# American Institute of Mathematical Sciences

August  2020, 14(4): 583-606. doi: 10.3934/ipi.2020027

## Image restoration from noisy incomplete frequency data by alternative iteration scheme

 1 School of Mathematics, Southeast University, Nanjing, 210096, China 2 College of Sciences, Nanjing Agricultural University, Nanjing, 210095, China 3 School of Mathematics/S.T.Yau Center of Southeast University, Southeast University, Nanjing, 210096, China

* Corresponding author: Prof. Dr. Jijun Liu

Received  March 2019 Revised  February 2020 Published  May 2020

Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with different smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.

Citation: Xiaoman Liu, Jijun Liu. Image restoration from noisy incomplete frequency data by alternative iteration scheme. Inverse Problems and Imaging, 2020, 14 (4) : 583-606. doi: 10.3934/ipi.2020027
##### References:
 [1] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Vol. 147, Applied Mathematical Sciences, 2$^nd$ edition, Springer, New York, 2006. [2] F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), 137-141.  doi: 10.1007/BF01386217. [3] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. [4] S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2011), 1-122. [5] E. J. Candès and J. Romberg, Sparsity and incoherence in compressive sampling, Inverse Problems, 23 (2007), 969–985. doi: 10.1088/0266-5611/23/3/008. [6] E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), 489–509. doi: 10.1109/TIT.2005.862083. [7] A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89-97.  doi: 10.1023/B:JMIV.0000011321.19549.88. [8] I. Daubechies, M. Defrise and C. De Mol, 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. [9] D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289–1306. doi: 10.1109/TIT.2006.871582. [10] W. J. Fu, Penalized regressions: The bridge versus the lasso, J. Comput. Graph. Statist., 7 (1998), 397-416.  doi: 10.2307/1390712. [11] T. Goldstein and S. Osher, The split Bregman method for $L1$-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.  doi: 10.1137/080725891. [12] P. J. Huber, Robust estimation of a location parameter, Ann. Math. Statist., 35 (1964), 73–101. doi: 10.1214/aoms/1177703732. [13] E. M. Kalmoun, An investigation of smooth TV-like regularization in the context of the optical flow problem, J. Imaging, 4 (2018), 31. doi: 10.3390/jimaging4020031. [14] X. Liu and J. Liu, On image restoration from random sampling noisy frequency data with regularization, Inverse Probl. Sci. Eng., 27 (2019), 1765-1789.  doi: 10.1080/17415977.2018.1557655. [15] K. Madsen and H. B. Nielsen, A finite smoothing algorithm for linear $I_1$ estimation, SIAM J. Optim., 3 (1993), 223–235. doi: 10.1137/0803010. [16] M. K. Ng, R. H. Chan and W.-C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21 (1999), 851–866. doi: 10.1137/S1064827598341384. [17] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), 460-489.  doi: 10.1137/040605412. [18] S. Perkins, K. Lacker and J. Theiler, Grafting: Fast, incremental feature selection by gradient descent in function space, J. Mach. Learn. Res., 3 (2003), 1333-1356.  doi: 10.1162/153244303322753698. [19] G. Plonka and J. Ma, Curvelet-wavelet regularized split Bregman iteration for compressed sensing, Int. J. Wavelets Multiresolut. Inf. Process., 9 (2011), 79-110.  doi: 10.1142/S0219691311003955. [20] M. Schmidt, G. Fung and R. Rosales, Fast optimization methods for $L1$ regularization: A comparative study and two new approaches, in Machine Learning: ECML 2007, Lecture Notes in Computer Science, 4701, Springer, Berlin Heidelberg, 2007,286–297. doi: 10.1007/978-3-540-74958-5_28. [21] S. K. Shevade and S. S. Keerthi, A simple and efficient algorithm for gene selection using sparse logistic regression, Bioinformatics, 19 (2003), 2246-2253.  doi: 10.1093/bioinformatics/btg308. [22] X. Shu and N. Ahuja, Hybrid compressive sampling via a new total variation TVL1, in Computer Vision – ECCV 2010, Lecture Notes in Computer Science, 6316, Springer, Berlin, Heidelberg, 2010,393–404. doi: 10.1007/978-3-642-15567-3_29. [23] W. Sun and Y.-X. Yuan, Optimization Theory and Methods, Optimization and Its Applications, 1, 1$^st$ edition, Springer, New York, 2006. [24] C. R. Vogel, Computational Methods for Inverse Problems, Frontiers in Applied Mathematics, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717570. [25] Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1 (2008), 248–272. doi: 10.1137/080724265. [26] J. Yang, W. Yin, Y. Zhang and Y. Wang, A fast algorithm for edge-preserving variational multichannel image restoration, SIAM J. Imaging Sci., 2 (2009), 569-592.  doi: 10.1137/080730421. [27] J. Yang, Y. Zhang and W. Yin, A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data, 2008. [28] J. Yang, Y. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), 2842-2865.  doi: 10.1137/080732894. [29] J. Yang, Y. Zhang and W. Yin, A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data, IEEE Journal of Selected Topics in Signal Processing, 4 (2010), 288–297. [30] W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $I_1$-minimization with applications to compressed sensing, SIAM J. Imaging Sci., 1 (2008), 143-168.  doi: 10.1137/070703983. [31] X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276.  doi: 10.1137/090746379. [32] Y. Zhu and I.-L. Chern, Convergence of the alternating minimization method for sparse MR image reconstruction, J. Inform. Comput. Sci., 8 (2011), 2067-2075. [33] Y. Zhu and X. Liu, A fast method for L1-L2 modeling for MR image compressive sensing, J. Inverse Ill-Posed Probl., 23 (2015), 211–218. doi: 10.1515/jiip-2013-0046. [34] Y. Zhu, Y. Shi, B. Zhang and X. Yu, Weighted-average alternating minimization method for magnetic resonance image reconstruction based on compressive sensing, Inverse Probl. Imaging, 8 (2014), 925-937.  doi: 10.3934/ipi.2014.8.925.

show all references

##### References:
 [1] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Vol. 147, Applied Mathematical Sciences, 2$^nd$ edition, Springer, New York, 2006. [2] F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), 137-141.  doi: 10.1007/BF01386217. [3] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. [4] S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2011), 1-122. [5] E. J. Candès and J. Romberg, Sparsity and incoherence in compressive sampling, Inverse Problems, 23 (2007), 969–985. doi: 10.1088/0266-5611/23/3/008. [6] E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), 489–509. doi: 10.1109/TIT.2005.862083. [7] A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89-97.  doi: 10.1023/B:JMIV.0000011321.19549.88. [8] I. Daubechies, M. Defrise and C. De Mol, 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. [9] D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289–1306. doi: 10.1109/TIT.2006.871582. [10] W. J. Fu, Penalized regressions: The bridge versus the lasso, J. Comput. Graph. Statist., 7 (1998), 397-416.  doi: 10.2307/1390712. [11] T. Goldstein and S. Osher, The split Bregman method for $L1$-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.  doi: 10.1137/080725891. [12] P. J. Huber, Robust estimation of a location parameter, Ann. Math. Statist., 35 (1964), 73–101. doi: 10.1214/aoms/1177703732. [13] E. M. Kalmoun, An investigation of smooth TV-like regularization in the context of the optical flow problem, J. Imaging, 4 (2018), 31. doi: 10.3390/jimaging4020031. [14] X. Liu and J. Liu, On image restoration from random sampling noisy frequency data with regularization, Inverse Probl. Sci. Eng., 27 (2019), 1765-1789.  doi: 10.1080/17415977.2018.1557655. [15] K. Madsen and H. B. Nielsen, A finite smoothing algorithm for linear $I_1$ estimation, SIAM J. Optim., 3 (1993), 223–235. doi: 10.1137/0803010. [16] M. K. Ng, R. H. Chan and W.-C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21 (1999), 851–866. doi: 10.1137/S1064827598341384. [17] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), 460-489.  doi: 10.1137/040605412. [18] S. Perkins, K. Lacker and J. Theiler, Grafting: Fast, incremental feature selection by gradient descent in function space, J. Mach. Learn. Res., 3 (2003), 1333-1356.  doi: 10.1162/153244303322753698. [19] G. Plonka and J. Ma, Curvelet-wavelet regularized split Bregman iteration for compressed sensing, Int. J. Wavelets Multiresolut. Inf. Process., 9 (2011), 79-110.  doi: 10.1142/S0219691311003955. [20] M. Schmidt, G. Fung and R. Rosales, Fast optimization methods for $L1$ regularization: A comparative study and two new approaches, in Machine Learning: ECML 2007, Lecture Notes in Computer Science, 4701, Springer, Berlin Heidelberg, 2007,286–297. doi: 10.1007/978-3-540-74958-5_28. [21] S. K. Shevade and S. S. Keerthi, A simple and efficient algorithm for gene selection using sparse logistic regression, Bioinformatics, 19 (2003), 2246-2253.  doi: 10.1093/bioinformatics/btg308. [22] X. Shu and N. Ahuja, Hybrid compressive sampling via a new total variation TVL1, in Computer Vision – ECCV 2010, Lecture Notes in Computer Science, 6316, Springer, Berlin, Heidelberg, 2010,393–404. doi: 10.1007/978-3-642-15567-3_29. [23] W. Sun and Y.-X. Yuan, Optimization Theory and Methods, Optimization and Its Applications, 1, 1$^st$ edition, Springer, New York, 2006. [24] C. R. Vogel, Computational Methods for Inverse Problems, Frontiers in Applied Mathematics, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717570. [25] Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1 (2008), 248–272. doi: 10.1137/080724265. [26] J. Yang, W. Yin, Y. Zhang and Y. Wang, A fast algorithm for edge-preserving variational multichannel image restoration, SIAM J. Imaging Sci., 2 (2009), 569-592.  doi: 10.1137/080730421. [27] J. Yang, Y. Zhang and W. Yin, A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data, 2008. [28] J. Yang, Y. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), 2842-2865.  doi: 10.1137/080732894. [29] J. Yang, Y. Zhang and W. Yin, A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data, IEEE Journal of Selected Topics in Signal Processing, 4 (2010), 288–297. [30] W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $I_1$-minimization with applications to compressed sensing, SIAM J. Imaging Sci., 1 (2008), 143-168.  doi: 10.1137/070703983. [31] X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276.  doi: 10.1137/090746379. [32] Y. Zhu and I.-L. Chern, Convergence of the alternating minimization method for sparse MR image reconstruction, J. Inform. Comput. Sci., 8 (2011), 2067-2075. [33] Y. Zhu and X. Liu, A fast method for L1-L2 modeling for MR image compressive sensing, J. Inverse Ill-Posed Probl., 23 (2015), 211–218. doi: 10.1515/jiip-2013-0046. [34] Y. Zhu, Y. Shi, B. Zhang and X. Yu, Weighted-average alternating minimization method for magnetic resonance image reconstruction based on compressive sensing, Inverse Probl. Imaging, 8 (2014), 925-937.  doi: 10.3934/ipi.2014.8.925.
Object images: (A) circles under black background; (B) circles under gray background; (C) a phantom from Matlab; (D) an MRI chest image
Masks: (A) random sampling with 20 rows and 20 columns; (B) random sampling with 40 rows and 40 columns; (C) radial sampling with 22 lines; (D) radial sampling with 44 lines
Reconstructions by random band sampling. From left to right: exact images, images by back projections, images by DM, images by RecPF, images by C-SMRM, images by H-SMRM
Reconstructions by radial sampling. From left to right: exact images, images by back projections, images by DM, images by RecPF, images by C-SMRM, images by H-SMRM
Errors by random band sampling: $\|\mathbf{f}^{(k+1)}-\mathbf{f}^{(k)}\|_{l^2}$ (top line); $\|\mathbf{f}^{(k+1)}-\mathbf{f}^*\|_{l^2}$ (bottom line). The first column is for C-SMRM method, while the second column is for H-SMRM method
Error distributions by radial sampling. $\|\mathbf{f}^{(k+1)}-\mathbf{f}^{(k)}\|_{l^2}$ (top line), $\|\mathbf{f}^{(k+1)}-\mathbf{f}^*\|_{l^2}$ (bottom line). The first column is for C-SMRM, the second column is for H-SMRM
Computational costs for random band sampling
 image scheme ISNR(dB) ReErr(%) CPU time(s) IterNum circles direct 3.0437 17.0221 1.5734 40 RecPF 14.6056 1.0492 0.2680 27 C-SMRM 14.2219 1.1462 1.9248 100 H-SMRM 14.7205 1.0265 1.3517 100 grayscale direct 1.8183 5.7418 9.6594 40 RecPF 11.9131 0.5245 1.0943 21 C-SMRM 11.5221 0.5627 1.9255 100 H-SMRM 13.5264 0.2621 39.1639 100 phantom direct 1.7383 8.7405 2.1778 40 RecPF 11.3009 1.9514 0.3689 38 C-SMRM 11.4490 1.8190 1.2895 100 H-SMRM 14.4623 0.8047 50.9566 100 chest direct 2.6484 3.8296 2.1683 40 RecPF 11.4839 1.0310 0.2189 21 C-SMRM 11.5464 0.9734 1.7518 100 H-SMRM 17.4996 0.0164 45.6364 100
 image scheme ISNR(dB) ReErr(%) CPU time(s) IterNum circles direct 3.0437 17.0221 1.5734 40 RecPF 14.6056 1.0492 0.2680 27 C-SMRM 14.2219 1.1462 1.9248 100 H-SMRM 14.7205 1.0265 1.3517 100 grayscale direct 1.8183 5.7418 9.6594 40 RecPF 11.9131 0.5245 1.0943 21 C-SMRM 11.5221 0.5627 1.9255 100 H-SMRM 13.5264 0.2621 39.1639 100 phantom direct 1.7383 8.7405 2.1778 40 RecPF 11.3009 1.9514 0.3689 38 C-SMRM 11.4490 1.8190 1.2895 100 H-SMRM 14.4623 0.8047 50.9566 100 chest direct 2.6484 3.8296 2.1683 40 RecPF 11.4839 1.0310 0.2189 21 C-SMRM 11.5464 0.9734 1.7518 100 H-SMRM 17.4996 0.0164 45.6364 100
 image scheme ISNR(dB) ReErr(%) CPU time(s) IterNum phantom direct 3.8014 11.1883 2.4127 40 RecPF 12.3966 2.7976 0.4137 36 C-SMRM 12.0596 3.0233 1.3909 100 H-SMRM 13.0561 2.2250 45.5967 100 chest direct 4.0305 4.3061 2.2478 40 RecPF 2.6372 4.5132 0.2848 22 C-SMRM 11.4866 0.6724 1.3651 100 H-SMRM 12.5369 3.5845 37.5576 100
 image scheme ISNR(dB) ReErr(%) CPU time(s) IterNum phantom direct 3.8014 11.1883 2.4127 40 RecPF 12.3966 2.7976 0.4137 36 C-SMRM 12.0596 3.0233 1.3909 100 H-SMRM 13.0561 2.2250 45.5967 100 chest direct 4.0305 4.3061 2.2478 40 RecPF 2.6372 4.5132 0.2848 22 C-SMRM 11.4866 0.6724 1.3651 100 H-SMRM 12.5369 3.5845 37.5576 100
 [1] Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 [2] Ruiqiang He, Xiangchu Feng, Xiaolong Zhu, Hua Huang, Bingzhe Wei. RWRM: Residual Wasserstein regularization model for image restoration. Inverse Problems and Imaging, 2021, 15 (6) : 1307-1332. doi: 10.3934/ipi.2020069 [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] Bartomeu Coll, Joan Duran, Catalina Sbert. Half-linear regularization for nonconvex image restoration models. Inverse Problems and Imaging, 2015, 9 (2) : 337-370. doi: 10.3934/ipi.2015.9.337 [5] Nicolas Lermé, François Malgouyres, Dominique Hamoir, Emmanuelle Thouin. Bayesian image restoration for mosaic active imaging. Inverse Problems and Imaging, 2014, 8 (3) : 733-760. doi: 10.3934/ipi.2014.8.733 [6] Amir Averbuch, Pekka Neittaanmäki, Valery Zheludev. Periodic spline-based frames for image restoration. Inverse Problems and Imaging, 2015, 9 (3) : 661-707. doi: 10.3934/ipi.2015.9.661 [7] Ying Zhang, Xuhua Ren, Bryan Alexander Clifford, Qian Wang, Xiaoqun Zhang. Image fusion network for dual-modal restoration. Inverse Problems and Imaging, 2021, 15 (6) : 1409-1419. doi: 10.3934/ipi.2021067 [8] Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems and Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523 [9] 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 [10] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 [11] Zhong-Zhi Bai. On convergence of the inner-outer iteration method for computing PageRank. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 855-862. doi: 10.3934/naco.2012.2.855 [12] Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial and Management Optimization, 2021, 17 (2) : 805-825. doi: 10.3934/jimo.2019135 [13] Dan Li, Li-Ping Pang, Fang-Fang Guo, Zun-Quan Xia. An alternating linearization method with inexact data for bilevel nonsmooth convex optimization. Journal of Industrial and Management Optimization, 2014, 10 (3) : 859-869. doi: 10.3934/jimo.2014.10.859 [14] Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial and Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067 [15] Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems and Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149 [16] Stefan Kindermann, Andreas Neubauer. On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Problems and Imaging, 2008, 2 (2) : 291-299. doi: 10.3934/ipi.2008.2.291 [17] Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007 [18] Petteri Harjulehto, Peter Hästö, Juha Tiirola. Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models. Inverse Problems and Imaging, 2015, 9 (3) : 835-851. doi: 10.3934/ipi.2015.9.835 [19] C. M. Elliott, B. Gawron, S. Maier-Paape, E. S. Van Vleck. Discrete dynamics for convex and non-convex smoothing functionals in PDE based image restoration. Communications on Pure and Applied Analysis, 2006, 5 (1) : 181-200. doi: 10.3934/cpaa.2006.5.181 [20] Jia Li, Zuowei Shen, Rujie Yin, Xiaoqun Zhang. A reweighted $l^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise. Inverse Problems and Imaging, 2015, 9 (3) : 875-894. doi: 10.3934/ipi.2015.9.875

2020 Impact Factor: 1.639