  This article studies the problem of image restoration of observed images
corrupted by impulse noise and mixed Gaussian impulse noise. Since the pixels
damaged by impulse noise contain no information about the true image, how to
find this set correctly is a very important problem. We propose two methods
based on blind inpainting and $\ell_0$ minimization that can simultaneously
find the damaged pixels and restore the image. By iteratively restoring the
image and updating the set of damaged pixels, these methods have better
performance than other methods, as shown in the experiments. In addition, we
provide convergence analysis for these methods, these algorithms will converge
to coordinatewise minimum points. In addition, they will converge to local
minimum points (or with probability one) with some modifications in the
algorithms.
