Split Bregman Methods and Frame Based Image Restoration

Split Bregman methods introduced in [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2(2009), pp. 323–343] have been demonstrated to be efficient tools for solving total variation norm minimization problems, which arise from partial differential equation based image restoration such as image denoising and magnetic resonance imaging reconstruction from sparse samples. In this paper, we prove the convergence of the split Bregman iterations, where the number of inner iterations is fixed to be one. Furthermore, we show that these split Bregman iterations can be used to solve minimization problems arising from the analysis based approach for image restoration in the literature. We apply these split Bregman iterations to the analysis based image restoration approach whose analysis operator is derived from tight framelets constructed in [A. Ron and Z. Shen, J. Funct. Anal., 148(1997), pp. 408–447]. This gives a set of new frame based image restoration algorithms that cover several topics in image restorations, such as image denoising, deblurring, inpainting, and cartoon-texture image decomposition. Several numerical simulation results are provided.