A direct variational approach to a problem arising in image reconstruction
Summary: 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 \[ \int|\nabla u|\Biggl(\alpha+ \beta\Biggl|\text{div}{\nabla u\over|\nabla u|}\Biggr|^p\Biggr),\qquad\alpha, \beta> 0,\;p\geq 1, \] which takes into account the perimeter of the level sets of \(u\) as well as the \(L^p\) norm of the mean curvature along their boundaries. We investigate the properties of this variational model and the existence of minimizing functions in BV. We also address related issues for integral varifolds with generalized mean curvature in \(L^p\).
scientific article; zbMATH DE number 1941729