A direct variational approach to a problem arising in image reconstruction

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 ∫|∇ u|(α+β|div∇ u/|∇ u||^p),α,β>0,\ p≥ 1, which takes into account the perimeter of the level sets of u as well as the Lᵖ 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 .