  Compressed sensing aims at reconstructing sparse signals from significantly
reduced number of samples, and a popular reconstruction approach is
$\ell_1$-norm minimization. In this correspondence, a method called orthonormal
expansion is presented to reformulate the basis pursuit problem for noiseless
compressed sensing. Two algorithms are proposed based on convex optimization:
one exactly solves the problem and the other is a relaxed version of the first
one. The latter can be considered as a modified iterative soft thresholding
algorithm and is easy to implement. Numerical simulation shows that, in dealing
with noise-free measurements of sparse signals, the relaxed version is
accurate, fast and competitive to the recent state-of-the-art algorithms. Its
practical application is demonstrated in a more general case where signals of
interest are approximately sparse and measurements are contaminated with noise.
