# Osher

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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 | + | 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\mathbb{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> |

## Current revision

## Contents |

# Stanley Osher

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 is a recipient of the 2007 USACM Computational and Applied Sciences Award, he was awarded Docteur Honoris Causa from École Normale Supérieure de Cachan France in 2006, and elected to the National Academy of Sciences in 2005. Stanley Osher has received the SIAM Kleinman Prize in 2005, the SIAM ICIAM Pioneer Prize in 2003, the NASA Public Service Group Achievement Award, and the Japan Society of Mechanical Engineers Computational Mechanics Award. He was an invited speaker at the International Congress of Mathematicians in 1994.

Stanley Osher is currently Director of Special Projects at the Institute for Pure and Applied Mathematics (IPAM) at the University of California 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 matrix with and

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

## Reflections

*We had a week of movie industry people coming into UCLA giving lectures about what they did and, in fact, imaging science was highlighted by the American Mathematical Society one year. We invited people from the local movie industry and they were interested in what we were doing. The water in "Titanic", which won many Academy awards, was very bad. It was old-fashioned stuff. Their people came to talk about that, and so we decided we could do better. In recent movies, the water is much more realistic. The first movie that actually used sophisticated water was "Ants". Now level sets are used in movies like "Shrek" and "Terminator". My former student Ron Fedkiw is doing this movie animation stuff very well. *

*Sometimes it's serendipity. Things just happen. I was lucky. In all my years of doing science, I managed to work with the right people who knew what the problems were. For example, I knew nothing about image processing at all after my PhD. Then this guy Rudin came over to me and asked about some work I had done in fluid dynamics on supersonic flow and shock waves. I asked him what he wanted to know and I got fired up. He was a computer scientist and he realized that shock waves had something to do with imaging. It was a fantastic observation and our collaboration worked out well.*

*I was living in Los Angeles when the city went up in smoke. There was a big riot in Los Angeles after this guy* [Rodney King] *was beaten up by the police. The riot resulted in people being arrested for looting and beating up passers-by. There was a video recording of the bad guys beating up truck driver Denny and it showed a speck on the arm of a man throwing a brick at Denny. It turned out that I had a friend who knew the District Attorney or somebody, and I was then doing video image enhancement with my colleague L. Rudin. We were able to resolve the speck into a rose tattoo and it was a great application of what we were doing. After the Denny case trial* **[**that tattoo led to conviction of the suspect**]** *we had a lot of media publicity. Eventually I sold my share of the company to Rudin. He has a package on video image enhancement which is used by police around the world.*

*You learn things, you read stuff and learn new ideas; and you are fired up. Sometimes you deliver something different from what you have found. You have a vague idea that something interesting is going to come up. You wander around and something happens. Then you get very excited. It's like opening a door and you don't know what good things are behind it. You're not sure where it's going to end and what level of success it's going to be. It's very exciting. Everyday I can't wait to go to work. People often asked me, “What kind of life is this that work is so important?” People go on vacation. My work is vacation.*

*The basic idea is to try to make order out of this life that we live. Everyday you encounter things; and it's a messy world. The goal is to take this mess that we see and somehow "mathematize" it, and then make a prediction.* Stanley Osher