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 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

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 = {Yu-Wen Fang, Xiao-Mei Huo and You-Wei Wen}, 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 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.

}, 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 AU - Yu-Wen Fang, Xiao-Mei Huo & You-Wei Wen JO - East Asian Journal on Applied Mathematics VL - 1 SP - 21 EP - 34 PY - 2018 DA - 2018/02 SN - 4 DO - http://dor.org/10.4208/eajam.150413.260913a UR - https://global-sci.org/intro/eajam/10818.html KW - Total Variation (TV) KW - wavelet KW - regularization KW - MR image AB -

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.

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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