  Total Variation (TV) is an effective and popular prior model in the field of
regularization-based image processing. This paper focuses on total variation
for removing impulse noise in image restoration. This type of noise frequently
arises in data acquisition and transmission due to many reasons, e.g. a faulty
sensor or analog-to-digital converter errors. Removing this noise is an
important task in image restoration. State-of-the-art methods such as Adaptive
Outlier Pursuit(AOP) \cite{yan2013restoration}, which is based on TV with
$\ell_{02}$-norm data fidelity, only give sub-optimal performance. In this
paper, we propose a new sparse optimization method, called $\ell_0TV$-PADMM,
which solves the TV-based restoration problem with $\ell_0$-norm data fidelity.
To effectively deal with the resulting non-convex non-smooth optimization
problem, we first reformulate it as an equivalent biconvex Mathematical Program
with Equilibrium Constraints (MPEC), and then solve it using a proximal
Alternating Direction Method of Multipliers (PADMM). Our $\ell_0TV$-PADMM
method finds a desirable solution to the original $\ell_0$-norm optimization
problem and is proven to be convergent under mild conditions. We apply
$\ell_0TV$-PADMM to the problems of image denoising and deblurring in the
presence of impulse noise. Our extensive experiments demonstrate that
$\ell_0TV$-PADMM outperforms state-of-the-art image restoration methods.
