Frame-based block sparse compressed sensing via $$l_2/l_1$$-synthesis

In this paper, we consider the frame-based block sparse signal recovery via a $$l_2/l_1$$ l 2 / l 1 -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 $$l_2/l_1$$ l 2 / l 1 -synthesis method to exactly recover signals that are block sparse in frame D , but also a sufficient and necessary condition for the $$l_2/l_1$$ l 2 / l 1 -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 $$l_2/l_1$$ l 2 / l 1 -synthesis. In addition, we also characterize the theoretical performance of recovering signals via the $$l_2/l_1$$ l 2 / l 1 -synthesis in the case of the measurements are disturbed.