Compressive Sensing (CS) is an emerging methodology in signal and image processing that utilizes sparsity in signal representations to reduce the number of linear measurements needed for signal encoding. In CS, L1-minimization plays a central role in signal decoding. So far, the theory of CS has largely been built on the notion of Restricted Isometry Property or RIP, which unfortunately does not preserve a fundamental row-transformation invariance. We present a non-RIP analysis, including some new extensions, that preserve this invariance. We also discuss the choice of L1-models under practical and noisy environments. Finally, we introduce a 3-line algorithm, derived from the classic alternating direction method approach, that can efficiently solve six different L1 models in CS.
A Matlab package for solving six L1-minimization models is available at: http://www.caam.rice.edu/~optimization/L1/YALL1/
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