Bregman iterative regularization (1967) was introduced by
Osher, Burger, Goldfarb, Xu and Yin as a device for improving 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 and Yin requires two lines
of MATLAB code and is remarkably efficient. This means we rapidly and
easily solve the problem:
minimize over u in R^n {myu||u||_1 +1/2 (||Au-f||_2)2}
for a given k by n matrix A, with k much less than n and f in R^k
By some beautiful results of Candes, Tao, Donoho and collaborators, this L1 minimization gives the sparsest solution u, under reasonable assumptions.
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