An iteratively approximated gradient projection algorithm for sparse signal reconstruction

In this paper we developed an iteratively approximated gradient projection algorithm for ℓ1ℓ1-minimization problems arising from sparse signal reconstruction in compressive sensing. By introducing a relaxed variable, the noisy problem can be transformed into the problem with equality constraints. The nonsmooth ℓ1ℓ1 term was tackled by variable-splitting techniques. Thus the problem was transformed into a quadratic programming problem. All linear variables in the objective function were imposed on ℓ2ℓ2 regularization. Based on ideas of quasi-Lagrangian functions and partial duality, a reduced quadratic programming problem can be obtained iteratively. At each iteration, we applied gradient projection methods with approximated gradients to get the next iterates. The computational experiments show the proposed method is very effective.