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

TL;DR

This paper introduces new sufficient conditions based on the $D$-restricted isometry property that guarantee stable signal recovery via $l_1$-analysis when signals are nearly sparse with respect to a tight frame, improving existing bounds.

Contribution

The paper establishes novel $D$-restricted isometry conditions for stable recovery using $l_1$-analysis, extending results to nearly sparse signals with tight frames.

Findings

Recovery condition $ ext{delta}_{ts}<t/(4-t)$ for $0<t<4/3$ ensures stable reconstruction.

Improved bound $ ext{delta}_s<1/3$ for the case $t=1$ over previous results.

Sharp bounds are achieved when $D=I$, aligning with known optimal results.

Abstract

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 $\delta_s<1/3$.