After introducing us to the very fast GPSR (new version 5.0), Mário Figueiredo is at it again. José Bioucas-Dias and Mário Figueiredo introduces us to TwIST (Two-step Iterative Shrinkage/Thresholding Algorithm for Linear Inverse Problems) that seems to be both speedy and capable of handling very large unknown vectors found when dealing with images. The main page has examples that include compressed sensing reconstruction examples.
References that explain the algorithms are A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration and Two-step algorithms for linear inverse problems with non-quadratic regularization" both by José Bioucas-Dias, Mário Figueiredo. MATLAB code is available here.
Source: Rice Compressed Sensing page.
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