New Bounds for Restricted Isometry Constants With Coherent Tight Frames

This paper discusses reconstruction of a signal from undersampled data in the situation that the signal is sparse or approximately sparse in terms of a (possibly) highly overcomplete and coherent tight frameDvia thel1-analysis optimization problem. Some new sufficient conditions on theD-restricted isometry property are given to guarantee stable recovery of signals which are nearly sparse in terms ofD, from undersampled data with minimall1-norm of transform coefficients. One of the main results of this paper shows that if theD-restricted isometry constant δsof the measurement matrixAsatisfies δss-sparse in terms ofDare guaranteed to be stably recovered via thel1-analysis optimization problem. We point out that with the lemmas and the proof techniques developed in this paper, most of the sufficient conditions on the standard restricted isometry property for stable recovery of nearly sparse signals via standardl1-minimization, can be similarly extended to the general case of theD-restricted isometry property for stable recovery of signals that are nearly sparse in terms ofDvia thel1-analysis optimization problem, yielding weaker conditions than previously available in the literature.