# Osher

(Difference between revisions)
Jump to: navigation, search
 Revision as of 16:57, 8 August 2008 (edit) (→Stanley Osher, University of California, Los Angeles)← Previous diff Revision as of 14:53, 13 August 2008 (edit) (undo)Next diff → Line 1: Line 1: = Stanley Osher = = Stanley Osher = [[Image:Osher2.jpg|thumb|right|450px|Stanley Osher, ca. 2008]] [[Image:Osher2.jpg|thumb|right|450px|Stanley Osher, ca. 2008]] + + [http://www.math.ucla.edu/~sjo Stanley Osher] has made fundamental contributions to applied mathematics, computational science, and scientific computing, and has cofounded three companies based on his research. He has applied level set methods for partial differential equations to the field of image processing, to video image enhancement, and movie animation. He has been featured in international media such as Science News, Die Zeit, and Los Angeles Times. + + Stanley Osher received the NASA Public Service Group Achievement Award, the Japan Society of Mechanical Engineers Computational Mechanics Award, and the SIAM Pioneer Prize. He was an invited speaker at the International Congress of Mathematicians 1994. + + Stanley Osher is currently Director of Special Projects at the Institute for Pure and Applied Mathematics (IPAM) at the University of California at Los Angeles, and Director of Applied Mathematics.

# Stanley Osher

Stanley Osher, ca. 2008

Stanley Osher has made fundamental contributions to applied mathematics, computational science, and scientific computing, and has cofounded three companies based on his research. He has applied level set methods for partial differential equations to the field of image processing, to video image enhancement, and movie animation. He has been featured in international media such as Science News, Die Zeit, and Los Angeles Times.

Stanley Osher received the NASA Public Service Group Achievement Award, the Japan Society of Mechanical Engineers Computational Mechanics Award, and the SIAM Pioneer Prize. He was an invited speaker at the International Congress of Mathematicians 1994.

Stanley Osher is currently Director of Special Projects at the Institute for Pure and Applied Mathematics (IPAM) at the University of California at Los Angeles, and Director of Applied Mathematics.

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

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

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

Bregman iterative regularization (1967) was introduced by Osher, Burger, Goldfarb, Xu, & Yin as a device for improving total variation (TV)-based image restoration (2004) and was used by Xu & Osher in (2006) to analyze and improve wavelet shrinkage. In recent work by Yin, Osher, Goldfarb, & 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, & 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 $LaTeX: f\!\in\!\mathbb{R}^k$

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

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