Volume 4, Issue 1
An Algorithm for the Proximity Operator in Hybrid TV-Wavelet Regularization, with Application to MR Image Reconstruction

Yu-Wen Fang, Xiao-Mei Huo & You-Wei Wen

East Asian J. Appl. Math., 4 (2014), pp. 21-34.

Published online: 2018-02

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  • Abstract

Total variation (TV) and wavelet $L_1$ 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 $L_1$ 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

Total Variation (TV), wavelet, regularization, MR image.

  • AMS Subject Headings

65K10, 68U10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-4-21, author = {}, title = {An Algorithm for the Proximity Operator in Hybrid TV-Wavelet Regularization, with Application to MR Image Reconstruction}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {4}, number = {1}, pages = {21--34}, abstract = {

Total variation (TV) and wavelet $L_1$ 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 $L_1$ 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.150413.260913a}, url = {http://global-sci.org/intro/article_detail/eajam/10818.html} }
TY - JOUR T1 - An Algorithm for the Proximity Operator in Hybrid TV-Wavelet Regularization, with Application to MR Image Reconstruction JO - East Asian Journal on Applied Mathematics VL - 1 SP - 21 EP - 34 PY - 2018 DA - 2018/02 SN - 4 DO - http://doi.org/10.4208/eajam.150413.260913a UR - https://global-sci.org/intro/article_detail/eajam/10818.html KW - Total Variation (TV), wavelet, regularization, MR image. AB -

Total variation (TV) and wavelet $L_1$ 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 $L_1$ 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.

Yu-Wen Fang, Xiao-Mei Huo & You-Wei Wen. (1970). An Algorithm for the Proximity Operator in Hybrid TV-Wavelet Regularization, with Application to MR Image Reconstruction. East Asian Journal on Applied Mathematics. 4 (1). 21-34. doi:10.4208/eajam.150413.260913a
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