Compressed Sensing with coherent tight frames via $l_q$-minimization for $0<q\leq1$

Our aim of this article is to reconstruct a signal from undersampled data in the situation that the signal is sparse in terms of a tight frame. We present a condition, which is independent of the coherence of the tight frame, to guarantee accurate recovery of signals which are sparse in the tight frame, from undersampled data with minimal l1-norm of transform coefficients. This improves the result in [1]. Also, the lq-minimization (0 < q < 1) approaches are introduced. We show that under a suitable condition, there exists a value q0 ∈ (0, 1] such that for any q ∈ (0, q0), each solution of the lq-minimization is approximately well to the true signal. In particular, when the tight frame is an identity matrix or an orthonormal basis, all results obtained in this paper appeared in [13] and [26].