), where $ \eta = 1, \sigma = 10^{-4} $, using test image Circles(Fig1(A))
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Figure 10.  The performance of BCA$ _f $ w.r.t. $ \alpha_w $ (in log scale), $ \alpha_p $ (in log scale), where $ \eta = 1, \sigma = 10^{-4} $, using test image Circles(Fig1(A))
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Figure 11.  The performance of BCA$ _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))
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Figure 12.  SNR changes w.r.t. the parameter $ \epsilon $ for BCA Algorithm, using test image Circles(Fig1(A))
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Figure 13.  Minimum value curves of $ w $ for BCA Algorithm. The test image is Circles(Fig1(A))
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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
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$ \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
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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

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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
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