Compressed Sensing with General Frames via Optimal-dual-based $\ell_1$-analysis

TL;DR

This paper enhances the understanding of compressed sensing with general frames by providing a weaker recovery condition, introducing an optimal-dual-based approach, and demonstrating its effectiveness through experiments.

Contribution

It introduces a new optimal-dual-based $ ext{l}_1$-analysis method for compressed sensing with general frames, along with an iterative algorithm for its implementation.

Findings

Weaker recovery condition for $ ext{l}_1$-analysis with general frames.

Proposed optimal-dual-based technique improves signal recovery.

Experimental results confirm the effectiveness of the new approach.

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 $\ell_1$-analysis. We expand in this article 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 alternative 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 analysis operators. An iterative algorithm is outlined for solving the optimal-dual-based $\ell_1$-analysis problem. The effectiveness of the proposed method and algorithm is demonstrated through several experiments.