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

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