New Bounds for Restricted Isometry Constants With Coherent Tight Frames

Abstract

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

Stable recovery of analysis based approaches

“…It was first proved in [3] that the condition δ 2k < 0.08 guarantees the stable recovery of the l 1 minimization model in (1.1) and the constant C 0 in (1.2) depends on the D-RIP constant δ 2k . This sufficient condition for stable recovery was later improved in [13,14].…”

Improved RIP Conditions for Compressed Sensing with Coherent Tight Frames

“…Let h =f − f, where f is the original signal andf is the solution to (3). As noted in [15,18], different from the proof in [8] for standard compressed sensing, we need to develop bounds on ‖ * h‖ 2 instead of ‖h‖ 2 . We write * h = ∑ =1 Δ , where 1 ≥ 2 ≥ ⋅ ⋅ ⋅ ≥ ≥ 0 and {Δ } =1 are indicator vectors (we have mentioned in the proof of Lemma 3) with different support.…”

“…Remark 9. It is obvious that the obtained condition + , < 1 is weaker than + 1.25 , < 1, (8/7) + (8/7) ,(8/7) < 1, and 1.25 + ,1.25 < 1 which were used in [15].…”

“…There are a number of practical applications in signal and image processing point to problems where signals are not sparse in terms of an orthogonal basis but in terms of an overcomplete and tight frame (see [12][13][14][15][16], and the references therein). In such contexts, the signal f can be expressed as f = x, where x ∈ R is a sparse or nearly sparse vector.…”

Non-convex block-sparse compressed sensing with coherent tight frames

“…In this paper, along with previous works on the non-convex q (0 < q < 1) strategy, we first propose an 2 / q -analysis method with 0 < q ≤ 1 to recover general signals that can be expressed as blocksparse signals in terms of . Our method is different from conventional CS methods, which only concern cases where the signals per se are sparse or block-sparse [18,[21][22][23], and also different from previous analysis methods [24][25][26], which only focus on the recovery of general signals that are expressed as non-block structured signals in terms of . Specifically, the proposed method can be described as:…”