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An Algorithm for the Proximity Operator in Hybrid TV-Wavelet Regularization, with Application to MR Image Reconstruction

Published online by Cambridge University Press:  28 May 2015

Yu-Wen Fang ,
Xiao-Mei Huo  and
You-Wei Wen
Yu-Wen Fang
Affiliation:
Faculty of Science, Kunming University of Science and Technology, Yunnan, China
Xiao-Mei Huo
Affiliation:
Faculty of Science, Kunming University of Science and Technology, Yunnan, China
You-Wei Wen*
Affiliation:
Faculty of Science, Kunming University of Science and Technology, Yunnan, China
*
Corresponding author. Email: [email protected]
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Abstract

Total variation (TV) and wavelet L1 norms have often been used as regularization terms in image restoration and reconstruction problems. However, TV regularization can introduce staircase effects and wavelet regularization some ringing artifacts, but hybrid TV and wavelet regularization can reduce or remove these drawbacks in the reconstructed images. We need to compute the proximal operator of hybrid regularizations, which is generally a sub-problem in the optimization procedure. Both TV and wavelet L1 regularisers are nonlinear and non-smooth, causing numerical difficulty. We propose a dual iterative approach to solve the minimization problem for hybrid regularizations and also prove the convergence of our proposed method, which we then apply to the problem of MR image reconstruction from highly random under-sampled k-space data. Numerical results show the efficiency and effectiveness of this proposed method.

Keywords

65K1068U10Total Variation (TV)waveletregularizationMR image
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 1 , February 2014 , pp. 21 - 34
Copyright
Copyright © Global-Science Press 2014

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References

[1]Avriel, M., Nonlinear Programming: Analysis and Methods, Dover Publications, 2003.Google Scholar
[2]Cao, Z. and Wen, Y., A Splitting Algorithm for MR Image Reconstruction from Sparse Sampling, pp. 329–334, CRC Pressi Llc, 2012.Google Scholar
[3]Chambolle, A., An algorithm for total variation minimization and applications, J. Math. Imaging Vision 20(1-2), 89–97 (2004).Google Scholar
[4]Chan, T., Golub, G., and Mulet, P., A nonlinear primaldual method for total variation-based image restoration, SIAM J. Sci. Comput. 20(6), 1964–1977 (1999).Google Scholar
[5]Chen, G. and Teboulle, M., A proximal-based decomposition method for convex minimization problems, Math. Program. Ser. A 64(1), 81–101 (1994).Google Scholar
[6]Chen, X., Nashed, Z., and Qi, L., Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal. 38(4), 1200–1216 (2000).CrossRefGoogle Scholar
[7]Combettes, P. and Pesquet, J., A proximal decomposition method for solving convex variational inverse problems, Inverse Problems 24(6), 065014 (2008).Google Scholar
[8]Combettes, P.L. and Wajs, V.R., Signal recovery by proximal forward-backward splitting, Multi-scale Model. Simul. 4(4), 1168–1200 (2005).Google Scholar
[9]Donoho, D., Nonlinear solution of linear inverse problems bywavelet-vaguelette decompositions, Appl. Comput. Harmon. Anal. 1, 100–115 (1995).Google Scholar
[10]Dupe, F., Fadili, J., and Starck, J., A proximal iteration for deconvolving poisson noisy images using sparse representations, IEEE Trans. Image Process. 18(2), 310–321 (2009).Google Scholar
[11]Eckstein, J. and Bertsekas, D., On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program. Ser. A 55(3), 293–318 (1992).Google Scholar
[12]He, L., Chang, T.C., Osher, S., Fang, T., and Speier, P., MR image reconstruction by using the iterative refinement method and nonlinear inverse scale space methods, UCLA CAM Report 6(35), 2006.Google Scholar
[13]Lustig, M., Donoho, D., and Pauly, J.M., Sparse MRI: The application of compressed sensing for rapid MR imaging, Magnetic Resonance in Medicine 58(6), 1182–1195 (2007).CrossRefGoogle ScholarPubMed
[14]Rockafellar, R., Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1(2), 97–116 (1976).Google Scholar
[15]Rockafellar, R., Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14(5), 877–898 (1976).Google Scholar
[16]Rudin, L., Osher, S., and Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D 60, 259–268 (1992).CrossRefGoogle Scholar
[17]Tseng, P., Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. Control Optim. 29(1), 119–138 (1991).Google Scholar
[18]Zeng, T., Li, X., and Ng, M., Alternating minimization method for total variation based wavelet shrinkage model Commun. Comput. Phys. 8(5), 976–994 (2010).Google Scholar