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An interesting topic in compressed sensing aims to recover signals withsparse representations in a dictionary. Recently the performance of theL1-analysis method has been a focus, while some fundamental problems for theL1-synthesis method are still unsolved. For example, what are the conditionsfor it to stably recover compressible signals under noise? Whether coherentdictionaries allow the existence of sensing matrices that guarantee goodperformances of the L1-synthesis method? To answer these questions, we build upa framework for the L1-synthesis method. In particular, we propose adictionary-based null space property DNSP which, to the best of our knowledge,is the first sufficient and necessary condition for the success of L1-synthesiswithout measurement noise. With this new property, we show that when thedictionary D is full spark, it cannot be too coherent otherwise the methodfails for all sensing matrices. We also prove that in the real case, DNSP isequivalent to the stability of L1-synthesis under noise.

A null space analysis of the L1 synthesis method in dictionary-based compressed sensing