SPLIT BREGMAN METHODS AND FRAME BASED IMAGERESTORATION

摘要

Split Bregman methods introduced in [T. Goldstein and S. Osher, SIAM J. ImagingSci., 2 (2009), pp. 323-343] have been demonstrated to be efficient tools for solving total variationnorm minimization problems, which arise from partial differential equation based image restorationsuch as image denoising and magnetic resonance imaging reconstruction from sparse samples. Inthis paper, we prove the convergence of the split Bregman iterations, where the number of inneriterations is fixed to be one. Furthermore, we show that these split Bregman iterations can be usedto solve minimization problems arising from the analysis based approach for image restoration inthe literature. We apply these split Bregman iterations to the analysis based image restoration ap-proach 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 algo-rithms that cover several topics in image restorations, such as image denoising, deblurring, inpainting,and cartoon-texture image decomposition. Several numerical simulation results are provided.