Nonconvex model for mixing noise with fractional-order regularization

Shaowen Yan; Guoxi Ni; Tieyong Zeng

doi:10.3934/ipi.2022041

Article Contents

2023, Volume 17, Issue 1: 275-296. Doi: 10.3934/ipi.2022041

This issue Previous Article Inverse transmission eigenvalue problem for fixed angular momentum Next Article Approximation of the elastic Dirichlet-to-Neumann map

Nonconvex model for mixing noise with fractional-order regularization

1.
Institute of Microelectronics of the Chinese Academy of Sciences, Beijing, China
2.
Institute of Applied Physics and Computational Mathematics, Beijing, China
3.
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong

^*Corresponding author: Guoxi Ni
^*Corresponding author: Guoxi Ni

Received: September 2021

Revised: June 2022

Early access: August 2022

Published: February 2023

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Abstract

In this paper, we propose a restoration model for image degraded by different kinds of blur and mixing Gaussian-impulse noise. Our model consist of a nonconvex Exponential-Type (ET) function, a $ L_{2} $-norm data-fitting term and a fractional-order Besov norm as the regularization term. We employ the proximal linearized minimization (PLM) algorithm and alternating direction method of multipliers (ADMM) algorithm to solve our proposed minimization model, convergence analysis is carried out. The experimental outcomes demonstrate that the restored images by our proposed method are better than those existing relative methods in terms of PSNR, SSIM values and visual quality.

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Mathematics Subject Classification: 94A08, 30H25.

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Figure 1. Original images. (a): 'Boat', (b): 'House', (c): 'Cameraman', (d): 'Goldhill', (e): 'Plate', (f): 'Bridge', (g): 'Barbara', (h): 'Mandrill'

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Figure 2. Comparison of results by different methods applied on images degraded by Average blur and mixed Gaussian, salt-and-pepper noise. First column (a, e, i, m): the degraded images; Second to fourth columns: the restoration results by AOP, L12TV and our proposed algorithm respectively

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Figure 3. Comparison of results by different methods applied on images degraded by Gaussian blur and mixed Gaussian, salt-and-pepper noise. First column (a, e, i, m): the degraded images; Second to fourth columns: the restoration results by AOP, L12TV and our proposed algorithm respectively

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Figure 4. Comparison of results by different methods applied on images degraded by Average blur and mixed Gaussian, random-valued noise. First column (a, e, i, m): the degraded images; Second to fourth columns: the restoration results by AOP, L12TV and our proposed algorithm respectively

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Figure 5. Comparison of results by different methods applied on images degraded by Gaussian blur and mixed Gaussian, random-valued noise. First column (a, e, i, m): the degraded images; Second to fourth columns: the restoration results by AOP, L12TV and our proposed algorithm respectively

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Table 1. PSNR(dB) and SSIM results for images corrupted by Average blur and mixing noise (Gaussian salt-and-pepper impulse)

Image	Blur/BSNR/SP	Evaluation	AOP	L12TV	Ours
Boat	AB(5, 5)/40/30%	PSNR	26.54	29.56	32.77
	OPSNR=10.09	SSIM	0.8481	0.8901	0.9274
	AB(7, 7)/30/50%	PSNR	24.15	25.84	27.22
	OPSNR=7.88	SSIM	0.7430	0.7796	0.8189
House	AB(5, 5)/40/30%	PSNR	30.18	33.84	36.45
	OPSNR=10.67	SSIM	0.8511	0.8879	0.9131
	AB(7, 7)/30/50%	PSNR	27.86	30.50	31.97
	OPSNR=8.43	SSIM	0.8105	0.8353	0.8554
Cameraman	AB(5, 5)/40/30%	PSNR	26.96	25.98	30.22
	OPSNR=10.18	SSIM	0.8512	0.8324	0.8904
	AB(7, 7)/30/50%	PSNR	24.71	24.34	25.57
	OPSNR=8.02	SSIM	0.7889	0.7671	0.7794
Goldhill	AB(5, 5)/40/30%	PSNR	25.94	27.25	30.74
	OPSNR=10.51	SSIM	0.7032	0.7335	0.8663
	AB(7, 7)/30/50%	PSNR	24.07	25.73	27.02
	OPSNR=8.32	SSIM	0.5954	0.6374	0.7074
Plate	AB(5, 5)/40/30%	PSNR	24.74	29.08	32.26
	OPSNR=10.05	SSIM	0.9067	0.9334	0.9479
	AB(7, 7)/30/50%	PSNR	22.29	24.34	26.29
	OPSNR=7.89	SSIM	0.8227	0.8352	0.8591
Bridge	AB(5, 5)/40/30%	PSNR	24.85	25.10	28.41
	OPSNR=10.34	SSIM	0.7398	0.7031	0.8611
	AB(7, 7)/30/50%	PSNR	23.33	23.84	24.86
	OPSNR=8.17	SSIM	0.6098	0.6002	0.6682
Barbara	AB(5, 5)/40/30%	PSNR	25.33	24.67	27.96
	OPSNR=10.49	SSIM	0.7546	0.7204	0.8355
	AB(7, 7)/30/50%	PSNR	23.83	24.01	24.78
	OPSNR=8.36	SSIM	0.6660	0.6628	0.6936
Mandrill	AB(5, 5)/40/30%	PSNR	23.34	22.27	26.59
	OPSNR=10.53	SSIM	0.6872	0.5949	0.8469
	AB(7, 7)/30/50%	PSNR	21.73	21.63	23.05
	OPSNR=8.44	SSIM	0.5249	0.5195	0.6368

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Table 2. PSNR(dB) and SSIM results for images corrupted by Gaussian blur and mixing noise (Gaussian and salt-and-pepper impulse)

Image	Blur/BSNR/SP	Evaluation	AOP	L12TV	Ours
Boat	GB(5, 2)/40/30%	PSNR	27.26	29.73	32.38
	OPSNR=10.11	SSIM	0.8689	0.8964	0.9255
	GB(7, 3)/30/50%	PSNR	24.82	26.01	27.36
	OPSNR=7.89	SSIM	0.7734	0.7842	0.8195
House	GB(5, 2)/40/30%	PSNR	30.67	33.91	36.29
	OPSNR=10.68	SSIM	0.8578	0.8910	0.9122
	GB(7, 3)/30/50%	PSNR	28.40	30.38	31.55
	OPSNR=8.44	SSIM	0.8169	0.8329	0.8518
Cameraman	GB(5, 2)/40/30%	PSNR	26.23	27.39	29.65
	OPSNR=10.21	SSIM	0.8460	0.8659	0.8937
	GB(7, 3)/30/50%	PSNR	24.47	24.41	25.29
	OPSNR=8.03	SSIM	0.7837	0.7692	0.7735
Goldhill	GB(5, 2)/40/30%	PSNR	26.13	28.43	30.38
	OPSNR=10.53	SSIM	0.7132	0.7940	0.8551
	GB(7, 3)/30/50%	PSNR	24.73	25.87	26.81
	OPSNR=8.33	SSIM	0.6120	0.6504	0.6986
Plate	GB(5, 2)/40/30%	PSNR	24.99	29.62	31.90
	OPSNR=10.09	SSIM	0.9129	0.9372	0.9493
	GB(7, 3)/30/50%	PSNR	22.41	24.74	26.38
	OPSNR=7.91	SSIM	0.8483	0.8529	0.8781
Bridge	GB(5, 2)/40/30%	PSNR	25.09	25.84	28.34
	OPSNR=10.36	SSIM	0.7444	0.7570	0.8580
	GB(7, 3)/30/50%	PSNR	23.51	23.91	24.82
	OPSNR=8.18	SSIM	0.6143	0.6118	0.6686
Barbara	GB(5, 2)/40/30%	PSNR	24.52	24.87	27.76
	OPSNR=10.50	SSIM	0.7236	0.7399	0.8261
	GB(7, 3)/30/50%	PSNR	23.92	24.08	24.68
	OPSNR=8.37	SSIM	0.6705	0.6669	0.6931
Mandrill	GB(5, 2)/40/30%	PSNR	22.91	23.09	26.52
	OPSNR=10.57	SSIM	0.6512	0.6783	0.8413
	GB(7, 3)/30/50%	PSNR	21.45	21.65	22.83
	OPSNR=8.45	SSIM	0.4988	0.5265	0.6160

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Table 3. PSNR(dB) and SSIM results for images corrupted by Average blur and mixing noise (Gaussian and random-valued impulse)

Image	Blur/BSNR/RV	Evaluation	AOP	L12TV	Ours
Boat	AB(5, 5)/40/30%	PSNR	25.13	29.17	32.21
	OPSNR=13.17	SSIM	0.7588	0.8852	0.9205
	AB(7, 7)/30/50%	PSNR	21.54	24.14	26.53
	OPSNR=11.02	SSIM	0.5977	0.7095	0.7660
House	AB(5, 5)/40/30%	PSNR	29.00	33.97	36.18
	OPSNR=14.40	SSIM	0.8149	0.8904	0.9111
	AB(7, 7)/30/50%	PSNR	24.80	29.21	31.22
	OPSNR=12.20	SSIM	0.7270	0.8192	0.8476
Cameraman	AB(5, 5)/40/30%	PSNR	24.26	26.79	29.66
	OPSNR=13.32	SSIM	0.7506	0.8518	0.8724
	AB(7, 7)/30/50%	PSNR	20.99	22.36	24.91
	OPSNR=11.23	SSIM	0.6189	0.6823	0.7013
Goldhill	AB(5, 5)/40/30%	PSNR	25.11	28.39	30.26
	OPSNR=14.07	SSIM	0.6238	0.7866	0.8547
	AB(7, 7)/30/50%	PSNR	22.44	24.87	26.45
	OPSNR=11.91	SSIM	0.4618	0.6034	0.6782
Plate	AB(5, 5)/40/30%	PSNR	23.11	29.30	31.62
	OPSNR=13.07	SSIM	0.8340	0.9315	0.9388
	AB(7, 7)/30/50%	PSNR	19.42	22.80	25.37
	OPSNR=10.95	SSIM	0.6408	0.7765	0.8130
Bridge	AB(5, 5)/40/30%	PSNR	23.58	26.35	28.01
	OPSNR=13.69	SSIM	0.6007	0.7798	0.8493
	AB(7, 7)/30/50%	PSNR	21.20	22.95	24.39
	OPSNR=11.60	SSIM	0.4046	0.5466	0.6416
Barbara	AB(5, 5)/40/30%	PSNR	23.94	24.83	27.85
	OPSNR=14.04	SSIM	0.6576	0.7294	0.8275
	AB(7, 7)/30/50%	PSNR	22.61	23.70	24.48
	OPSNR=12.03	SSIM	0.5762	0.6418	0.6445
Mandrill	AB(5, 5)/40/30%	PSNR	21.81	22.82	25.70
	OPSNR=14.13	SSIM	0.5348	0.6464	0.8178
	AB(7, 7)/30/50%	PSNR	20.14	21.22	22.48
	OPSNR=12.22	SSIM	0.3396	0.4700	0.5959

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Table 4. PSNR(dB) and SSIM results for images corrupted by Gaussian blur and mixing noise (Gaussian and random-valued impulse)

Image	Blur/BSNR/RV	Evaluation	AOP	L12TV	Ours
Boat	GB(5, 2)/40/30%	PSNR	25.38	29.10	31.90
	OPSNR=13.21	SSIM	0.7679	0.8853	0.9191
	GB(7, 3)/30/50%	PSNR	21.92	24.28	26.85
	OPSNR=11.05	SSIM	0.6009	0.7141	0.8044
House	GB(5, 2)/40/30%	PSNR	29.39	33.52	35.83
	OPSNR=14.43	SSIM	0.8212	0.8857	0.9081
	GB(7, 3)/30/50%	PSNR	25.17	29.00	30.91
	OPSNR=12.21	SSIM	0.7330	0.8172	0.8399
Cameraman	GB(5, 2)/40/30%	PSNR	24.41	26.75	29.19
	OPSNR=13.38	SSIM	0.7589	0.8524	0.8870
	GB(7, 3)/30/50%	PSNR	21.11	22.40	24.83
	OPSNR=11.26	SSIM	0.6229	0.6819	0.7766
Goldhill	GB(5, 2)/40/30%	PSNR	25.30	28.31	29.93
	OPSNR=14.11	SSIM	0.6374	0.7945	0.8441
	GB(7, 3)/30/50%	PSNR	22.68	24.84	26.52
	OPSNR=11.92	SSIM	0.4746	0.6062	0.6875
Plate	GB(5, 2)/40/30%	PSNR	23.19	29.32	31.47
	OPSNR=13.15	SSIM	0.8402	0.9323	0.9451
	GB(7, 3)/30/50%	PSNR	19.71	23.22	25.71
	OPSNR=11.00	SSIM	0.6448	0.7939	0.8657
Bridge	GB(5, 2)/40/30%	PSNR	23.77	26.00	27.94
	OPSNR=13.75	SSIM	0.6231	0.7668	0.8462
	GB(7, 3)/30/50%	PSNR	21.31	22.94	24.48
	OPSNR=11.62	SSIM	0.4158	0.5495	0.6523
Barbara	GB(5, 2)/40/30%	PSNR	23.95	24.62	26.81
	OPSNR=14.07	SSIM	0.6619	0.7206	0.8073
	GB(7, 3)/30/50%	PSNR	22.66	23.73	24.47
	OPSNR=12.03	SSIM	0.5799	0.6450	0.6865
Mandrill	GB(5, 2)/40/30%	PSNR	21.92	23.54	25.68
	OPSNR=14.21	SSIM	0.5644	0.7108	0.8138
	GB(7, 3)/30/50%	PSNR	20.20	21.36	22.34
	OPSNR=12.24	SSIM	0.3498	0.4962	0.5787

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