  Recovery of arbitrarily positioned samples that are missing in sparse signals
recently attracted significant research interest. Sparse signals with heavily
corrupted arbitrary positioned samples could be analyzed in the same way as
compressive sensed signals by omitting the corrupted samples and considering
them as unavailable during the recovery process. The reconstruction of missing
samples is done by using one of the well known reconstruction algorithms. In
this paper we will propose a very simple and efficient adaptive variable step
algorithm, applied directly to the concentration measures, without
reformulating the reconstruction problem within the standard linear programming
form. Direct application of the gradient approach to the nondifferentiable
forms of measures lead us to introduce a variable step size algorithm. A
criterion for changing adaptive algorithm parameters is presented. The results
are illustrated on the examples with sparse signals, including approximately
sparse signals and noisy sparse signals.
