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 frame D via the l₁-analysis optimization problem. Some new sufficient conditions on the D -restricted isometry property are given to guarantee stable recovery of signals which are nearly sparse in terms of D, from undersampled data with minimal l₁-norm of transform coefficients. One of the main results of this paper shows that if the D-restricted isometry constant δs of the measurement matrix A satisfies δs <; 0.307 , then signals which are approximately s-sparse in terms of D are guaranteed to be stably recovered via the l₁-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 standard l₁ -minimization, can be similarly extended to the general case of the D-restricted isometry property for stable recovery of signals that are nearly sparse in terms of D via the l₁-analysis optimization problem, yielding weaker conditions than previously available in the literature.