Dictionary-sparse and disjointed recovery

We consider recovery of signals whose coefficient vectors with respect to a redundant dictionary are simultaneously sparse and disjointed - such signals are referred to as analysis-sparse and analysis-disjointed. We determine the order of a sufficient number of linear measurements needed to recover such signals via an iterative hard thresholding algorithm. The sufficient number of measurements compares with the sufficient number of measurements from which one may recover a classical sparse and disjointed vector. We then consider approximately analysis-sparse and analysis-disjointed signals and obtain the order of sufficient number of measurements in that scenario as well.