Common ISAR radar images and signals can be reconstructed from much fewer samples than the sampling theorem requires since they are usually sparse. Unavailable randomly positioned samples can result from heavily corrupted parts of the signal. Since these samples can be omitted and declared as unavailable, the application of the compressive sensing methods in the recovery of heavily corrupted signal and radar images is possible. A\ simple direct method for the recovery of unavailable signal samples and the calculation of the restored ISAR image is reviewed. An analysis of the noise influence is performed. For fast maneuvering ISAR targets the sparsity property is lost since the ISAR image is blurred. A nonparametric quadratic time-frequency representations based method is used to restore the ISAR image sparsity. However, the linear relation between the signal and the sparsity domain transformation is lost. A recently proposed gradient recovery algorithm is adapted for this kind of analysis. It does not require the linear relation of the signal and its sparsity domain transformation in the process of unavailable data recovery. The presented methods and results are tested on several numerical examples proving the expected accuracy and improvements.