Frame-based block sparse compressed sensing via [Formula omitted]-synthesis

In this paper, we consider the frame-based block sparse signal recovery via a [Formula omitted]-synthesis method. A new kind of null space property based on the given dictionary D(block D-NSP) is proposed. It is proved that sensing matrices satisfying the block D-NSP is not just a sufficient and necessary condition for the [Formula omitted]-synthesis method to exactly recover signals that are block sparse in frame D, but also a sufficient and necessary condition for the [Formula omitted]-synthesis to stably recover signals which are block-compressible in frame D. To the best of our knowledge, this new property is the first sufficient and necessary condition for successful signal recovery via the [Formula omitted]-synthesis. In addition, we also characterize the theoretical performance of recovering signals via the [Formula omitted]-synthesis in the case of the measurements are disturbed.