A Robust and Fast Combination Algorithms of Split Bregman Method for Deblurring and Denoising

Singular Value Thresholding Algorithms for Low-rank Matrix Completion Jianfeng Cai, University of California at Los Angles, USA Low-rank matrix completion refers to recovering a low-rank matrix from a sampling of its entries. It routinely comes up whenever one collects partially filled out surveys, and one would like to infer the many missing entries. Matrix completion is a natural extension of compressed sensing. Candes and his co-authors proved that one can solve the low-rank matrix completion problem exactly by minimizing a nuclear norm (the L1-norm of the vector of singular values) function subject to linear constraints. In this talk, I will present singular value thresholding algorithms for the nuclear norm minimization arising from low-rank matrix completion. A Robust and Fast Combination Algorithms of Split Bregman Method for Deblurring and Denoising Qianshun Chang, Chinese Academy of Sciences, China In this paper, we propose two efficient algorithms for split Bregman method for deblurring and denoising. The split Bregman method is used to convert nonlinear TV model into linear systems. Then, the FFT method is applied to solve the linear system with the blurring operator. Another method is to combine algorithm of algebraic multigrid method and Krylov acceleration method for deblurring and denoising. For the linear system, we have deduced convergence analysis to determine an auxiliary linear term that significantly stabilizes and accelerates the outer iteration of the linear system. The inclusion of the linear stabilizing term plays a crucial role in our combination algorithm. The iterative convergence is proved. The two algorithms are efficient and robust. Our algorithms also prove to work efficiently over a wide range of parameters. We have conducted extensive numerical experiments. The result shows that our algorithms are efficient and robust for μ ranging from O(1) to the pure blurring limit μ = 100 and various strong blur operators such as the out of focus and truncated Gaussian. The results of numerical experiments are given and compared with some published papers. Nonhomogeneous Wavelet Systems and Frequency-based Framelets Bin Han, University of Alberta, Canada Linked with discretization of continuous wavelet transforms, most wavelets (including framelets) studied in the literature are homogeneous wavelet systems generated by square integrable functions. However, in this talk, we show that nonhomogeneous wavelet systems, in particular with wavelet generators being distributions in the frequency domain, play a fundamental role by naturally linking many aspects of wavelet analysis together. For example, we show that every dual framelet filter bank is naturally associated with a pair of frequency-based nonhomogeneous dual framelets in the distribution space, for which we have a complete and simple characterization. Directional framelets in high dimensions will be also mentioned in this talk.