New Sufficient Conditions of Signal Recovery With Tight Frames via ${l}_1$ -Analysis Approach

This paper discusses the recovery of signals that are nearly sparse with respect to a tight frame D by means of the l1-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 fulfills the condition δts <; t/(4 – t) for 0 <; t <; 4/3, then signals which are approximately sparse with respect to D can be stably recovered by the l1-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 l1-analysis method can stably reconstruct the sparse signal in terms of tight frames.

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