# American Institute of Mathematical Sciences

October  2018, 12(5): 1103-1120. doi: 10.3934/ipi.2018046

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

 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

Received  November 2016 Revised  June 2018 Published  July 2018

Fund Project: 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.

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.

Citation: You-Wei Wen, Raymond Honfu Chan. Using generalized cross validation to select regularization parameter for total variation regularization problems. Inverse Problems & Imaging, 2018, 12 (5) : 1103-1120. doi: 10.3934/ipi.2018046
##### References:
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Image Process., 18 (2009), 12-26.  doi: 10.1109/TIP.2008.2007354.  Google Scholar [6] D. Bertsekas, A. Nedic and E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003.  Google Scholar [7] J. Bioucas-Dias, Bayesian wavelet-based image deconvolution: A GEM algorithm exploiting a class of heavy-tailed priors, IEEE Trans. Image Process., 15 (2006), 937-951.  doi: 10.1109/TIP.2005.863972.  Google Scholar [8] P. Blomgren and T. Chan, Modular solvers for image restoration problems using the discrepancy principle, Numer. Linear Algebra Appl., 9 (2002), 347-358.  doi: 10.1002/nla.278.  Google Scholar [9] S Bonettini and V Ruggiero, An alternating extragradient method for total variation-based image restoration from Poisson data, Inverse Problems, 27 (2011), 095001, 26pp. doi: 10.1088/0266-5611/27/9/095001.  Google Scholar [10] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar [11] P. Craven and G. Wahba, Smoothing noisy data with spline functions, Numerische Mathematik, 31 (1978), 377-403.   Google Scholar [12] I. Daubechies, M. Defrise and C. De Mol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.  Google Scholar [13] M. Figueiredo, J. M. Bioucas-Dias and M. V. Afonso, Fast frame-based image deconvolution using variable splitting and constrained optimization, In Proc. IEEE/SP 15th Workshop Statistical Signal Processing SSP ’09, (2009), 109-112. Google Scholar [14] M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), 906-916.  doi: 10.1109/TIP.2003.814255.  Google Scholar [15] N. Galatsanos and A. Katsaggelos, Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, IEEE Trans. Image Process., 1 (1992), 322-336.   Google Scholar [16] D. Girard, The fast monte-carlo cross-validation and cl procedures-comments, new results and application to image recovery problems, Computational statistics, 10 (1995), 205-258.   Google Scholar [17] G. H. Golub, M. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.  doi: 10.2307/1268518.  Google Scholar [18] E. Haber and D. Oldenburg, A gcv based method for nonlinear ill-posed problems, Computational Geosciences, 4 (2000), 41-63.  doi: 10.1023/A:1011599530422.  Google Scholar [19] P. Hansen and D. O'Leary, The use of the $L$-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput., 14 (1993), 1487-1503.  doi: 10.1137/0914086.  Google Scholar [20] P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34 (1992), 561-580.  doi: 10.1137/1034115.  Google Scholar [21] H. Liao, F. Li and M. Ng, Selection of regularization parameter in total variation image restoration, J. Opt. Soc. Am. A, 26 (2009), 2311-2320.  doi: 10.1364/JOSAA.26.002311.  Google Scholar [22] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998.  Google Scholar [23] V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984. Translated from the Russian by A. B. Aries, Translation edited by Z. Nashed. doi: 10.1007/978-1-4612-5280-1.  Google Scholar [24] M. Ng, R. Chan and W. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21 (1999), 851-866.  doi: 10.1137/S1064827598341384.  Google Scholar [25] M. Ng, P. Weiss and X. Yuan, Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM J. Sci. Comput., 32 (2010), 2710-2736.  doi: 10.1137/090774823.  Google Scholar [26] N Nguyen, P. Milanfar and G. Golub, Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement, IEEE Trans. Image Process., 10 (2001), 1299-1308.  doi: 10.1109/83.941854.  Google Scholar [27] J. P. Oliveira, J. M. Bioucas-Dias and M. A. T. Figueiredo, Adaptive total variation image deblurring: A majorization--minimization approach, Signal Processing, 89 (2009), 1683-1693.   Google Scholar [28] F. O'Sullivan and G. Wahba, A cross validated bayesian retrieval algorithm for nonlinear remote sensing experiments, Journal of Computational Physics, 59 (1985), 441-455.   Google Scholar [29] S. Ramani, Z. Liu, J. Rosen, J. Nielsen and J. Fessler, Regularization parameter selection for nonlinear iterative image restoration and mri reconstruction using gcv and sure-based methods, IEEE Trans. Image Process., 21 (2012), 3659-3672.  doi: 10.1109/TIP.2012.2195015.  Google Scholar [30] 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 [31] A. Tikhonov, Solution of incorrectly formulated problems and regularization method, Soviet Math. Dokl, 4 (1963), 1035-1038.   Google Scholar [32] P. Weiss, L. Blanc-Féraud and G. Aubert, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. Sci. Comput., 31 (2009), 2047-2080.  doi: 10.1137/070696143.  Google Scholar [33] Y. Wen and R. Chan, Parameter selection for total-variation-based image restoration using discrepancy principle, IEEE Trans. Image Process., 21 (2012), 1770-1781.  doi: 10.1109/TIP.2011.2181401.  Google Scholar [34] Y. Wen, R. Chan and A. Yip, A primal-dual method for total variation-based wavelet domain inpainting, IEEE Trans. Image Process., 21 (2012), 106-114.  doi: 10.1109/TIP.2011.2159983.  Google Scholar [35] M. Zhu, Fast Numerical Algorithms for Total Variation Based Image Restoration, PhD thesis, University of California, Los Angeles, 2008.  Google Scholar [36] M. Zhu and T. Chan, An efficient primal-dual hybrid gradient algorithm for total variation image restoration, UCLA CAM Report, 08-34, 2007. Google Scholar

show all references

##### References:
 [1] M. Afonso, J. Bioucas-Dias and M. Figueiredo, An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems, IEEE Trans. Image Process., 20 (2011), 681-695.  doi: 10.1109/TIP.2010.2076294.  Google Scholar [2] H. Andrew and B. Hunt, Digital Image Restoration, Prentice-Hall, Englewood Cliffs, NJ, 1977. Google Scholar [3] J. F. Aujol and G. Gilboa, Constrained and SNR-based solutions for TV-Hilbert space image denoising, J. Math. Imag. Vision, 26 (2006), 217-237.  doi: 10.1007/s10851-006-7801-6.  Google Scholar [4] S. Babacan, R. Molina and A. Katsaggelos, Parameter estimation in TV image restoration using variational distribution approximation, IEEE Trans. Image Process., 17 (2008), 326-339.  doi: 10.1109/TIP.2007.916051.  Google Scholar [5] S. Babacan, R. Molina and A. Katsaggelos, Variational bayesian blind deconvolution using a total variation prior, IEEE Trans. Image Process., 18 (2009), 12-26.  doi: 10.1109/TIP.2008.2007354.  Google Scholar [6] D. Bertsekas, A. Nedic and E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003.  Google Scholar [7] J. Bioucas-Dias, Bayesian wavelet-based image deconvolution: A GEM algorithm exploiting a class of heavy-tailed priors, IEEE Trans. Image Process., 15 (2006), 937-951.  doi: 10.1109/TIP.2005.863972.  Google Scholar [8] P. Blomgren and T. Chan, Modular solvers for image restoration problems using the discrepancy principle, Numer. Linear Algebra Appl., 9 (2002), 347-358.  doi: 10.1002/nla.278.  Google Scholar [9] S Bonettini and V Ruggiero, An alternating extragradient method for total variation-based image restoration from Poisson data, Inverse Problems, 27 (2011), 095001, 26pp. doi: 10.1088/0266-5611/27/9/095001.  Google Scholar [10] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar [11] P. Craven and G. Wahba, Smoothing noisy data with spline functions, Numerische Mathematik, 31 (1978), 377-403.   Google Scholar [12] I. Daubechies, M. Defrise and C. De Mol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457.  doi: 10.1002/cpa.20042.  Google Scholar [13] M. Figueiredo, J. M. Bioucas-Dias and M. V. Afonso, Fast frame-based image deconvolution using variable splitting and constrained optimization, In Proc. IEEE/SP 15th Workshop Statistical Signal Processing SSP ’09, (2009), 109-112. Google Scholar [14] M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), 906-916.  doi: 10.1109/TIP.2003.814255.  Google Scholar [15] N. Galatsanos and A. Katsaggelos, Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, IEEE Trans. Image Process., 1 (1992), 322-336.   Google Scholar [16] D. Girard, The fast monte-carlo cross-validation and cl procedures-comments, new results and application to image recovery problems, Computational statistics, 10 (1995), 205-258.   Google Scholar [17] G. H. Golub, M. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.  doi: 10.2307/1268518.  Google Scholar [18] E. Haber and D. Oldenburg, A gcv based method for nonlinear ill-posed problems, Computational Geosciences, 4 (2000), 41-63.  doi: 10.1023/A:1011599530422.  Google Scholar [19] P. Hansen and D. O'Leary, The use of the $L$-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput., 14 (1993), 1487-1503.  doi: 10.1137/0914086.  Google Scholar [20] P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34 (1992), 561-580.  doi: 10.1137/1034115.  Google Scholar [21] H. Liao, F. Li and M. Ng, Selection of regularization parameter in total variation image restoration, J. Opt. Soc. Am. A, 26 (2009), 2311-2320.  doi: 10.1364/JOSAA.26.002311.  Google Scholar [22] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998.  Google Scholar [23] V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984. Translated from the Russian by A. B. Aries, Translation edited by Z. Nashed. doi: 10.1007/978-1-4612-5280-1.  Google Scholar [24] M. Ng, R. Chan and W. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21 (1999), 851-866.  doi: 10.1137/S1064827598341384.  Google Scholar [25] M. Ng, P. Weiss and X. Yuan, Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM J. Sci. Comput., 32 (2010), 2710-2736.  doi: 10.1137/090774823.  Google Scholar [26] N Nguyen, P. Milanfar and G. Golub, Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement, IEEE Trans. Image Process., 10 (2001), 1299-1308.  doi: 10.1109/83.941854.  Google Scholar [27] J. P. Oliveira, J. M. Bioucas-Dias and M. A. T. Figueiredo, Adaptive total variation image deblurring: A majorization--minimization approach, Signal Processing, 89 (2009), 1683-1693.   Google Scholar [28] F. O'Sullivan and G. Wahba, A cross validated bayesian retrieval algorithm for nonlinear remote sensing experiments, Journal of Computational Physics, 59 (1985), 441-455.   Google Scholar [29] S. Ramani, Z. Liu, J. Rosen, J. Nielsen and J. Fessler, Regularization parameter selection for nonlinear iterative image restoration and mri reconstruction using gcv and sure-based methods, IEEE Trans. Image Process., 21 (2012), 3659-3672.  doi: 10.1109/TIP.2012.2195015.  Google Scholar [30] 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 [31] A. Tikhonov, Solution of incorrectly formulated problems and regularization method, Soviet Math. Dokl, 4 (1963), 1035-1038.   Google Scholar [32] P. Weiss, L. Blanc-Féraud and G. Aubert, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. Sci. Comput., 31 (2009), 2047-2080.  doi: 10.1137/070696143.  Google Scholar [33] Y. Wen and R. Chan, Parameter selection for total-variation-based image restoration using discrepancy principle, IEEE Trans. Image Process., 21 (2012), 1770-1781.  doi: 10.1109/TIP.2011.2181401.  Google Scholar [34] Y. Wen, R. Chan and A. Yip, A primal-dual method for total variation-based wavelet domain inpainting, IEEE Trans. Image Process., 21 (2012), 106-114.  doi: 10.1109/TIP.2011.2159983.  Google Scholar [35] M. Zhu, Fast Numerical Algorithms for Total Variation Based Image Restoration, PhD thesis, University of California, Los Angeles, 2008.  Google Scholar [36] M. Zhu and T. Chan, An efficient primal-dual hybrid gradient algorithm for total variation image restoration, UCLA CAM Report, 08-34, 2007. Google Scholar
Original goldhill image with size $256\times 256$ and man image with size $512\times 512$
ISNR value versus the regularization parameter $\alpha$ for man image degraded by the eight blurs listed in Table 1. Here the noise variance $\sigma = 2, 4, 6, 8$
ISNR value versus the regularization parameter $\alpha$ for man image degraded by the eight blurs listed in Table 1. Here the noise variance $\sigma = 10, 20, 30, 40$
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 $512 \times 768$
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 $\texttt{fspecial('motion', 20, 45)}$
 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 $\texttt{fspecial('motion', 20, 45)}$
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
 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
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
 $\sigma$ 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
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
 $\sigma$ 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
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
 $\sigma$ 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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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