Fast Linearized Bregman Iteration for Compressive Sensing and Sparse   Denoising
  We propose and analyze an extremely fast, efficient, and simple method for solving the problem:min{parallel to u parallel to(1) : Au = f, u is an element of R-n}.This method was first described in [J. Darbon and S. Osher, preprint, 2007], with more details in [W. Yin, S. Osher, D. Goldfarb and J. Darbon, SIAM J. Imaging Sciences, 1(1), 143-168, 2008] and rigorous theory given in [J. Cai, S. Osher and Z. Shen, Math. Comp., to appear, 2008, see also UCLA CAM Report 08-06] and [J. Cai, S. Osher and Z. Shen, UCLA CAM Report, 08-52, 2008]. The motivation was compressive sensing, which now has a vast and exciting history, which seems to have started with Candes, et. al. [E. Candes, J. Romberg and T. Tao, 52(2), 489-509, 2006] and Donoho, [D. L. Donoho, IEEE Trans. Inform. Theory, 52, 1289-1306, 2006]. See [W. Yin, S. Osher, D. Goldfarb and J. Darbon, SIAM J. Imaging Sciences 1(1), 143-168, 2008] and [J. Cai, S. Osher and Z. Shen, Math. Comp., to appear, 2008, see also UCLA CAM Report, 08-06] and [J. Cai, S. Osher and Z. Shen, UCLA CAM Report, 08-52, 2008] for a large set of references. Our method introduces an improvement called "kicking" of the very efficient method of [J. Darbon and S. Osher, preprint, 2007] and [W. Yin, S. Osher, D. Goldfarb and J. Darbon, SIAM J. Imaging Sciences, 1(1), 143-168, 2008] and also applies it to the problem of denoising of undersampled signals. The use of Bregman iteration for denoising of images began in [S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, Multiscale Model. Simul, 4(2), 460-489, 2005] and led to improved results for total variation based methods. Here we apply it to denoise signals, especially essentially sparse signals, which might even be undersampled. 
Stanley Osher, Yu Mao, Bin Dong, Wotao Yin
math.OC cs.IT math.IT