Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal

Jie Zhang; Yuping Duan; Yue Lu; Michael K. Ng; Huibin Chang

doi:10.3934/ipi.2020071

Article Contents

2021, Volume 15, Issue 2: 339-366. Doi: 10.3934/ipi.2020071

This issue Previous Article A new variational approach based on level-set function for convex hull problem with outliers Next Article Duality between range and no-response tests and its application for inverse problems

Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal

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
^* Corresponding author: Huibin Chang

Received: January 2020

Revised: September 2020

Early access: November 2020

Published: April 2021

Abstract / Introduction Full Text(HTML) Figure(13) / Table(3) Related Papers Cited by

Abstract

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:

Full Text(HTML)

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

Download: Full-size image 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} $

Download: Full-size image 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} $

Download: Full-size image 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

Download: Full-size image PowerPoint slide

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

Download: Full-size image 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} $

Download: Full-size image 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

Download: Full-size image 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))

Download: Full-size image 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))

Download: Full-size image 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))

Download: Full-size image 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))

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image 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

| Show Table

DownLoad: CSV

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
	1	$ 10^{-1} $	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
	1	$ 10^{-4} $	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
	4	$ 10^{-1} $	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
	4	$ 10^{-4} $	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
	16	$ 10^{-1} $	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
	16	$ 10^{-4} $	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
	Average		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
	1	$ 10^{-1} $	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
	1	$ 10^{-4} $	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
	4	$ 10^{-1} $	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
	4	$ 10^{-4} $	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
	16	$ 10^{-1} $	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
	16	$ 10^{-4} $	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
	Average		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
	1	$ 10^{-1} $	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
	1	$ 10^{-4} $	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
	4	$ 10^{-1} $	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
	4	$ 10^{-4} $	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
	16	$ 10^{-1} $	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
	16	$ 10^{-4} $	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
	Average		0.1455	0.5564	0.6233	0.4940	0.6211	0.6290	0.6033	0.6555	0.6618

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.