Image restoration under sparsity constraints has received increased attention in recent years. This problem can be formulated as a nondifferentiable convex optimization problem whose solution is challenging. In this work, the non-differentiability of the objective is addressed by reformulating the image restoration problem as a nonnegatively constrained quadratic program which is then solved by a specialized Newton projection method where the search direction computation only requires matrix-vector operations. A comparative study with state-of-the-art methods is performed in order to illustrate the efficiency and effectiveness of the proposed approach.