Artificial Intelligence


New Sufficient Conditions of Signal Recovery with Tight Frames Via $l_1$-Analysis

Authors: Jianwen Huang, Jianjun Wang, Feng Zhang, Wendong Wang

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 $\Phi$ fulfils the condition $\delta_{ts}<t/(4-t)$ 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 \cite{Cai and Zhang 2014}. When $t=1$, the present bound improves the condition $\delta_s<0.307$ from Lin et al.'s reuslt to $\delta_s<1/3$. In addition, numerical simulations are conducted to indicate that the $l_1$-analysis method can stably reconstruct the sparse signal in terms of tight frames.

Comments: 18 Pages.

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Submission history

[v1] 2017-10-29 06:43:09
[v2] 2017-11-09 05:34:27

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