Choosing The Best Interpolation Data in Images with Noise

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