Image restoration from noisy incomplete frequency data by alternative iteration scheme

Xiaoman Liu; Jijun Liu

doi:10.3934/ipi.2020027

Article Contents

2020, Volume 14, Issue 4: 583-606. Doi: 10.3934/ipi.2020027

This issue Previous Article Next Article Joint reconstruction in low dose multi-energy CT

Image restoration from noisy incomplete frequency data by alternative iteration scheme

Xiaoman Liu^1,2, and
Jijun Liu^3, ,

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
^* Corresponding author: Prof. Dr. Jijun Liu

Received: February 28, 2019

Revised: January 31, 2020

Published: May 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 65M32, 68U10; Secondary: 65K10, 65T50.

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Figure 1. Object images: (A) circles under black background; (B) circles under gray background; (C) a phantom from Matlab; (D) an MRI chest image

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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

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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

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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

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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

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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

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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

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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

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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.