Optimal D-RIP bounds in compressed sensing

Zhang, Rui; Li, Song

doi:10.1007/s10114-015-4234-4

Cited by 55 publications

(8 citation statements)

References 23 publications

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“…We prove that under the condition δ ts < t/(4 − t) for 0 < t < 4/3, any signals f which are sparse with respect to a tight frame D can be accurately and stably recovered via (1.1). When t = 1, our main results are consistent with Theorem 3.1 and Theorem 4.1 in [15]. Besides, in the situation of D = I (for the standard compressing sensing), we obtain same results as the main results in [17] and the bound on the constant δ ts is sharp, referred to see [5].…”

Section: Introductionsupporting

confidence: 86%

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New sufficient conditions of signal recovery with tight frames via $l_1$-analysis

Huang¹,

Wang²,

Zhang³

et al. 2017

Preprint

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The paper discusses the recovery of signals in the case that signals are nearly sparse with respect to a tight frame D by means of the l 1 -analysis approach. We establish several new sufficient conditions regarding the D-restricted isometry property to ensure stable reconstruction of signals that are approximately sparse with respect to D. It is shown that if the measurement matrix Φ fulfils the condition δ ts < t/(4 − t) for 0 < t < 4/3, then signals which are approximately sparse with respect to D can be stably recovered by the l 1 -analysis method. In the case of D = I, the bound is sharp, see Cai and Zhang's work [5]. When t = 1, the present bound improves the condition δ s < 0.307 from Lin et al.'s reuslt to δ s < 1/3.

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Section: Introductionsupporting

confidence: 86%

“…Liu et al [12] employed the assumption 9δ 2s + 4δ 4s < 5 to assure recovery under the general frame. Zhang and Li [15] refined the bound to δ 2s < √ 2/2 ≈ 0.707 and δ s < 1/3 ≈ 0.333. Furthermore, Chen and Li [16] gave a high order condition on the D-RIP for the recovery of signal.…”

Section: Introductionmentioning

confidence: 94%

New sufficient conditions of signal recovery with tight frames via $l_1$-analysis

Huang¹,

Wang²,

Zhang³

et al. 2017

Preprint

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“…Remark 12. As stated in [20], < 1/3 is the best condition for stable recovery of nearly sparse signal in terms of via 1analysis. Based on Corollary 11, our presented condition , + < 1 is mostly weaker than < 1/3.…”

Section: Improved Rip Conditionsmentioning

confidence: 96%

“…Moreover, they also obtained that < 0.307, which is the first sufficient condition on , is sufficient for 1 -analysis to guarantee the stable recovery of nearly -sparse (in terms of ) signals. In a recent paper [20], the condition < 0.307 was improved to < 1/3.…”

Section: Introductionmentioning

confidence: 99%

Improved RIP Conditions for Compressed Sensing with Coherent Tight Frames

Wang

2017

Discrete Dynamics in Nature and Society

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This paper establishes new sufficient conditions on the restricted isometry property (RIP) for compressed sensing with coherent tight frames. One of our main results shows that the RIP (adapted to ) condition + , < 1 guarantees the stable recovery of all signals that are nearly -sparse in terms of a coherent tight frame via the 1 -analysis method, which improves the existing ones in the literature. Hindawi

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“…Our problem is to recover sparse dictionary signals from a set of noisy measurements like the framework proposed by Zhang and Li [43]. Let xR n be a k sparse input vector, where k is the known level of sparsity.…”

Section: Problem Formulationmentioning

confidence: 99%

Design and Application of a Greedy Pursuit Algorithm Adapted to Overcomplete Dictionary for Sparse Signal Recovery

Zhao¹,

Zhu²,

Wu³

2020

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Compressive sensing (CS) is a novel paradigm to recover a sparse signal in compressed domain. In some overcomplete dictionaries, most practical signals are sparse rather than orthonormal. Signal space greedy method can derive the optimal or near-optimal projections, making it possible to identify a few most relevant dictionary atoms of an arbitrary signal. More practically, such projections can be processed by standard CS recovery algorithms. This paper proposes a signal space subspace pursuit (SSSP) method to compute spare signal representations with overcomplete dictionaries, whenever the sensing matrix satisfies the restricted isometry property adapted to dictionary (D-RIP). Specifically, theoretical guarantees were provided to recover the signals from their measurements with overwhelming probability, as long as the sensing matrix satisfies the D-RIP. In addition, a thorough analysis was performed to minimize the number of measurements required for such guarantees. Simulation results demonstrate the validity of our hypothetical theory, as well as the superiority of the proposed approach.

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Optimal D-RIP bounds in compressed sensing

Cited by 55 publications

References 23 publications

New sufficient conditions of signal recovery with tight frames via $l_1$-analysis

New sufficient conditions of signal recovery with tight frames via $l_1$-analysis

Improved RIP Conditions for Compressed Sensing with Coherent Tight Frames

Design and Application of a Greedy Pursuit Algorithm Adapted to Overcomplete Dictionary for Sparse Signal Recovery

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