New Sufficient Conditions of Signal Recovery With Tight Frames via <inline-formula> <tex-math notation="LaTeX">${l}_1$ </tex-math> </inline-formula>-Analysis Approach

Abstract

This paper discusses the recovery of signals that 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 $ fulfills the condition $\delta _{ts} for $0 , then signals which are approximately sparse with respect to $D$ can be stably recovered by the $l_{1}$ -analysis approach. In the case of $D=I$ , the bound is sharp (see Cai and Zhang’s work). 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.