In the L-estimation and compressive sensing some arbitrarily positioned samples of the signal are either so heavily corrupted by disturbances that it is better to omit them in the analysis or they are unavailable. If the considered signal with missing samples is sparse then we are still able to reconstruct these samples by using the well know reconstruction algorithms. In this paper we will illustrate different measures for the signal concentration and propose a simple adaptive algorithm, applied on these measures, without reformulating the reconstruction problem within the standard linear programming form. Direct application of the gradient on nondifferentiable forms of measures lead to an efficient variable step size algorithm. The results are illustrated on the examples.