Robust Sparse Recovery in Impulsive Noise via M-Estimator and Non-Convex Regularization

Robust sparse recovery aims at recovering a sparse signal or image from its compressed and contaminated measurements. Under the impulsive noise condition, the performance of traditional sparse recovery algorithms may deteriorate seriously for exploiting ℓ
₂
-norm to model the non-Gaussian noise. In this paper, a novel formulation which combines the M-estimator and the non-convex regularization term is presented to address the issue of robust sparse recovery in the impulsive noise environment. Since the ℓ
₂
-norm is highly sensitive to the large outliers appearing in impulse interference, we replace it with the robust M-estimators that have exhibited the powerful capability of suppressing impulsive noise in various scenarios. Meanwhile, the non-convex regularization is capable of overcoming the biased estimation problem induced by the convex ℓ
₁
-norm regularization and thus can obtain more accurate reconstruction results. Furthermore, to solve the resulting non-convex formulation, an efficient first-order algorithm with low computational complexity is developed by utilizing the alternating direction method of multipliers framework and the half-quadratic optimization. The reconstruction experiments under the circumstance of impulsive noise are conducted to demonstrate the superior performance of the proposed algorithm over several typical sparse recovery algorithms.