Frame-based block sparse compressed sensing via l₂/l₁-synthesis

In this paper, we consider the frame-based block sparse signal recovery via a l₂/l₁ l2/l1 -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 l₂/l₁ l2/l1 -synthesis method to exactly recover signals that are block sparse in frame D , but also a sufficient and necessary condition for the l₂/l₁ l2/l1 -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 l₂/l₁ l2/l1 -synthesis. In addition, we also characterize the theoretical performance of recovering signals via the l₂/l₁ l2/l1 -synthesis in the case of the measurements are disturbed.