Using generalized cross validation to select regularization parameter for total variation regularization problems

You-Wei Wen; Raymond Honfu Chan

doi:10.3934/ipi.2018046

Article Contents

2018, Volume 12, Issue 5: 1103-1120. Doi: 10.3934/ipi.2018046

This issue Previous Article Mitigating the influence of the boundary on PDE-based covariance operators Next Article Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations

Using generalized cross validation to select regularization parameter for total variation regularization problems

You-Wei Wen^1, , and
Raymond Honfu Chan^2,

1.
Key Laboratory of High Performance Computing and Stochastic Information Processing, College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China
2.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China

* Corresponding author: wenyouwei@gmail.com
* Corresponding author: wenyouwei@gmail.com

Received: October 31, 2016

Revised: May 31, 2018

Published: September 2018

The first author is supported by NSFC Grant No. 11361030, the Construct Program of the Key Discipline in Hunan Province, and the SRF of Hunan Provincial Education Department Grant No.17A128. The second author is supported by the HKRGC Grant No. CUHK14306316, HKRGC CRF Grant C1007-15G, HKRGC AoE Grant AoE/M-05/12, CUHK DAG No. 4053211, and CUHK FIS Grant No. 1907303.

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Abstract

The regularization approach is used widely in image restoration problems. The visual quality of the restored image depends highly on the regularization parameter. In this paper, we develop an automatic way to choose a good regularization parameter for total variation (TV) image restoration problems. It is based on the generalized cross validation (GCV) approach and hence no knowledge of noise variance is required. Due to the lack of the closed-form solution of the TV regularization problem, difficulty arises in finding the minimizer of the GCV function directly. We reformulate the TV regularization problem as a minimax problem and then apply a first-order primal-dual method to solve it. The primal subproblem is rearranged so that it becomes a special Tikhonov regularization problem for which the minimizer of the GCV function is readily computable. Hence we can determine the best regularization parameter in each iteration of the primal-dual method. The regularization parameter for the original TV regularization problem is then obtained by an averaging scheme. In essence, our method needs only to solve the TV regulation problem twice: one to determine the regularization parameter and one to restore the image with that parameter. Numerical results show that our method gives near optimal parameter, and excellent performance when compared with other state-of-the-art adaptive image restoration algorithms.

Keywords:

Mathematics Subject Classification: 65F10, 65F22, 65K10.

Citation:

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Figure 1. Original goldhill image with size $256\times 256$ and man image with size $512\times 512$

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Figure 2. ISNR value versus the regularization parameter $\alpha$ for man image degraded by the eight blurs listed in Table 1. Here the noise variance $\sigma = 2, 4, 6, 8$

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Figure 3. ISNR value versus the regularization parameter $\alpha$ for man image degraded by the eight blurs listed in Table 1. Here the noise variance $\sigma = 10, 20, 30, 40$

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Figure 4. The test images: macaws, motors, sailboat at pier, tropical island, lighthouse in Maine, P51 Mustang, Portland Head Light, barn and pond, mountain chalet. The sizes of the images are all $512 \times 768$

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Table 1. The point spread functions of the blurs used in the tests

Type	Function
PSF1	$\texttt{fspecial('average', 3)} $
PSF2	$\texttt{fspecial('average', 9)} $
PSF3	$ \texttt{fspecial('gaussian', 3, 1)}$
PSF4	$\texttt{fspecial('gaussian', 9, 3)}$
PSF5	$\texttt{fspecial('disk', 2)}$
PSF6	$\texttt{fspecial('disk', 4)}$
PSF7	$[1, 4, 6, 4, 1]'\times [1, 4, 6, 4, 1]/256$
PSF8	$\texttt{fspecial('motion', 20, 45)}$

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Table 2. Regularization parameter $\alpha$ obtained by our approach for difference step sizes $t$ and $s = \frac{1}{16t}$

		Goldhill			Man
PSF	σ	t = 0.1	t = 0.5	t = 1	t = 0.1	t = 0.5	t = 1
PSF1	2	4.21	4.21	4.21	5.27	5.27	5.27
	4	1.05	1.05	1.05	1.65	1.65	1.65
	6	0.71	0.71	0.71	0.78	0.78	0.78
	8	0.49	0.49	0.49	0.50	0.50	0.50
PSF2	2	17.68	17.68	17.68	7.43	7.43	7.43
	4	5.53	5.53	5.53	3.58	3.58	3.58
	6	2.47	2.47	2.47	1.66	1.66	1.66
	8	1.39	1.39	1.39	0.96	0.96	0.96
PSF3	2	6.93	6.93	6.93	2.00	2.00	2.00
	4	1.20	1.20	1.20	1.99	1.99	1.99
	6	0.66	0.66	0.66	1.13	1.13	1.13
	8	0.46	0.46	0.46	0.82	0.82	0.82
PSF4	2	9.41	9.41	9.41	5.66	5.66	5.66
	4	2.66	2.66	2.66	2.43	2.43	2.43
	6	1.39	1.39	1.39	1.16	1.16	1.16
	8	0.91	0.91	0.91	0.74	0.74	0.74
PSF5	2	5.41	5.41	5.41	6.28	6.28	6.28
	4	1.25	1.25	1.25	1.30	1.30	1.30
	6	0.69	0.69	0.69	0.72	0.72	0.72
	8	0.50	0.50	0.50	0.50	0.50	0.50
PSF6	2	7.28	7.28	7.28	6.33	6.33	6.33
	4	2.65	2.65	2.65	2.19	2.19	2.19
	6	1.35	1.35	1.35	1.08	1.08	1.08
	8	0.78	0.78	0.78	0.72	0.72	0.72
PSF7	2	4.22	4.22	4.22	6.51	6.51	6.51
	4	1.23	1.23	1.23	1.33	1.33	1.33
	6	0.76	0.76	0.76	0.70	0.70	0.70
	8	0.50	0.50	0.50	0.48	0.48	0.48

| Show Table

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Table 3. ISNR results for macaws image

	$\sigma$	ADMM	SALSA	ADMM-O	SALSA-O	GCV-L	Ours	Difference
PSF1	2	2.31	1.10	2.33	1.12	$ \underline {{\mathit{3.30}}} $	3.68	0.38
	4	2.69	0.41	2.66	0.37	$ \underline {{\mathit{3.41}}} $	3.71	0.30
	6	3.72	0.50	3.65	0.44	$ \underline {{\mathit{4.12}}} $	4.56	0.44
	8	4.78	0.87	4.68	0.79	$ \underline {{\mathit{4.82}}} $	5.54	0.72
PSF2	2	2.12	1.20	2.10	1.18	$ \underline {{\mathit{2.86}}} $	3.47	0.61
	4	2.10	0.64	2.04	0.60	$ \underline {{\mathit{2.49}}} $	2.66	0.17
	6	2.55	0.56	2.49	0.52	$ \underline {{\mathit{2.78}}} $	3.04	0.26
	8	3.21	0.95	3.13	0.95	$ \underline {{\mathit{3.32}}} $	3.61	0.29
PSF3	2	1.94	0.64	1.98	0.71	$ \underline {{\mathit{2.70}}} $	2.94	0.24
	4	2.63	0.23	2.61	0.20	$ \underline {{\mathit{3.50}}} $	3.62	0.12
	6	3.82	0.46	3.76	0.40	$ \underline {{\mathit{4.30}}} $	4.69	0.39
	8	4.98	0.90	4.88	0.82	$ \underline {{\mathit{5.00}}} $	5.75	0.75
PSF4	2	1.53	0.71	1.51	0.68	$ \underline {{\mathit{2.05}}} $	2.32	0.27
	4	1.74	0.25	1.70	0.22	$ \underline {{\mathit{2.04}}} $	2.30	0.26
	6	2.34	0.28	2.28	0.23	$ \underline {{\mathit{2.51}}} $	2.79	0.28
	8	3.10	0.71	3.02	0.71	$ \underline {{\mathit{3.21}}} $	3.47	0.26
PSF5	2	2.13	1.06	2.12	1.04	$ \underline {{\mathit{2.73}}} $	3.33	0.60
	4	2.42	0.24	2.38	0.19	$ \underline {{\mathit{3.05}}} $	3.35	0.30
	6	3.27	0.28	3.19	0.22	$ \underline {{\mathit{3.77}}} $	4.11	0.34
	8	4.25	0.64	4.13	0.56	$ \underline {{\mathit{4.61}}} $	5.03	0.42
PSF6	2	1.92	1.00	1.89	0.98	$ \underline {{\mathit{2.62}}} $	3.09	0.47
	4	1.99	0.46	1.95	0.42	$ \underline {{\mathit{2.38}}} $	2.70	0.32
	6	2.55	0.45	2.49	0.41	$ \underline {{\mathit{2.73}}} $	3.07	0.34
	8	3.30	0.83	3.22	0.83	$ \underline {{\mathit{3.39}}} $	3.73	0.34
PSF7	2	2.09	0.92	2.08	0.90	$ \underline {{\mathit{2.79}}} $	3.45	0.66
	4	2.46	0.10	2.40	0.03	$ \underline {{\mathit{3.21}}} $	3.47	0.26
	6	3.41	0.19	3.31	0.11	$ \underline {{\mathit{3.97}}} $	4.29	0.32
	8	4.43	0.59	4.30	0.50	$ \underline {{\mathit{4.78}}} $	5.25	0.47
PSF8	2	3.26	1.82	3.34	1.88	$ \underline {{\mathit{3.78}}} $	4.41	0.63
	4	$ \underline {{\mathit{2.87}}} $	0.95	2.86	0.94	2.76	3.43	0.56
	6	$\underline {{\mathit{3.05}}} $	0.70	3.01	0.67	2.52	3.94	0.89
	8	$ \underline {{\mathit{3.49}}} $	0.85	3.41	0.85	2.84	4.16	0.67

| Show Table

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Table 4. ISNR results for motorcross bikes image

	$\sigma$	ADMM	SALSA	ADMM-O	SALSA-O	GCV-L	Ours	Difference
PSF1	2	2.68	1.75	2.71	1.79	$ \underline {{\mathit{3.47}}} $	3.85	0.38
	4	2.57	0.80	2.54	0.77	$ \underline {{\mathit{3.17}}} $	3.41	0.24
	6	3.20	0.65	3.14	0.59	$ \underline {{\mathit{3.61}}} $	3.87	0.26
	8	4.03	0.76	3.95	0.69	$ \underline {{\mathit{4.19}}} $	4.59	0.40
PSF2	2	3.64	2.23	3.59	2.18	$ \underline {{\mathit{4.88}}} $	5.42	0.54
	4	2.95	0.80	2.87	0.77	$ \underline {{\mathit{3.40}}} $	4.59	1.19
	6	2.83	0.47	2.70	0.43	$ \underline {{\mathit{3.04}}} $	4.19	1.15
	8	3.01	0.42	2.84	0.38	$ \underline {{\mathit{3.11}}} $	4.18	1.07
PSF3	2	1.89	1.03	1.95	1.09	$ \underline {{\mathit{2.90}}} $	3.12	0.22
	4	2.18	0.39	2.17	0.39	2.98	$ \underline {{\mathit{2.94}}} $	$-0.04$
	6	3.06	0.46	3.02	0.41	$ \underline {{\mathit{3.60}}} $	3.64	0.04
	8	4.04	0.69	3.98	0.63	$ \underline {{\mathit{4.25}}} $	4.49	0.24
PSF4	2	1.79	1.01	1.75	0.98	$ \underline {{\mathit{2.75}}} $	3.56	0.81
	4	1.58	0.25	1.53	0.22	$ \underline {{\mathit{2.16}}} $	2.66	0.50
	6	1.82	0.05	1.74	0.01	$ \underline {{\mathit{2.21}}} $	2.70	0.49
	8	2.26	0.09	2.14	0.04	$ \underline {{\mathit{2.55}}} $	2.94	0.39
PSF5	2	2.36	1.64	2.36	1.63	$ \underline {{\mathit{2.94}}} $	3.57	0.63
	4	2.36	0.69	2.32	0.65	$ \underline {{\mathit{2.92}}} $	3.19	0.27
	6	2.93	0.39	2.86	0.33	$ \underline {{\mathit{3.34}}} $	3.65	0.31
	8	3.69	0.44	3.60	0.36	$ \underline {{\mathit{3.98}}} $	4.31	0.33
PSF6	2	2.53	1.48	2.49	1.44	$ \underline {{\mathit{3.42}}} $	4.65	1.23
	4	2.07	0.49	2.00	0.45	$ \underline {{\mathit{2.63}}} $	3.46	0.83
	6	2.20	0.21	2.11	0.16	$ \underline {{\mathit{2.58}}} $	3.19	0.61
	8	2.62	0.23	2.51	0.17	$ \underline {{\mathit{2.83}}} $	3.41	0.58
PSF7	2	2.22	1.42	2.21	1.41	$ \underline {{\mathit{3.10}}} $	3.39	0.29
	4	2.25	0.50	2.20	0.45	$ \underline {{\mathit{2.92}}} $	3.12	0.20
	6	2.88	0.28	2.82	0.21	$ \underline {{\mathit{3.39}}} $	3.60	0.21
	8	3.71	0.37	3.62	0.29	$ \underline {{\mathit{4.05}}} $	4.35	0.30
PSF8	2	3.92	2.35	4.07	2.50	$ \underline {{\mathit{4.44}}} $	5.73	1.29
	4	2.96	0.98	2.97	0.98	$ \underline {{\mathit{3.16}}} $	4.90	1.74
	6	$ \underline {{\mathit{2.76}}} $	0.62	2.70	0.60	2.49	4.13	1.37
	8	$ \underline {{\mathit{2.83}}} $	0.54	2.72	0.51	2.21	3.87	1.04

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Table 5. ISNR results for sailboat at pier image

	$\sigma$	ADMM	SALSA	ADMM-O	SALSA-O	GCV-L	Ours	Difference
PSF1	2	2.43	1.67	2.51	1.77	$ \underline {{\mathit{2.97}}} $	3.73	0.76
	4	1.79	0.52	1.79	0.52	$ \underline {{\mathit{2.16}}} $	2.55	0.39
	6	1.93	0.26	1.90	0.23	$ \underline {{\mathit{2.25}}} $	2.49	0.24
	8	2.39	0.33	2.35	0.28	$ \underline {{\mathit{2.65}}} $	2.84	0.19
PSF2	2	2.37	1.59	2.34	1.56	3.23	$ \underline {{\mathit{3.12}}} $	$-0.11$
	4	1.95	0.74	1.90	0.71	$ \underline {{\mathit{2.32}}} $	2.61	0.29
	6	1.99	0.47	1.92	0.44	$ \underline {{\mathit{2.14}}} $	2.71	0.57
	8	2.24	0.41	2.15	0.37	$ \underline {{\mathit{2.30}}} $	2.73	0.43
PSF3	2	1.61	0.85	1.74	1.00	$ \underline {{\mathit{2.36}}} $	2.98	0.62
	4	1.24	-0.04	1.27	-0.00	$ \underline {{\mathit{1.78}}} $	2.07	0.29
	6	1.58	-0.16	1.57	-0.17	$ \underline {{\mathit{2.06}}} $	2.20	0.14
	8	2.18	0.03	2.15	-0.01	$ \underline {{\mathit{2.57}}} $	2.68	0.11
PSF4	2	1.22	0.79	1.20	0.77	$ \underline {{\mathit{1.72}}} $	2.04	0.32
	4	1.18	0.34	1.15	0.31	$ \underline {{\mathit{1.41}}} $	1.77	0.36
	6	1.42	0.18	1.37	0.15	$ \underline {{\mathit{1.53}}} $	1.83	0.30
	8	1.81	0.18	1.75	0.14	$ \underline {{\mathit{1.85}}} $	2.14	0.29
PSF5	2	1.72	1.14	1.74	1.16	$ \underline {{\mathit{2.25}}} $	2.94	0.69
	4	1.45	0.37	1.43	0.35	$ \underline {{\mathit{1.74}}} $	2.03	0.29
	6	1.68	0.16	1.65	0.12	$ \underline {{\mathit{1.92}}} $	2.11	0.19
	8	2.17	0.21	2.12	0.16	$ \underline {{\mathit{2.38}}} $	2.41	0.03
PSF6	2	1.78	1.15	1.76	1.14	$ \underline {{\mathit{2.47}}} $	3.35	0.88
	4	1.52	0.57	1.48	0.54	$ \underline {{\mathit{1.80}}} $	2.32	0.52
	6	1.68	0.38	1.64	0.34	$ \underline {{\mathit{1.81}}} $	2.12	0.31
	8	2.03	0.36	1.97	0.32	$ \underline {{\mathit{2.05}}} $	2.42	0.37
PSF7	2	1.59	0.95	1.59	0.96	$ \underline {{\mathit{2.15}}} $	2.69	0.54
	4	1.26	0.06	1.24	0.04	$ \underline {{\mathit{1.62}}} $	1.96	0.34
	6	1.52	-0.13	1.47	-0.18	$ \underline {{\mathit{1.82}}} $	2.04	0.22
	8	2.04	-0.03	1.98	-0.09	$ \underline {{\mathit{2.32}}} $	2.44	0.12
PSF8	2	2.84	1.94	3.12	2.25	$ \underline {{\mathit{3.40}}} $	3.97	0.57
	4	2.36	0.94	$ \underline {{\mathit{2.50}}} $	1.01	2.47	3.47	0.97
	6	2.30	0.58	$ \underline {{\mathit{2.34}}} $	0.60	2.08	3.09	0.75
	8	$ \underline {{\mathit{2.43}}} $	0.44	$ \underline {{\mathit{2.43}}} $	0.44	1.98	3.03	0.60

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Table 6. ISNR results for tropical island image

PSF	$\sigma$	ADMM	SALSA	ADMM-O	SALSA-O	GCV-L	Ours	Difference
PSF1	2	1.67	1.04	1.70	1.08	$ \underline {{\mathit{2.21}}} $	2.95	0.74
	4	1.42	0.18	1.41	0.16	$ \underline {{\mathit{1.89}}} $	2.20	0.31
	6	1.94	0.24	1.91	0.20	$ \underline {{\mathit{2.37}}} $	2.48	0.11
	8	2.74	0.58	2.69	0.53	$ \underline {{\mathit{3.05}}} $	3.18	0.13
PSF2	2	1.42	0.85	1.40	0.83	$ \underline {{\mathit{2.43}}} $	2.50	0.07
	4	1.44	0.44	1.41	0.42	$ \underline {{\mathit{1.77}}} $	2.10	0.33
	6	1.87	0.45	1.84	0.41	$ \underline {{\mathit{2.05}}} $	2.34	0.29
	8	2.49	0.96	2.45	0.96	$ \underline {{\mathit{2.57}}} $	2.80	0.23
PSF3	2	0.85	0.19	0.91	0.27	$ \underline {{\mathit{1.48}}} $	2.15	0.67
	4	0.87	-0.40	0.86	-0.40	$ \underline {{\mathit{1.46}}} $	1.70	0.24
	6	1.62	-0.17	1.59	-0.21	$ \underline {{\mathit{2.15}}} $	2.17	0.02
	8	2.55	0.30	2.51	0.25	$ \underline {{\mathit{2.95}}} $	3.03	0.08
PSF4	2	0.80	0.39	0.78	0.37	$ \underline {{\mathit{1.04}}} $	1.34	0.30
	4	1.03	0.16	1.01	0.13	$ \underline {{\mathit{1.08}}} $	1.38	0.30
	6	1.58	0.24	1.54	0.20	$ \underline {{\mathit{1.66}}} $	1.84	0.18
	8	2.28	0.74	2.23	0.74	$ \underline {{\mathit{2.35}}} $	2.51	0.16
PSF5	2	0.97	0.49	0.97	0.49	$ \underline {{\mathit{1.29}}} $	2.02	0.73
	4	1.06	0.01	1.04	-0.02	$ \underline {{\mathit{1.43}}} $	1.65	0.22
	6	1.67	0.12	1.64	0.07	$ \underline {{\mathit{2.02}}} $	2.12	0.10
	8	2.50	0.46	2.44	0.40	$ \underline {{\mathit{2.80}}} $	2.84	0.04
PSF6	2	1.09	0.64	1.07	0.63	$ \underline {{\mathit{1.44}}} $	2.14	0.70
	4	1.26	0.34	1.23	0.32	$ \underline {{\mathit{1.41}}} $	1.70	0.29
	6	1.76	0.40	1.73	0.36	$ \underline {{\mathit{1.84}}} $	2.03	0.19
	8	2.45	0.83	2.40	0.83	$ \underline {{\mathit{2.50}}} $	2.70	0.20
PSF7	2	0.77	0.22	0.76	0.21	$ \underline {{\mathit{1.30}}} $	1.85	0.55
	4	0.82	-0.35	0.79	-0.38	$ \underline {{\mathit{1.25}}} $	1.50	0.25
	6	1.48	-0.17	1.43	-0.22	$ \underline {{\mathit{1.90}}} $	1.97	0.07
	8	2.36	0.24	2.30	0.18	$ \underline {{\mathit{2.72}}} $	2.73	0.01
PSF8	2	1.74	1.04	1.82	1.15	$ \underline {{\mathit{2.06}}} $	2.57	0.51
	4	1.68	0.52	$ \underline {{\mathit{1.70}}} $	0.53	1.67	2.07	0.37
	6	$ \underline {{\mathit{1.99}}} $	0.43	1.97	0.42	1.82	2.48	0.49
	8	$ \underline {{\mathit{2.51}}} $	0.92	2.47	0.92	2.29	2.88	0.37

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Table 7. ISNR results for lighthouse in Maine image

PSF	$\sigma$	ADMM	SALSA	ADMM-O	SALSA-O	GCV-L	Ours	Difference
PSF1	2	3.14	2.36	3.21	2.45	$ \underline {{\mathit{3.55}}} $	4.63	1.08
	4	2.15	0.95	2.14	0.94	$ \underline {{\mathit{2.51}}} $	3.03	0.52
	6	2.05	0.44	2.02	0.40	$ \underline {{\mathit{2.39}}} $	2.63	0.24
	8	2.33	0.27	2.28	0.22	$ \underline {{\mathit{2.63}}} $	2.69	0.06
PSF2	2	3.44	2.60	3.40	2.55	$ \underline {{\mathit{5.07}}} $	5.18	0.11
	4	2.69	1.17	2.62	1.15	3.63	$ \underline {{\mathit{3.02}}} $	$-0.61$
	6	$ \underline {{\mathit{2.40}}} $	0.82	2.29	0.79	2.98	2.07	$-0.91$
	8	2.37	0.63	2.27	0.60	$ \underline {{\mathit{2.59}}} $	2.86	0.27
PSF3	2	1.86	1.34	1.96	1.47	$ \underline {{\mathit{2.78}}} $	3.36	0.58
	4	1.39	0.27	1.41	0.29	$ \underline {{\mathit{2.04}}} $	2.19	0.15
	6	1.54	-0.05	1.53	-0.07	$ \underline {{\mathit{2.12}}} $	2.13	0.01
	8	2.00	-0.07	1.96	-0.11	$ \underline {{\mathit{2.50}}} $	2.53	0.03
PSF4	2	1.39	1.00	1.37	0.98	$ \underline {{\mathit{2.13}}} $	3.23	1.10
	4	1.29	0.48	1.27	0.46	$ \underline {{\mathit{1.50}}} $	1.70	0.20
	6	1.40	0.30	1.36	0.27	$ \underline {{\mathit{1.48}}} $	1.75	0.27
	8	1.65	0.23	1.60	0.20	$ \underline {{\mathit{1.66}}} $	1.91	0.25
PSF5	2	2.10	1.66	2.10	1.67	$ \underline {{\mathit{2.72}}} $	3.23	0.51
	4	1.71	0.63	1.69	0.60	$ \underline {{\mathit{2.09}}} $	2.31	0.22
	6	1.73	0.13	1.68	0.09	$ \underline {{\mathit{2.04}}} $	2.36	0.32
	8	1.98	-0.01	1.92	-0.06	$ \underline {{\mathit{2.25}}} $	2.52	0.27
PSF6	2	1.99	1.36	1.84	1.34	$ \underline {{\mathit{3.39}}} $	3.74	0.35
	4	1.55	0.70	1.52	0.68	$ \underline {{\mathit{1.85}}} $	2.46	0.61
	6	1.63	0.45	1.60	0.42	$ \underline {{\mathit{1.75}}} $	1.97	0.22
	8	1.86	0.37	1.81	0.34	$ \underline {{\mathit{1.87}}} $	2.10	0.23
PSF7	2	1.93	1.45	1.93	1.46	$ \underline {{\mathit{2.50}}} $	3.02	0.52
	4	1.49	0.40	1.47	0.37	$ \underline {{\mathit{1.97}}} $	2.21	0.24
	6	1.57	-0.09	1.53	-0.13	$ \underline {{\mathit{1.97}}} $	2.10	0.13
	8	1.89	-0.23	1.83	-0.28	$ \underline {{\mathit{2.24}}} $	2.38	0.14
PSF8	2	2.92	2.08	3.19	2.38	$ \underline {{\mathit{3.63}}} $	3.97	0.34
	4	2.42	0.81	2.50	0.86	$ \underline {{\mathit{2.51}}} $	3.49	0.98
	6	2.20	0.38	$ \underline {{\mathit{2.22}}} $	0.39	2.01	3.00	0.78
	8	$ \underline {{\mathit{2.15}}} $	0.24	2.10	0.23	1.67	2.86	0.71

| Show Table

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Table 8. ISNR results for P51 Mustang image

PSF	$\sigma$	ADMM	SALSA	ADMM-O	SALSA-O	GCV-L	Ours	Difference
PSF1	2	3.04	2.33	3.09	2.41	$ \underline {{\mathit{3.90}}} $	4.49	0.59
	4	2.74	1.15	2.74	1.14	$ \underline {{\mathit{3.24}}} $	3.78	0.54
	6	3.20	0.77	3.15	0.72	$ \underline {{\mathit{3.40}}} $	3.93	0.53
	8	$ \underline {{\mathit{3.90}}} $	0.82	3.83	0.75	3.83	4.49	0.59
PSF2	2	3.58	2.45	3.55	2.41	$ \underline {{\mathit{4.22}}} $	5.32	1.10
	4	3.27	1.25	3.23	1.23	$ \underline {{\mathit{3.31}}} $	4.42	1.11
	6	$ \underline {{\mathit{3.36}}} $	0.78	3.30	0.73	3.10	4.13	0.77
	8	$ \underline {{\mathit{3.66}}} $	0.66	3.58	0.62	3.21	4.27	0.61
PSF3	2	2.27	1.58	2.36	1.70	$ \underline {{\mathit{3.32}}} $	3.74	0.42
	4	2.26	0.63	2.28	0.65	$ \underline {{\mathit{2.93}}} $	3.24	0.31
	6	2.91	0.43	2.89	0.40	$ \underline {{\mathit{3.30}}} $	3.66	0.36
	8	3.76	0.61	3.71	0.56	$ \underline {{\mathit{3.84}}} $	4.36	0.52
PSF4	2	2.73	1.55	2.71	1.52	$ \underline {{\mathit{2.78}}} $	3.64	0.86
	4	$ \underline {{\mathit{2.65}}} $	0.77	2.62	0.73	2.30	3.29	0.64
	6	$ \underline {{\mathit{2.88}}} $	0.44	2.83	0.40	2.43	3.44	0.56
	8	$ \underline {{\mathit{3.29}}} $	0.41	3.21	0.37	2.75	3.70	0.41
PSF5	2	2.65	2.03	2.66	2.03	$ \underline {{\mathit{3.04}}} $	4.05	1.01
	4	2.57	0.90	2.54	0.86	$ \underline {{\mathit{2.75}}} $	3.40	0.65
	6	$ \underline {{\mathit{3.00}}} $	0.52	2.95	0.46	2.99	3.61	0.61
	8	$ \underline {{\mathit{3.65}}} $	0.58	3.57	0.52	3.51	4.23	0.58
PSF6	2	3.10	1.95	3.08	1.93	$ \underline {{\mathit{3.50}}} $	4.90	1.40
	4	$ \underline {{\mathit{2.90}}} $	0.99	2.87	0.96	2.68	3.70	0.80
	6	$ \underline {{\mathit{3.09}}} $	0.63	3.04	0.59	2.73	3.68	0.59
	8	$ \underline {{\mathit{3.49}}} $	0.59	3.42	0.55	2.95	3.98	0.49
PSF7	2	2.43	1.81	2.43	1.81	$ \underline {{\mathit{2.93}}} $	3.75	0.82
	4	2.35	0.64	2.32	0.60	$ \underline {{\mathit{2.72}}} $	3.24	0.52
	6	2.83	0.30	2.78	0.25	$ \underline {{\mathit{3.01}}} $	3.56	0.55
	8	3.56	0.41	3.48	0.34	$ \underline {{\mathit{3.57}}} $	4.16	0.59
PSF8	2	4.94	3.29	5.09	3.55	$ \underline {{\mathit{5.22}}} $	6.71	1.49
	4	4.37	1.65	$ \underline {{\mathit{4.42}}} $	1.68	3.94	5.35	0.93
	6	$ \underline {{\mathit{4.19}}} $	1.01	4.17	1.00	3.27	5.32	1.13
	8	$ \underline {{\mathit{4.22}}} $	0.72	4.17	0.71	2.89	5.17	0.95

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Table 9. ISNR results for Portland head light image

PSF	$\sigma$	ADMM	SALSA	ADMM-O	SALSA-O	GCV-L	Ours	Difference
PSF1	2	2.50	1.76	2.61	1.90	$ \underline {{\mathit{3.08}}} $	3.98	0.90
	4	1.79	0.54	1.81	0.56	$ \underline {{\mathit{2.21}}} $	2.58	0.37
	6	1.79	0.18	1.78	0.17	$ \underline {{\mathit{2.18}}} $	2.39	0.21
	8	2.13	0.19	2.10	0.16	$ \underline {{\mathit{2.49}}} $	2.63	0.14
PSF2	2	2.37	1.65	2.34	1.62	$ \underline {{\mathit{3.28}}} $	3.82	0.54
	4	1.89	0.74	1.84	0.71	2.32	$ \underline {{\mathit{2.31}}} $	$-0.01$
	6	1.81	0.45	1.74	0.42	$ \underline {{\mathit{2.09}}} $	2.60	0.51
	8	1.96	0.36	1.87	0.32	$ \underline {{\mathit{2.17}}} $	2.59	0.42
PSF3	2	1.71	0.94	1.90	1.16	$ \underline {{\mathit{2.57}}} $	3.27	0.70
	4	1.24	-0.04	1.30	0.04	$ \underline {{\mathit{1.88}}} $	2.15	0.27
	6	1.44	-0.25	1.45	-0.24	$ \underline {{\mathit{2.01}}} $	2.10	0.09
	8	1.92	-0.14	1.90	-0.16	$ \underline {{\mathit{2.42}}} $	2.45	0.03
PSF4	2	1.23	0.85	1.22	0.84	$ \underline {{\mathit{1.86}}} $	2.35	0.49
	4	1.12	0.35	1.10	0.33	$ \underline {{\mathit{1.49}}} $	1.70	0.21
	6	1.27	0.14	1.23	0.11	$ \underline {{\mathit{1.52}}} $	1.74	0.22
	8	1.56	0.10	1.51	0.06	$ \underline {{\mathit{1.74}}} $	1.91	0.17
PSF5	2	1.71	1.18	1.74	1.21	$ \underline {{\mathit{2.21}}} $	2.85	0.64
	4	1.37	0.36	1.35	0.35	$ \underline {{\mathit{1.76}}} $	1.99	0.23
	6	1.50	0.12	1.47	0.09	$ \underline {{\mathit{1.84}}} $	1.92	0.08
	8	1.88	0.11	1.83	0.06	$ \underline {{\mathit{2.19}}} $	2.27	0.08
PSF6	2	1.88	1.29	1.85	1.27	$ \underline {{\mathit{3.15}}} $	3.50	0.35
	4	1.51	0.61	1.48	0.59	$ \underline {{\mathit{1.93}}} $	2.38	0.45
	6	1.57	0.36	1.52	0.33	$ \underline {{\mathit{1.84}}} $	2.15	0.31
	8	1.83	0.29	1.77	0.25	$ \underline {{\mathit{2.00}}} $	2.25	0.25
PSF7	2	1.64	1.03	1.66	1.06	$ \underline {{\mathit{2.17}}} $	2.79	0.62
	4	1.23	0.08	1.21	0.07	$ \underline {{\mathit{1.69}}} $	1.94	0.25
	6	1.36	-0.19	1.32	-0.22	$ \underline {{\mathit{1.76}}} $	1.77	0.01
	8	1.75	-0.15	1.70	-0.20	2.13	$ \underline {{\mathit{2.11}}} $	$-0.02$
PSF8	2	2.90	2.07	3.31	2.52	$ \underline {{\mathit{3.74}}} $	4.23	0.49
	4	2.41	1.08	2.59	1.19	$ \underline {{\mathit{2.70}}} $	3.65	0.95
	6	2.27	0.71	$ \underline {{\mathit{2.35}}} $	0.75	2.25	3.16	0.81
	8	2.33	0.54	$ \underline {{\mathit{2.35}}} $	0.55	2.01	3.01	0.66

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Table 10. ISNR results for barn and pond image

PSF	$\sigma$	ADMM	SALSA	ADMM-O	SALSA-O	GCV-L	Ours	Difference
PSF1	2	2.31	1.59	2.37	1.68	$ \underline {{\mathit{2.92}}} $	3.46	0.54
	4	1.80	0.46	1.80	0.46	$ \underline {{\mathit{2.26}}} $	2.65	0.39
	6	2.08	0.22	2.05	0.18	$ \underline {{\mathit{2.47}}} $	2.72	0.25
	8	2.67	0.34	2.61	0.29	$ \underline {{\mathit{2.96}}} $	3.23	0.27
PSF2	2	2.15	1.35	2.13	1.33	3.11	$ \underline {{\mathit{3.08}}} $	$-0.03$
	4	1.78	0.58	1.74	0.55	$ \underline {{\mathit{2.30}}} $	2.85	0.55
	6	1.89	0.41	1.82	0.37	$ \underline {{\mathit{2.19}}} $	2.74	0.55
	8	2.22	0.44	2.13	0.40	$ \underline {{\mathit{2.43}}} $	2.85	0.42
PSF3	2	1.47	0.72	1.60	0.88	$ \underline {{\mathit{2.23}}} $	2.82	0.59
	4	1.22	-0.08	1.25	-0.04	$ \underline {{\mathit{1.90}}} $	2.14	0.24
	6	1.73	-0.11	1.72	-0.12	$ \underline {{\mathit{2.32}}} $	2.37	0.05
	8	2.48	0.13	2.45	0.09	$ \underline {{\mathit{2.94}}} $	3.03	0.09
PSF4	2	0.97	0.55	0.95	0.53	$ \underline {{\mathit{1.64}}} $	2.16	0.52
	4	0.98	0.18	0.96	0.15	$ \underline {{\mathit{1.32}}} $	1.55	0.23
	6	1.31	0.11	1.27	0.08	$ \underline {{\mathit{1.55}}} $	1.70	0.15
	8	1.80	0.21	1.75	0.16	$ \underline {{\mathit{1.99}}} $	2.14	0.15
PSF5	2	1.66	1.07	1.66	1.09	$ \underline {{\mathit{2.10}}} $	2.87	0.77
	4	1.45	0.21	1.42	0.18	$ \underline {{\mathit{1.82}}} $	2.21	0.39
	6	1.74	0.04	1.69	-0.01	$ \underline {{\mathit{2.10}}} $	2.20	0.10
	8	2.29	0.17	2.22	0.12	$ \underline {{\mathit{2.65}}} $	2.84	0.19
PSF6	2	1.57	0.93	1.55	0.91	$ \underline {{\mathit{2.54}}} $	3.19	0.65
	4	1.35	0.39	1.32	0.37	$ \underline {{\mathit{1.74}}} $	2.24	0.50
	6	1.58	0.29	1.53	0.26	$ \underline {{\mathit{1.82}}} $	2.27	0.45
	8	2.03	0.37	1.97	0.33	$ \underline {{\mathit{2.19}}} $	2.45	0.26
PSF7	2	1.45	0.87	1.46	0.89	$ \underline {{\mathit{1.90}}} $	2.65	0.75
	4	1.26	-0.05	1.24	-0.08	$ \underline {{\mathit{1.74}}} $	2.02	0.28
	6	1.64	-0.22	1.59	-0.26	$ \underline {{\mathit{2.06}}} $	2.25	0.19
	8	2.24	-0.03	2.18	-0.09	$ \underline {{\mathit{2.65}}} $	2.82	0.17
PSF8	2	2.44	1.64	2.67	1.90	$ \underline {{\mathit{3.17}}} $	3.68	0.51
	4	2.01	0.79	2.11	0.85	$ \underline {{\mathit{2.28}}} $	3.19	0.91
	6	2.04	0.53	$ \underline {{\mathit{2.07}}} $	0.54	1.98	2.88	0.81
	8	$ \underline {{\mathit{2.30}}} $	0.50	2.29	0.49	2.11	2.93	0.63

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Table 11. ISNR results for mountain chalet image

PSF	$\sigma$	ADMM	SALSA	ADMM-O	SALSA-O	GCV-L	Ours	Difference
PSF1	2	3.00	2.23	3.22	2.49	$ \underline {{\mathit{3.71}}} $	4.78	1.07
	4	2.02	0.85	2.08	0.92	$ \underline {{\mathit{2.41}}} $	3.04	0.63
	6	1.72	0.32	1.74	0.33	$ \underline {{\mathit{2.03}}} $	2.42	0.39
	8	1.78	0.16	1.77	0.15	$ \underline {{\mathit{2.07}}} $	2.34	0.27
PSF2	2	2.32	1.79	2.30	1.76	$ \underline {{\mathit{3.00}}} $	3.29	0.29
	4	1.90	0.93	1.86	0.91	$ \underline {{\mathit{2.32}}} $	2.61	0.29
	6	1.80	0.59	1.75	0.57	$ \underline {{\mathit{1.94}}} $	2.40	0.46
	8	1.89	0.43	1.83	0.40	$ \underline {{\mathit{1.94}}} $	2.35	0.41
PSF3	2	2.15	1.35	2.45	1.70	$ \underline {{\mathit{3.11}}} $	3.95	0.84
	4	1.39	0.22	1.53	0.35	$ \underline {{\mathit{2.01}}} $	2.34	0.33
	6	1.26	-0.22	1.32	-0.17	$ \underline {{\mathit{1.77}}} $	2.04	0.27
	8	1.45	-0.31	1.46	-0.30	$ \underline {{\mathit{1.92}}} $	1.95	0.03
PSF4	2	1.22	0.89	1.21	0.88	$ \underline {{\mathit{1.70}}} $	2.00	0.30
	4	1.13	0.48	1.12	0.46	$ \underline {{\mathit{1.31}}} $	1.49	0.18
	6	1.22	0.27	1.19	0.25	$ \underline {{\mathit{1.28}}} $	1.53	0.25
	8	1.41	0.18	1.37	0.15	$ \underline {{\mathit{1.42}}} $	1.64	0.22
PSF5	2	1.88	1.39	1.94	1.45	$ \underline {{\mathit{2.59}}} $	3.33	0.74
	4	1.42	0.58	1.42	0.58	$ \underline {{\mathit{1.73}}} $	2.06	0.33
	6	1.38	0.22	1.36	0.20	$ \underline {{\mathit{1.60}}} $	1.86	0.26
	8	1.55	0.09	1.52	0.06	$ \underline {{\mathit{1.74}}} $	1.92	0.18
PSF6	2	1.85	1.36	1.85	1.35	$ \underline {{\mathit{2.89}}} $	3.59	0.70
	4	1.50	0.74	1.48	0.73	$ \underline {{\mathit{1.78}}} $	2.35	0.57
	6	1.50	0.50	1.47	0.48	$ \underline {{\mathit{1.63}}} $	1.94	0.31
	8	1.65	0.39	1.61	0.36	$ \underline {{\mathit{1.68}}} $	2.00	0.32
PSF7	2	1.84	1.26	1.89	1.33	$ \underline {{\mathit{2.60}}} $	3.07	0.47
	4	1.27	0.26	1.26	0.26	$ \underline {{\mathit{1.67}}} $	2.03	0.36
	6	1.16	-0.14	1.14	-0.16	$ \underline {{\mathit{1.48}}} $	1.85	0.37
	8	1.33	-0.25	1.30	-0.28	$ \underline {{\mathit{1.62}}} $	1.78	0.16
PSF8	2	3.05	2.28	3.48	2.80	$ \underline {{\mathit{3.78}}} $	4.12	0.34
	4	2.59	1.30	$ \underline {{\mathit{2.79}}} $	1.46	2.77	3.63	0.84
	6	2.41	0.85	$ \underline {{\mathit{2.52}}} $	0.91	2.29	3.16	0.64
	8	2.38	0.63	$ \underline {{\mathit{2.43}}} $	0.65	2.00	2.97	0.54

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