  We consider the very challenging task of restoring images (i) which have a
large number of missing pixels, (ii) whose existing pixels are corrupted by
noise and (iii) the ideal image to be restored contains both cartoon and
texture elements. The combination of these three properties makes this inverse
problem a very difficult one. The solution proposed in this manuscript is based
on directional global three-part decomposition (DG3PD) [ThaiGottschlich2016]
with directional total variation norm, directional G-norm and
$\ell_\infty$-norm in curvelet domain as key ingredients of the model. Image
decomposition by DG3PD enables a decoupled inpainting and denoising of the
cartoon and texture components. A comparison to existing approaches for
inpainting and denoising shows the advantages of the proposed method. Moreover,
we regard the image restoration problem from the viewpoint of a Bayesian
framework and we discuss the connections between the proposed solution by
function space and related image representation by harmonic analysis and
pyramid decomposition.
