  We consider some iterative methods for finding the best interpolation data in
the images compression with noise. The interpolation data consists of the set
of pixels and their grey/color values. The aim in the iterative approach is to
allow the change of the data dynamically during the inpainting process for a
reconstruction of the image that includes the enhancement and denoising
effects. The governing PDE model of this approach is a fully parabolic problem
where the set of stored pixels is time dependent. We consider the semi-discrete
dynamical system associated to the model which gives rise to an iterative
method where the stored data are modified during the iterations for best
outcomes. Finding the compression sets follows from a shape-based analysis
within the $\Gamma$-convergence tools, in particular well suited topological
asymptotic and a ``fat pixels'' approach are considered to obtain an analytic
characterization of the optimal sets in the sense of shape optimization theory.
We perform the analysis and derive several iterative algorithms that we
implement and compare to obtain the most efficient strategies of compression
and inpainting for noisy images. Some numerical computations are presented to
confirm the theoretical findings. Finally, we propose a modified model that
allows the inpainting data to change with the iteration and compare the
resulting new method to the ``probabilistic'' ones from the state-of-the-art.
