  We consider a shape optimization based method 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 an
$L^p$-norm between original images and their reconstructed counterparts using
linear diffusion PDE-based inpainting. Reformulating the problem as a
constrained optimization over sets (shapes), we derive the topological
asymptotic expansion of the considered shape functionals with respect to the
insertion of small ball (a single pixel) using the adjoint method. Based on the
achieved distributed topological shape derivatives, we propose a numerical
approach to determine the optimal set and present numerical experiments
showing, the efficiency of our method. Numerical computations are presented
that confirm the usefulness of our theoretical findings for PDE-based image
compression.
