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 & 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.  Google Scholar

[2]

F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), 137-141.  doi: 10.1007/BF01386217.  Google Scholar

[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.  Google Scholar

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S. BoydN. ParikhE. ChuB. 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.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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[18]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

W. Sun and Y.-X. Yuan, Optimization Theory and Methods, Optimization and Its Applications, 1, 1$^st$ edition, Springer, New York, 2006. Google Scholar

[24]

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[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.  Google Scholar

[26]

J. YangW. YinY. 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.  Google Scholar

[27]

J. Yang, Y. Zhang and W. Yin, A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data, 2008. Google Scholar

[28]

J. YangY. 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.  Google Scholar

[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. Google Scholar

[30]

W. YinS. OsherD. 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.  Google Scholar

[31]

X. ZhangM. BurgerX. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276.  doi: 10.1137/090746379.  Google Scholar

[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.   Google Scholar

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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.  Google Scholar

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Y. ZhuY. ShiB. 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.  Google Scholar

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.  Google Scholar

[2]

F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), 137-141.  doi: 10.1007/BF01386217.  Google Scholar

[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.  Google Scholar

[4]

S. BoydN. ParikhE. ChuB. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289–1306. doi: 10.1109/TIT.2006.871582.  Google Scholar

[10]

W. J. Fu, Penalized regressions: The bridge versus the lasso, J. Comput. Graph. Statist., 7 (1998), 397-416.  doi: 10.2307/1390712.  Google Scholar

[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.  Google Scholar

[12]

P. J. Huber, Robust estimation of a location parameter, Ann. Math. Statist., 35 (1964), 73–101. doi: 10.1214/aoms/1177703732.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. OsherM. BurgerD. GoldfarbJ. 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.  Google Scholar

[18]

S. PerkinsK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

W. Sun and Y.-X. Yuan, Optimization Theory and Methods, Optimization and Its Applications, 1, 1$^st$ edition, Springer, New York, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. YangW. YinY. 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.  Google Scholar

[27]

J. Yang, Y. Zhang and W. Yin, A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data, 2008. Google Scholar

[28]

J. YangY. 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.  Google Scholar

[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. Google Scholar

[30]

W. YinS. OsherD. 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.  Google Scholar

[31]

X. ZhangM. BurgerX. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276.  doi: 10.1137/090746379.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[34]

Y. ZhuY. ShiB. 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.  Google Scholar

Figure 1.  Object images: (A) circles under black background; (B) circles under gray background; (C) a phantom from Matlab; (D) an MRI chest image
Figure 2.  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
Figure 3.  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
Figure 5.  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
Figure 4.  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
Figure 6.  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
Table 1.  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
Table 2.  Computational costs for radial sampling
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
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