Compressed sensing with coherent tight frames via $l_q$-minimization for $0 < q \leq 1$

Abstract

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 l 1 -norm of transform coefficients. This improves the result in [1]. Also, the l q -minimization (0 < q < 1) approaches are introduced. We show that under a suitable condition, there exists a value q 0 ∈ (0, 1] such that for any q ∈ (0, q 0 ), each solution of the l q -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].

New Bounds for Restricted Isometry Constants With Coherent Tight Frames

“…The proof makes use of the ideas from [7], [10], [14], [29]. As in [7], using the fact that is a tight frame, we will develop bounds on instead of .…”

“…( ) with , can recover a signal that is (approximately) sparse in terms of with a small or zero error. Later, the -RIP condition was improved to , and in some special cases, to [29]. Liu et al [31] have used the sufficient condition for stable recovery of signal that is (approximately) sparse in terms of via the ABP.…”

Iteratively reweighted algorithm for signals recovery with coherent tight frame

“…Li et al improved the restricted isometric constant δ 2 s with tight frame D . They showed that, if δ 2 s < 0.493, the result also holds.…”

“…Before the proof of Theorem , we first introduce the following lemma (see inequality (2.2) of Li and Lin.…”

“…In Needell and Tropp, if δ 7 s = 0.5, one obtains an estimate from Equation with C 0 = 30 and C 1 = 62. Similarly, Li et al obtained a smaller estimate of C 0 and C 1 with C 0 ≈5.06 and C 1 ≈10.57.…”