# Stanley Osher

Stanley Osher, ca. 2008

## Bregman Iterative Algorithms for L1 Minimization with Applications to Compressed Sensing

### The effectiveness of Bregman iteration as applied to compressed sensing and image restoration

#### Stanley Osher, University of California at Los Angeles

Bregman iterative regularization (1967) was introduced by Osher, Burger, Goldfarb, Xu, and Yin as a device for improving total variation (TV)-based image restoration (2004) and was used by Xu and Osher in (2006) to analyze and improve wavelet shrinkage. In recent work by Yin, Osher, Goldfarb and Darbon, we devised simple and extremely efficient methods for solving the basis pursuit problem which is used in compressed sensing. A linearized version, done by Osher, Dong, Mao and Yin, requires two lines of MATLAB code and is remarkably efficient. This means we rapidly and easily solve the problem: for a given $LaTeX: k\!\times\!n$ matrix $LaTeX: \,A\,$ with $LaTeX: k\!\ll\!n$ and $LaTeX: f\!\in\!R^k$

$LaTeX: \mbox{minimize}_{u\in R^n}~\mu||u||_1+{\textstyle\frac{1}{2}}||Au-f||_2^2$

By some beautiful results of Candes, Tao, Donoho, and collaborators, this L1 minimization gives the sparsest solution $LaTeX: \,u\,$ under reasonable assumptions.