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An Efficient Proximity Point Algorithm for Total-Variation-Based Image Restoration

Published online by Cambridge University Press:  03 June 2015

Wei Zhu ,
Shi Shu  and
Lizhi Cheng
Wei Zhu*
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China
Shi Shu*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, Hunan, China
Lizhi Cheng*
Affiliation:
Department of Mathematics and Computational Science, College of Science, National University of Defense Technology, Changsha 410073, Hunan, China
*
Email: [email protected]
Corresponding author. Email: [email protected]
Email: [email protected]
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Abstract

In this paper, we propose a fast proximity point algorithm and apply it to total variation (TV) based image restoration. The novel method is derived from the idea of establishing a general proximity point operator framework based on which new first-order schemes for total variation (TV) based image restoration have been proposed. Many current algorithms for TV-based image restoration, such as Chambolle’s projection algorithm, the split Bregman algorithm, the Bermúdez-Moreno algorithm, the Jia-Zhao denoising algorithm, and the fixed point algorithm, can be viewed as special cases of the new first-order schemes. Moreover, the convergence of the new algorithm has been analyzed at length. Finally, we make comparisons with the split Bregman algorithm which is one of the best algorithms for solving TV-based image restoration at present. Numerical experiments illustrate the efficiency of the proposed algorithms.

Keywords

68U1065F2265K10Proximity point operatorimage restorationtotal variationfirst-order schemes
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 2 , April 2014 , pp. 145 - 164
Copyright
Copyright © Global-Science Press 2014

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References

[1]Rudin, L., Osher, S., and Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259–268.Google Scholar
[2]Aubert, G. and Kornprobst, P., Mathematical problems in image processing, Volume 147 of Applied Mathematical Sciences, Springer-Verlag, 2002.CrossRefGoogle Scholar
[3]Chan, T. AND Shen, J., Image processing and analysis-Variational, PDE, Wavelet, and Stochastic Methods, SIAM Publisher, 2005.Google Scholar
[4]Chambolle, A., An algorithm for total variation minimization and applications, J. Math. Imag. Vision, 20 (2004), pp. 89–97.Google Scholar
[5]Chambolle, A., Total variation minimization and a class of binary MRF models, In EMMCVPR 05, Volume 3757 of Lecture Notes in Computer Sciences, pp. 136–152,2005.Google Scholar
[6]Goldstein, T. and Osher, S., The split Bregman method for regularization problems, SIAM J. Imag. Sci., 2 (2009), pp. 323–343.CrossRefGoogle Scholar
[7]Zhang, X., Burger, M., AND Osher, S., A unified primal-dual algorithm framework based on Bregman iteration, J. Sci. Comput., 46(1) (2011), pp. 20–46.CrossRefGoogle Scholar
[8]Nesterov, Y., Smooth minimization of non-smooth functions, Math. Program. A, 103(1) (2005), pp.127–152.CrossRefGoogle Scholar
[9]Weiss, P., Aubert, G., and Blanc-Feraud, L., Efficient schemes for total variation minimization under constraints in image processing, SIAM J. Sci. Comput., 31(3) (2010), pp. 2047–2080.Google Scholar
[10]BermúDez, A. AND Moreno, C., Duality methods for solving variational inequalities, Comput. Math. Appl., 7 (1981), pp. 43–58.CrossRefGoogle Scholar
[11]Micchelli, C. A, Shen, L., and Xu, Y., Proximity algorithms for image models: denoising, Inverse Problem, 27 (2011), pp. 045009.Google Scholar
[12]Cai, J. F., Osher, S. and Shen, Z., Split Bregman methods and frame based image restoration, Multiscale Model. Simul., 8(2) (2009), pp. 337–369.Google Scholar
[13]Goldstein, T., Bresson, X., and Osher, S., Geometric applications of the split Bregman method: segmentation and surface reconstruction, J. Sci. Comput., 45(1-3) (2010), pp. 272–293.Google Scholar
[14]Gu, M., Lim, L. H., AND Wu, C. J., PARNES: a rapidly convergent algorithm for accurate recovery of sparse and approximately sparse signal, arXiv: 0911.0492,2009.Google Scholar
[15]Becker, S., Bobin, J., and Candes, E. J., Nesta: a fast and accurate first-order method for sparse recovery, arXiv: 0904.3367, 2009.Google Scholar
[16]Beck, A. and Teboulle, M., A fast iterative shrinkage-thresholding algorithm with application to wavelet-based image deblurring acoustics, speech, and signal processing, IEEE Int. Conference on, 10 (2009), pp. 693–696.Google Scholar
[17]Aujol, J. F., Some first-order algorithms for total variation based image restoration, J. Math. Imag. Vision, 34(3) (2010), pp. 307–327.Google Scholar
[18]Jia, R. Q. AND Zhao, H., A fast algorithm for the total variation model of image denoising, Adv. Comput. Math., 33 (2010), pp. 231–241.CrossRefGoogle Scholar
[19]Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mapping, Bulletin American Mathematical Society, 73 (1967), pp. 591–597.CrossRefGoogle Scholar
[20]Afonso, M. V., Bioucas-Dias, J. M., AND Figueiredo, M., Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process., 19(9) (2010), pp. 2345–2356.Google Scholar
[21]He, L., Chang, T. C. and Osher, S. et al, MR image reconstruction by using the iterative refinement method and nonlinear inverse scale space methods, UCLA CAM Report(06-35), 2006.Google Scholar
[22]Yang, J., Zhang, Y., and Yin, W., A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data, IEEE Journal of Selected Topics in Signal Processing, 4 (2009), pp. 288–297.Google Scholar
[23]Zhang, H., Cheng, L. Z., and Li, J. P., Reweighted minimization model for MR image reconstruction with split bregman method, Sci. China Ser., 54(1) (2011), pp. 1–10.CrossRefGoogle Scholar
[24]Becker, S., Candes, E. J., and Grant, M., Templates for convex cone problems with applications to sparse signal recovery, arXiv: 1009.2065, 2010.Google Scholar
[25]Moreau, J., Proximiteet dualitedans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), pp. 273–299.CrossRefGoogle Scholar
[26]Combettes, P. and Wajs, V., Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), pp. 1168–1200.Google Scholar
[27]Combettes, P. and Pesquet, J., A proximity decomposition method for solving convex variational inverse problem, Inverse Problem, 24(6) (2008), pp. 65014–65040.CrossRefGoogle Scholar
[28]Combettes, P. and Pesquet, J., A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), pp. 564–574.CrossRefGoogle Scholar
[29]Bertsekas, D., Nedić, A., and Ozdaglar, A. E., Convex Analysis and Optimization, Athena Scientific and Tsinghua University Press, pp: 227, 2006.Google Scholar
[30]Wang, Y., Yang, J., Yin, W., and Zhang, Y., A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imag. Sci., 1 (2008), pp. 248–272.CrossRefGoogle Scholar
[31]Cheng, D., Zhang, H., and Cheng, L., A fast fixed point algorithm for total variation deblurring and segmentation, J. Math. Imag. Vision, 43(3) (2012), pp. 167–179.CrossRefGoogle Scholar
[32]Figueiredo, M. and Nowak, R., An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), pp. 906–916.Google Scholar