# 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 and Imaging, 2018, 12 (5) : 1103-1120. doi: 10.3934/ipi.2018046
##### 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. [2] H. Andrew and B. Hunt, Digital Image Restoration, Prentice-Hall, Englewood Cliffs, NJ, 1977. [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. [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. [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. [6] D. Bertsekas, A. Nedic and E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003. [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. [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. [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. [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. [11] P. Craven and G. Wahba, Smoothing noisy data with spline functions, Numerische Mathematik, 31 (1978), 377-403. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998. [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. [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. [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. [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. [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. [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. [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. [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. [31] A. Tikhonov, Solution of incorrectly formulated problems and regularization method, Soviet Math. Dokl, 4 (1963), 1035-1038. [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. [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. [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. [35] M. Zhu, Fast Numerical Algorithms for Total Variation Based Image Restoration, PhD thesis, University of California, Los Angeles, 2008. [36] M. Zhu and T. Chan, An efficient primal-dual hybrid gradient algorithm for total variation image restoration, UCLA CAM Report, 08-34, 2007.

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. [2] H. Andrew and B. Hunt, Digital Image Restoration, Prentice-Hall, Englewood Cliffs, NJ, 1977. [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. [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. [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. [6] D. Bertsekas, A. Nedic and E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003. [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. [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. [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. [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. [11] P. Craven and G. Wahba, Smoothing noisy data with spline functions, Numerische Mathematik, 31 (1978), 377-403. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998. [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. [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. [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. [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. [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. [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. [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. [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. [31] A. Tikhonov, Solution of incorrectly formulated problems and regularization method, Soviet Math. Dokl, 4 (1963), 1035-1038. [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. [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. [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. [35] M. Zhu, Fast Numerical Algorithms for Total Variation Based Image Restoration, PhD thesis, University of California, Los Angeles, 2008. [36] M. Zhu and T. Chan, An efficient primal-dual hybrid gradient algorithm for total variation image restoration, UCLA CAM Report, 08-34, 2007.
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
 [1] J. Mead. $\chi^2$ test for total variation regularization parameter selection. Inverse Problems and Imaging, 2020, 14 (3) : 401-421. doi: 10.3934/ipi.2020019 [2] Juan Carlos De los Reyes, Estefanía Loayza-Romero. Total generalized variation regularization in data assimilation for Burgers' equation. Inverse Problems and Imaging, 2019, 13 (4) : 755-786. doi: 10.3934/ipi.2019035 [3] Alina Toma, Bruno Sixou, Françoise Peyrin. Iterative choice of the optimal regularization parameter in TV image restoration. Inverse Problems and Imaging, 2015, 9 (4) : 1171-1191. doi: 10.3934/ipi.2015.9.1171 [4] Yuyuan Ouyang, Yunmei Chen, Ying Wu. Total variation and wavelet regularization of orientation distribution functions in diffusion MRI. Inverse Problems and Imaging, 2013, 7 (2) : 565-583. doi: 10.3934/ipi.2013.7.565 [5] Zhiwei Tian, Yanyan Shi, Meng Wang, Xiaolong Kong, Lei Li, Feng Fu. A wavelet frame constrained total generalized variation model for imaging conductivity distribution. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2021074 [6] Raymond H. Chan, Haixia Liang, Suhua Wei, Mila Nikolova, Xue-Cheng Tai. High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Problems and Imaging, 2015, 9 (1) : 55-77. doi: 10.3934/ipi.2015.9.55 [7] Xiaoqun Zhang, Tony F. Chan. Wavelet inpainting by nonlocal total variation. Inverse Problems and Imaging, 2010, 4 (1) : 191-210. doi: 10.3934/ipi.2010.4.191 [8] Rinaldo M. Colombo, Francesca Monti. Solutions with large total variation to nonconservative hyperbolic systems. Communications on Pure and Applied Analysis, 2010, 9 (1) : 47-60. doi: 10.3934/cpaa.2010.9.47 [9] Wei Wan, Haiyang Huang, Jun Liu. Local block operators and TV regularization based image inpainting. Inverse Problems and Imaging, 2018, 12 (6) : 1389-1410. doi: 10.3934/ipi.2018058 [10] Yuan Wang, Zhi-Feng Pang, Yuping Duan, Ke Chen. Image retinex based on the nonconvex TV-type regularization. Inverse Problems and Imaging, 2021, 15 (6) : 1381-1407. doi: 10.3934/ipi.2020050 [11] Xianchao Xiu, Ying Yang, Wanquan Liu, Lingchen Kong, Meijuan Shang. An improved total variation regularized RPCA for moving object detection with dynamic background. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1685-1698. doi: 10.3934/jimo.2019024 [12] Yunho Kim, Paul M. Thompson, Luminita A. Vese. HARDI data denoising using vectorial total variation and logarithmic barrier. Inverse Problems and Imaging, 2010, 4 (2) : 273-310. doi: 10.3934/ipi.2010.4.273 [13] Mujibur Rahman Chowdhury, Jun Zhang, Jing Qin, Yifei Lou. Poisson image denoising based on fractional-order total variation. Inverse Problems and Imaging, 2020, 14 (1) : 77-96. doi: 10.3934/ipi.2019064 [14] Haijuan Hu, Jacques Froment, Baoyan Wang, Xiequan Fan. Spatial-Frequency domain nonlocal total variation for image denoising. Inverse Problems and Imaging, 2020, 14 (6) : 1157-1184. doi: 10.3934/ipi.2020059 [15] Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066 [16] Wei Wang, Ling Pi, Michael K. Ng. Saturation-Value Total Variation model for chromatic aberration correction. Inverse Problems and Imaging, 2020, 14 (4) : 733-755. doi: 10.3934/ipi.2020034 [17] Yunhai Xiao, Junfeng Yang, Xiaoming Yuan. Alternating algorithms for total variation image reconstruction from random projections. Inverse Problems and Imaging, 2012, 6 (3) : 547-563. doi: 10.3934/ipi.2012.6.547 [18] Juan C. Moreno, V. B. Surya Prasath, João C. Neves. Color image processing by vectorial total variation with gradient channels coupling. Inverse Problems and Imaging, 2016, 10 (2) : 461-497. doi: 10.3934/ipi.2016008 [19] Liyan Ma, Lionel Moisan, Jian Yu, Tieyong Zeng. A stable method solving the total variation dictionary model with $L^\infty$ constraints. Inverse Problems and Imaging, 2014, 8 (2) : 507-535. doi: 10.3934/ipi.2014.8.507 [20] Lu Liu, Zhi-Feng Pang, Yuping Duan. Retinex based on exponent-type total variation scheme. Inverse Problems and Imaging, 2018, 12 (5) : 1199-1217. doi: 10.3934/ipi.2018050

2020 Impact Factor: 1.639