Compressed sensing aims at reconstructing sparse signals from significantlyreduced number of samples, and a popular reconstruction approach is$\ell_1$-norm minimization. In this correspondence, a method called orthonormalexpansion is presented to reformulate the basis pursuit problem for noiselesscompressed sensing. Two algorithms are proposed based on convex optimization:one exactly solves the problem and the other is a relaxed version of the firstone. The latter can be considered as a modified iterative soft thresholdingalgorithm and is easy to implement. Numerical simulation shows that, in dealingwith noise-free measurements of sparse signals, the relaxed version isaccurate, fast and competitive to the recent state-of-the-art algorithms. Itspractical application is demonstrated in a more general case where signals ofinterest are approximately sparse and measurements are contaminated with noise.
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