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

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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: December 31, 2019

Revised: August 31, 2020

Early access: November 2020

Published: April 2021

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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:

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

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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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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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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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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

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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

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