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On a variational problem arising in image reconstruction. (English) Zbl 1111.68732
Colli, Pierluigi (ed.) et al., Free boundary problems: theory and applications. Proceedings of a conference, Trento, Italy, June 2002. Basel: Birkhäuser (ISBN 3-7643-2193-8/hbk). ISNM, Int. Ser. Numer. Math. 147, 17-26 (2003).
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 \[ \int|\nabla u|\left(\alpha+\beta\left|\text{div} \frac{\nabla u}{|\nabla u|}\right|^p\right),\quad\alpha,\beta>0,\;p\geq 1 \] 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 \(L^p\).
For the entire collection see [Zbl 1027.00020].

68U10 Computing methodologies for image processing
49J10 Existence theories for free problems in two or more independent variables