The $ \ell_1 $-analysis with redundant dictionary in phase retrieval

This article presents new results concerning the recovery of a signal from magnitude only measurements where the signal is not sparse in an orthonormal basis but in a redundant dictionary. To solve this phaseless problem, we analyze the $\ell_1$-analysis model. Firstly we investigate the noiseless case with presenting a null space property of the measurement matrix under which the $\ell_1$-analysis model provide an exact recovery. Secondly we introduce a new property (S-DRIP) of the measurement matrix. By solving the $\ell_1$-analysis model, we prove that this property can guarantee a stable recovery of real signals that are nearly sparse in highly overcomplete dictionaries.