  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.
