Abstract
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.