Robust signal recovery approach for compressive sensing using unconstrained optimization

A robust signal recovery approach for compressive sensing using unconstrained minimization is proposed. The ℓ₁ penalty function of the constrained ℓ₁-regularized least-squares recovery problem is replaced by the smoothly clipped absolute deviation (SCAD) sparsity-promoting penalty function. A convex and differentiable local quadratic approximation for the SCAD function is employed to render the computation of the gradient and Hessian tractable. Unconstrained minimization of randomly selected wavelet coefficients is carried out using the Newton method with an inexact line search. Experimental results demonstrate that signals recovered using the proposed approach often exhibit reduced ℓ_∞ reconstruction error under increasingly additive Gaussian measurement noise when compared with signals recovered using the ℓ₁-Magic and gradient projection for sparse reconstruction (GPSR) methods. Conversely, the number of linear measurements required to represent a signal can be reduced. As shown through simulations, significant reduction in the reconstruction error can be achieved although the computational cost is increased.