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

• Main Authors: Jianwen Huang, Jianjun Wang, Feng Zhang, Wendong Wang
• Format: Article
• Language: English
• Published: IEEE 2018-01-01
• Series: IEEE Access
• Subjects: Compressed sensing; <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">l</italic>₁-analysis approach; restricted isometry property; sparse recovery; tight frames
• Online Access: https://ieeexplore.ieee.org/document/8355494/

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 &#x03A6; fulfills the condition &#x03B4;_ts &lt;; t/(4 - t) for 0 &lt;; t &lt;; 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.