East Asian Journal on Applied Mathematics Vol. 4, No. 1, pp. 21-34 doi: 10.4208/eajam.150413.260913a February 2014 An Algorithm for the Proximity Operator in Hybrid TV-Wavelet Regularization, with Application to MR Image Reconstruction Yu-Wen Fang, Xiao-Mei Huo and You-Wei Wen ∗ Faculty of Science, Kunming University of Science and Technology, Yunnan, China. Received 15 April 2013; Accepted (in revised version) 26 September 2013 Available online 24 February 2014 Abstract. Total variation (TV) and wavelet L 1 norms have often been used as regular- ization terms in image restoration and reconstruction problems. However, TV regular- ization 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 regular- izations, 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 regular- izations 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. AMS subject classifications: 65K10, 68U10 Key words: Total Variation (TV), wavelet, regularization, MR image. 1. Introduction In many image restoration or reconstruction problems, we need to solve a linear inverse problem of the form g = Kf + n , where g is the observed data, K is the system operator, f is the original image with size m × n and n is the random noise. It is well known that restoring an image is a very ill- conditioned process, and to alleviate this a regularization approach is generally used. The approach is to minimise the objective function, which is the weighted sum of the data- fitting term and the term containing some prior information about the original image. In many image processing problems, an image can be modelled as a piecewise smooth function, and simultaneously sparsely represented by a wavelet basis — e.g. Lustig et ∗ Corresponding author. Email address: (Y.-W. Wen) http://www.global-sci.org/eajam 21 c 2014 Global-Science Press