Signal and Image Reconstruction with Tight Frames via Unconstrained $\ell_1-\alpha \ell_2$-Analysis Minimizations

TL;DR

This paper proposes a novel unconstrained analysis model using $\,\ell_1-\alpha\ell_2$ minimization for signal and image reconstruction, demonstrating superior performance over existing methods in numerical experiments.

Contribution

It introduces a new nonconvex analysis model with theoretical recovery guarantees and develops an effective algorithm for signal and image reconstruction.

Findings

The proposed model outperforms existing methods in numerical experiments.

Recovery guarantees are established based on the restricted isometry property for frames.

The algorithm effectively reconstructs signals and images in compressed sensing MRI.

Abstract

In the paper, we introduce an unconstrained analysis model based on the $\ell_{1}-\alpha \ell_{2}$ $(0< \alpha \leq1)$ minimization for the signal and image reconstruction. We develop some new technology lemmas for tight frame, and the recovery guarantees based on the restricted isometry property adapted to frames. The effective algorithm is established for the proposed nonconvex analysis model. We illustrate the performance of the proposed model and algorithm for the signal and compressed sensing MRI reconstruction via extensive numerical experiments. And their performance is better than that of the existing methods.