# American Institute of Mathematical Sciences

August  2013, 7(3): 717-736. doi: 10.3934/ipi.2013.7.717

## Nonstationary iterated thresholding algorithms for image deblurring

 1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, China 2 Dipartimento di Scienza e Alta Tecnologia, Università dell'Insubria, Como 22100, Italy 3 Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, China

Received  April 2013 Revised  June 2013 Published  September 2013

We propose iterative thresholding algorithms based on the iterated Tikhonov method for image deblurring problems. Our method is similar in idea to the modified linearized Bregman algorithm (MLBA) so is easy to implement. In order to obtain good restorations, MLBA requires an accurate estimate of the regularization parameter $\alpha$ which is hard to get in real applications. Based on previous results in iterated Tikhonov method, we design two nonstationary iterative thresholding algorithms which give near optimal results without estimating $\alpha$. One of them is based on the iterative soft thresholding algorithm and the other is based on MLBA. We show that the nonstationary methods, if converge, will converge to the same minimizers of the stationary variants. Numerical results show that the accuracy and convergence of our nonstationary methods are very robust with respect to the changes in the parameters and the restoration results are comparable to those of MLBA with optimal $\alpha$.
Citation: Jie Huang, Marco Donatelli, Raymond H. Chan. Nonstationary iterated thresholding algorithms for image deblurring. Inverse Problems & Imaging, 2013, 7 (3) : 717-736. doi: 10.3934/ipi.2013.7.717
##### References:
 [1] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM J. Imaging Sci., 2 (2009), 183. doi: 10.1137/080716542. Google Scholar [2] M. Bertalmío, G. Sapiro, V. Caselles and C. Ballester, Image inpainting,, SIGGRAPH, 34 (2000), 417. doi: 10.1145/344779.344972. Google Scholar [3] M. Bertero and P. Boccacci, "Introduction to Inverse Problems in Imaging,", Institute of Physics Publishing, (1998). doi: 10.1887/0750304359. Google Scholar [4] N. Bose and K. Boo, High-resolution image reconstruction with multisensors,, International Journal of Imaging Systems and Technology, 9 (1998), 294. Google Scholar [5] L. M. Bregman, A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming,, (Russian) Z . Vycisl. Mat. i Mat. Fiz., 7 (1967), 620. Google Scholar [6] M. Brill and E. Schock, Iterative solution of ill-posed problems: A survey,, in, (1986), 13. Google Scholar [7] J. F. Cai, R. H. Chan and Z. Shen, A framelet-based image inpainting algorithm,, Appl. Comput. Harmon. Anal., 24 (2008), 131. doi: 10.1016/j.acha.2007.10.002. Google Scholar [8] J. F. Cai, S. Osher and Z. Shen, Linearized BRegman iterations for frame-based image deblurring,, SIAM J. Imaging Sci., 2 (2009), 226. doi: 10.1137/080733371. Google Scholar [9] J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration,, Multiscale Model. Simul., 8 (2009), 337. doi: 10.1137/090753504. Google Scholar [10] J. F. Cai, S. Osher and Z. Shen, Convergence of the linearized Bregman iteration for $l_1$-norm minimization,, Math. Comput., 78 (2009), 2127. doi: 10.1090/S0025-5718-09-02242-X. Google Scholar [11] J. F. Cai, S. Osher and Z. Shen, Linearized Bregman iterations for compressed sensing,, Math. Comput., 78 (2009), 1515. doi: 10.1090/S0025-5718-08-02189-3. Google Scholar [12] E. J. Candés and J. Romberg, Practical signal recovery from random projections,, Wavelet Applications in Signal and Image Processing XI Proc. SPIE Conf. 5914 (2004)., 5914 (2004). Google Scholar [13] A. Chai and Z. Shen, Deconvolution: A wavelet frame approach,, Numer. Math., 106 (2007), 529. doi: 10.1007/s00211-007-0075-0. Google Scholar [14] A. Chambolle, R. A. De Vore, N. Y. Lee and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage,, IEEE Trans. Image Process., 7 (1998), 319. doi: 10.1109/83.661182. Google Scholar [15] R. H. Chan, T. F. Chan, L. Shen and Z. Shen, Wavelet algorithms for high-resolution image reconstruction,, SIAM J. Sci. Comput., 24 (2003), 1408. doi: 10.1137/S1064827500383123. Google Scholar [16] R. H. Chan, S. D. Riemenschneider, L. Shen and Z. Shen, Tight frame: An efficient way for high-resolution image reconstruction,, Appl. Comput. Harmon. Anal., 17 (2004), 91. doi: 10.1016/j.acha.2004.02.003. Google Scholar [17] R. H. Chan, Z. Shen and T. Xia, A framelet algorithm for enhancing video stills,, Appl. Comput. Harmon. Anal., 23 (2007), 153. doi: 10.1016/j.acha.2006.10.003. Google Scholar [18] T. Chan and J. H. Shen, "Image Processing and Analysis-Variational, PDE, Wavelet, and Stochastic Methods,", Society for Industrial and Applied Mathematics (SIAM), (2005). doi: 10.1137/1.9780898717877. Google Scholar [19] T. Chan, J. H. Shen and H. M. Zhou, Total variation wavelet inpainting,, J. Math. Imaging Vision, 25 (2006), 107. doi: 10.1007/s10851-006-5257-3. Google Scholar [20] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting,, Multiscale Model. Simul., 4 (2005), 1168. doi: 10.1137/050626090. Google Scholar [21] 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. doi: 10.1002/cpa.20042. Google Scholar [22] I. Daubechies, M. Fornasier and I. Loris, Accelerated projected gradient method for linear inverse problems with sparsity constraints,, J. Fourier Anal. Appl. 14 (2008), 14 (2008), 764. doi: 10.1007/s00041-008-9039-8. Google Scholar [23] I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames,, Appl. Comput. Harmon. Anal., 14 (2003), 1. doi: 10.1016/S1063-5203(02)00511-0. Google Scholar [24] I. Daubechies, G. Teschke and L. Vese, Iteratively solving linear inverse problems under general convex constraints,, Inverse Problems Imaging, 1 (2007), 29. doi: 10.3934/ipi.2007.1.29. Google Scholar [25] M. Donatelli, Fast transforms for high order boundary conditions in deconvolution problems,, BIT, 50 (2010), 559. doi: 10.1007/s10543-010-0266-4. Google Scholar [26] M. Donatelli, On nondecreasing sequences of regularization parameters for nonstationary iterated tikhonov,, Numer. Algor., 60 (2012), 651. doi: 10.1007/s11075-012-9593-7. Google Scholar [27] M. Donatelli and M. Hanke, Fast nonstationary preconditioned iterative methods for ill-posed problems, with application to image deblurring,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/9/095008. Google Scholar [28] M. Elad and A. Feuer, Restoration of a single superresolution image from several blurred, noisy and undersampled measured images,, IEEE Trans. Image Process, 6 (1997), 1646. doi: 10.1109/83.650118. Google Scholar [29] H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", Mathematics and its Applications, (1996). doi: 10.1007/978-94-009-1740-8. Google Scholar [30] M. J. Fadili and J. L. Starck, Sparse representations and Bayesian image inpainting,, Proc. SPARS'05, (2005). Google Scholar [31] A. G. Fakeev, A class of iterative processes for solving degenerate systems of linear algebraic equations,, U. S. S. R. Comput. Math. Math. Phys., 21 (1981), 15. Google Scholar [32] M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration,, IEEE Trans. Image Process., 12 (2003), 906. doi: 10.1109/TIP.2003.814255. Google Scholar [33] E. Hale, W. Yin and Y. Zhang, Fixed-point continuation for $l_1$-minimization: Methodology and convergence,, SIAM J. Optim., 19 (2008), 1107. doi: 10.1137/070698920. Google Scholar [34] M. Hanke and C. W. Groetsh, Nonstationary iterated tikhonov regularization,, J. Optim. Theory Appl., 98 (1998), 37. doi: 10.1023/A:1022680629327. Google Scholar [35] M. Hanke and P. C. Hansen, Regularization methods for large-scale problems,, Surveys Math. Indust., 3 (1993), 253. Google Scholar [36] P. C. Hansen, "Rank-Deficient and Discrete Ill-Posed Problems,", SIAM, (1997). doi: 10.1137/1.9780898719697. Google Scholar [37] J. T. King and D. Chillingworth, Approximation of generalized inverses by iterated regularization,, Numer. Func. Anal. Opt., 1 (1979), 499. doi: 10.1080/01630567908816031. Google Scholar [38] A. V. Kryanev, An iterative method for solving incorrectly posed problems,, U. S. S. R. Comput. Math. Math. Phys., 14 (1974), 25. doi: 10.1016/0041-5553(74)90133-5. Google Scholar [39] L. Landweber, An iteration formula for fredholm integral equations of the first kind,, Am. J. Math., 73 (1951), 615. doi: 10.2307/2372313. Google Scholar [40] I. Loris, M. Bertero, C. De Mol, R. Zanella and L. Zanni, Accelerating gradient projection methods for $l_1$-constrained signal recovery by steplength selection rules,, Appl. Comput. Harmon. Anal., 27 (2009), 247. doi: 10.1016/j.acha.2009.02.003. Google Scholar [41] S. Mallat, "A Wavelet Tour of Signal Processing,", 2nd edition, (1999). Google Scholar [42] V. A. Morozov, On the solution of functional equations by the method of regularization,, Dokl. Akad. Nauk SSSR 167 510-512 (Russian), 167 (1966), 510. Google Scholar [43] F. Natterer, "The Mathematics of Computerized Tomography,", Reprint of the 1986 original. Classics in Applied Mathematics, (1986). doi: 10.1137/1.9780898719284. Google Scholar [44] 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. doi: 10.1137/S1064827598341384. Google Scholar [45] S. Osher, Y. Mao, B. Dong and W. Yin, Fast linearized Bregman iteration for compressed sensing and sparse denoising,, Commun. Math. Sci., 8 (2010), 93. Google Scholar [46] M. Piana and M. Bertero, Projected landweber method and preconditioning,, Inverse Problems, 13 (1997), 441. doi: 10.1088/0266-5611/13/2/016. Google Scholar [47] O. N. Strand, Theory and methods related to the singular-function expansion and landweber's iteration for integral equations of the first kind,, SIAM J. Numer. Anal, 11 (1974), 798. doi: 10.1137/0711066. Google Scholar [48] A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method,, Soviet Math. Dokl., 4 (1963), 1035. Google Scholar [49] C. R. Vogel, "Computational Methods for Inverse Problems,", With a foreword by H. T. Banks. Frontiers in Applied Mathematics, (2002). doi: 10.1137/1.9780898717570. Google Scholar [50] W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $l_1$-minimization with applications to compressed sensing,, SIAM J. Imaging Sci., 1 (2008), 143. doi: 10.1137/070703983. Google Scholar

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##### References:
 [1] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM J. Imaging Sci., 2 (2009), 183. doi: 10.1137/080716542. Google Scholar [2] M. Bertalmío, G. Sapiro, V. Caselles and C. Ballester, Image inpainting,, SIGGRAPH, 34 (2000), 417. doi: 10.1145/344779.344972. Google Scholar [3] M. Bertero and P. Boccacci, "Introduction to Inverse Problems in Imaging,", Institute of Physics Publishing, (1998). doi: 10.1887/0750304359. Google Scholar [4] N. Bose and K. Boo, High-resolution image reconstruction with multisensors,, International Journal of Imaging Systems and Technology, 9 (1998), 294. Google Scholar [5] L. M. Bregman, A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming,, (Russian) Z . Vycisl. Mat. i Mat. Fiz., 7 (1967), 620. Google Scholar [6] M. Brill and E. Schock, Iterative solution of ill-posed problems: A survey,, in, (1986), 13. Google Scholar [7] J. F. Cai, R. H. Chan and Z. Shen, A framelet-based image inpainting algorithm,, Appl. Comput. Harmon. Anal., 24 (2008), 131. doi: 10.1016/j.acha.2007.10.002. Google Scholar [8] J. F. Cai, S. Osher and Z. Shen, Linearized BRegman iterations for frame-based image deblurring,, SIAM J. Imaging Sci., 2 (2009), 226. doi: 10.1137/080733371. Google Scholar [9] J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration,, Multiscale Model. Simul., 8 (2009), 337. doi: 10.1137/090753504. Google Scholar [10] J. F. Cai, S. Osher and Z. Shen, Convergence of the linearized Bregman iteration for $l_1$-norm minimization,, Math. Comput., 78 (2009), 2127. doi: 10.1090/S0025-5718-09-02242-X. Google Scholar [11] J. F. Cai, S. Osher and Z. Shen, Linearized Bregman iterations for compressed sensing,, Math. Comput., 78 (2009), 1515. doi: 10.1090/S0025-5718-08-02189-3. Google Scholar [12] E. J. Candés and J. Romberg, Practical signal recovery from random projections,, Wavelet Applications in Signal and Image Processing XI Proc. SPIE Conf. 5914 (2004)., 5914 (2004). Google Scholar [13] A. Chai and Z. Shen, Deconvolution: A wavelet frame approach,, Numer. Math., 106 (2007), 529. doi: 10.1007/s00211-007-0075-0. Google Scholar [14] A. Chambolle, R. A. De Vore, N. Y. Lee and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage,, IEEE Trans. Image Process., 7 (1998), 319. doi: 10.1109/83.661182. Google Scholar [15] R. H. Chan, T. F. Chan, L. Shen and Z. Shen, Wavelet algorithms for high-resolution image reconstruction,, SIAM J. Sci. Comput., 24 (2003), 1408. doi: 10.1137/S1064827500383123. Google Scholar [16] R. H. Chan, S. D. Riemenschneider, L. Shen and Z. Shen, Tight frame: An efficient way for high-resolution image reconstruction,, Appl. Comput. Harmon. Anal., 17 (2004), 91. doi: 10.1016/j.acha.2004.02.003. Google Scholar [17] R. H. Chan, Z. Shen and T. Xia, A framelet algorithm for enhancing video stills,, Appl. Comput. Harmon. Anal., 23 (2007), 153. doi: 10.1016/j.acha.2006.10.003. Google Scholar [18] T. Chan and J. H. Shen, "Image Processing and Analysis-Variational, PDE, Wavelet, and Stochastic Methods,", Society for Industrial and Applied Mathematics (SIAM), (2005). doi: 10.1137/1.9780898717877. Google Scholar [19] T. Chan, J. H. Shen and H. M. Zhou, Total variation wavelet inpainting,, J. Math. Imaging Vision, 25 (2006), 107. doi: 10.1007/s10851-006-5257-3. Google Scholar [20] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting,, Multiscale Model. Simul., 4 (2005), 1168. doi: 10.1137/050626090. Google Scholar [21] 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. doi: 10.1002/cpa.20042. Google Scholar [22] I. Daubechies, M. Fornasier and I. Loris, Accelerated projected gradient method for linear inverse problems with sparsity constraints,, J. Fourier Anal. Appl. 14 (2008), 14 (2008), 764. doi: 10.1007/s00041-008-9039-8. Google Scholar [23] I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames,, Appl. Comput. Harmon. Anal., 14 (2003), 1. doi: 10.1016/S1063-5203(02)00511-0. Google Scholar [24] I. Daubechies, G. Teschke and L. Vese, Iteratively solving linear inverse problems under general convex constraints,, Inverse Problems Imaging, 1 (2007), 29. doi: 10.3934/ipi.2007.1.29. Google Scholar [25] M. Donatelli, Fast transforms for high order boundary conditions in deconvolution problems,, BIT, 50 (2010), 559. doi: 10.1007/s10543-010-0266-4. Google Scholar [26] M. Donatelli, On nondecreasing sequences of regularization parameters for nonstationary iterated tikhonov,, Numer. Algor., 60 (2012), 651. doi: 10.1007/s11075-012-9593-7. Google Scholar [27] M. Donatelli and M. Hanke, Fast nonstationary preconditioned iterative methods for ill-posed problems, with application to image deblurring,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/9/095008. Google Scholar [28] M. Elad and A. Feuer, Restoration of a single superresolution image from several blurred, noisy and undersampled measured images,, IEEE Trans. Image Process, 6 (1997), 1646. doi: 10.1109/83.650118. Google Scholar [29] H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", Mathematics and its Applications, (1996). doi: 10.1007/978-94-009-1740-8. Google Scholar [30] M. J. Fadili and J. L. Starck, Sparse representations and Bayesian image inpainting,, Proc. SPARS'05, (2005). Google Scholar [31] A. G. Fakeev, A class of iterative processes for solving degenerate systems of linear algebraic equations,, U. S. S. R. Comput. Math. Math. Phys., 21 (1981), 15. Google Scholar [32] M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration,, IEEE Trans. Image Process., 12 (2003), 906. doi: 10.1109/TIP.2003.814255. Google Scholar [33] E. Hale, W. Yin and Y. Zhang, Fixed-point continuation for $l_1$-minimization: Methodology and convergence,, SIAM J. Optim., 19 (2008), 1107. doi: 10.1137/070698920. Google Scholar [34] M. Hanke and C. W. Groetsh, Nonstationary iterated tikhonov regularization,, J. Optim. Theory Appl., 98 (1998), 37. doi: 10.1023/A:1022680629327. Google Scholar [35] M. Hanke and P. C. Hansen, Regularization methods for large-scale problems,, Surveys Math. Indust., 3 (1993), 253. Google Scholar [36] P. C. Hansen, "Rank-Deficient and Discrete Ill-Posed Problems,", SIAM, (1997). doi: 10.1137/1.9780898719697. Google Scholar [37] J. T. King and D. Chillingworth, Approximation of generalized inverses by iterated regularization,, Numer. Func. Anal. Opt., 1 (1979), 499. doi: 10.1080/01630567908816031. Google Scholar [38] A. V. Kryanev, An iterative method for solving incorrectly posed problems,, U. S. S. R. Comput. Math. Math. Phys., 14 (1974), 25. doi: 10.1016/0041-5553(74)90133-5. Google Scholar [39] L. Landweber, An iteration formula for fredholm integral equations of the first kind,, Am. J. Math., 73 (1951), 615. doi: 10.2307/2372313. Google Scholar [40] I. Loris, M. Bertero, C. De Mol, R. Zanella and L. Zanni, Accelerating gradient projection methods for $l_1$-constrained signal recovery by steplength selection rules,, Appl. Comput. Harmon. Anal., 27 (2009), 247. doi: 10.1016/j.acha.2009.02.003. Google Scholar [41] S. Mallat, "A Wavelet Tour of Signal Processing,", 2nd edition, (1999). Google Scholar [42] V. A. Morozov, On the solution of functional equations by the method of regularization,, Dokl. Akad. Nauk SSSR 167 510-512 (Russian), 167 (1966), 510. Google Scholar [43] F. Natterer, "The Mathematics of Computerized Tomography,", Reprint of the 1986 original. Classics in Applied Mathematics, (1986). doi: 10.1137/1.9780898719284. Google Scholar [44] 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. doi: 10.1137/S1064827598341384. Google Scholar [45] S. Osher, Y. Mao, B. Dong and W. Yin, Fast linearized Bregman iteration for compressed sensing and sparse denoising,, Commun. Math. Sci., 8 (2010), 93. Google Scholar [46] M. Piana and M. Bertero, Projected landweber method and preconditioning,, Inverse Problems, 13 (1997), 441. doi: 10.1088/0266-5611/13/2/016. Google Scholar [47] O. N. Strand, Theory and methods related to the singular-function expansion and landweber's iteration for integral equations of the first kind,, SIAM J. Numer. Anal, 11 (1974), 798. doi: 10.1137/0711066. Google Scholar [48] A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method,, Soviet Math. Dokl., 4 (1963), 1035. Google Scholar [49] C. R. Vogel, "Computational Methods for Inverse Problems,", With a foreword by H. T. Banks. Frontiers in Applied Mathematics, (2002). doi: 10.1137/1.9780898717570. Google Scholar [50] W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $l_1$-minimization with applications to compressed sensing,, SIAM J. Imaging Sci., 1 (2008), 143. doi: 10.1137/070703983. Google Scholar
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