In this paper, we propose and study the use of alternating direction
algorithms for several $\ell_1$-norm minimization problems arising from sparse
solution recovery in compressive sensing, including the basis pursuit problem,
the basis-pursuit denoising problems of both unconstrained and constrained
forms, as well as others. We present and investigate two classes of algorithms
derived from either the primal or the dual forms of the $\ell_1$-problems.