A direct variational approach to a problem arising in image reconstruction

Abstract

We consider a variational approach to the problem of recovering missing parts in a panchromatic digital image. Representing the image by a scalar function $u$, we propose a model based on the relaxation of the energy

which takes into account the perimeter of the level sets of $u$ as well as the $L^p$ norm of the mean curvature along their boundaries. We investigate the properties of this variational model and the existence of minimizing functions in $\mathrm{BV}$. We also address related issues for integral varifolds with generalized mean curvature in $L^p$.