JournalsifbVol. 5 , No. 1DOI 10.4171/ifb/72

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

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 uu, we propose a model based on the relaxation of the energy

u(α+βdivuup),α,β>0, p1,\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 uu as well as the \lpp\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\BV. We also address related issues for integral varifolds with generalized mean curvature in \lpp\lp{p}.