Compressed Sensing With General Frames via Optimal-Dual-Based $\ell _{1}$-Analysis

Abstract

Compressed sensing with sparse frame representations is seen to have much greater range of practical applications than that with orthonormal bases. In such settings, one approach to recover the signal is known as ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -analysis. We expand in this paper the performance analysis of this approach by providing a weaker recovery condition than existing results in the literature. Our analysis is also broadly based on general frames and alter native dual frames (as analysis operators). As one application to such a general-dual-based approach and performance analysis, an optimal-dual-based technique is proposed to demonstrate the effectiveness of using alternative dual frames as ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -analysis operators. An iterative algorithm is outlined for solving the optimal-dual-based -analysis problem. The effectiveness of the proposed method and algorithm is demonstrated through several experiments.