# A direct variational approach to a problem arising in image reconstruction

### Luigi Ambrosio

Scuola Normale Superiore, Pisa, Italy### Simon Masnou

Université Pierre et Marie Curie, Paris, France

## 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

$\int|\nabla u|\biggl(\alpha+\beta\biggl|\operatorname{div}\frac{\nabla u}{|\nabla u|}\biggr|^p\biggr),\qquad\alpha,\beta>0,\ p\geq 1,$

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