  We introduce and discuss shape-based models for finding the best
interpolation data in the compression of images with noise. The aim is to
reconstruct missing regions by means of minimizing a data fitting term in the
$L^2$-norm between the images and their reconstructed counterparts using
time-dependent PDE inpainting. We analyze the proposed models in the framework
of the $\Gamma$-convergence from two different points of view. First, we
consider a continuous stationary PDE model, obtained by focusing on the first
iteration of the discretized time-dependent PDE, and get pointwise information
on the "relevance" of each pixel by a topological asymptotic method. Second, we
introduce a finite dimensional setting of the continuous model based on "fat
pixels" (balls with positive radius), and we study by $\Gamma$-convergence the
asymptotics when the radius vanishes. Numerical computations are presented that
confirm the usefulness of our theoretical findings for non-stationary PDE-based
image compression.
