Osher

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(→Stanley Osher, University of California, Los Angeles)
(→Stanley Osher, University of California, Los Angeles)
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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:
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 <math>k\!\times\!n</math> matrix <math>\,A\,</math> with <math>k\!\ll\!n</math> and <math>f\!\in\!R^k</math>
+
for a given <math>k\!\times\!n</math> matrix <math>\,A\,</math> with <math>k\!\ll\!n</math> and <math>f\!\in\!\mathbb{R}^k</math>
-
<math>\mbox{minimize}_{u\in R^n}~\mu||u||_1+{\textstyle\frac{1}{2}}||Au-f||_2^2</math>
+
<math>\mbox{minimize}_{u\in\mathbb{R}^n}~\mu\|u\|_1+{\textstyle\frac{1}{2}}\|Au-f\|_2^2</math>
By some beautiful results of Candes, Tao, and Donoho, this L1 minimization gives the sparsest solution <math>\,u\,</math> under reasonable assumptions.
By some beautiful results of Candes, Tao, and Donoho, this L1 minimization gives the sparsest solution <math>\,u\,</math> under reasonable assumptions.

Revision as of 16:57, 8 August 2008

Contents

Stanley Osher

Stanley Osher, ca. 2008
Stanley Osher, ca. 2008


Bregman Iterative Algorithms for L1 Minimization with Applications to Compressed Sensing

Presented by Stanley Osher with W. Yin, D. Goldfarb, & J. Darbon at the iCME Colloquium (CME 500), Stanford University, December 3, 2007

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.

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