  In this paper, we consider the efficient and robust reconstruction of signals
and images
  via $\ell_{1}-\alpha \ell_{2}~(0<\alpha\leq 1)$ minimization in impulsive
noise case.
  To achieve this goal, we introduce two new models: the $\ell_1-\alpha\ell_2$
minimization with $\ell_1$ constraint, which is called
$\ell_1-\alpha\ell_2$-LAD, the $\ell_1-\alpha\ell_2$ minimization with Dantzig
selector constraint, which is called $\ell_1-\alpha\ell_2$-DS.
  We first show that sparse signals or nearly sparse signals can be exactly or
stably recovered via $\ell_{1}-\alpha\ell_{2}$ minimization under some
conditions based on the restricted $1$-isometry property ($\ell_1$-RIP).
Second, for $\ell_1-\alpha\ell_2$-LAD model, we introduce unconstrained
$\ell_1-\alpha\ell_2$ minimization model denoting $\ell_1-\alpha\ell_2$-PLAD
and propose $\ell_1-\alpha\ell_2$LA algorithm to solve the
$\ell_1-\alpha\ell_2$-PLAD.
  Last, numerical experiments %on success rates of sparse signal recovery
  demonstrate that when the sensing matrix is ill-conditioned (i.e., the
coherence of
  the matrix is larger than 0.99), the $\ell_1-\alpha\ell_2$LA method
  is better than the existing convex and non-convex compressed sensing solvers
for the recovery of sparse signals. And for the magnetic resonance imaging
(MRI) reconstruction with impulsive noise, we show that
  the $\ell_1-\alpha\ell_2$LA method has better performance than
state-of-the-art methods via numerical experiments.
