Ambrosio, Luigi; Masnou, Simon A direct variational approach to a problem arising in image reconstruction. (English) Zbl 1029.49037 Interfaces Free Bound. 5, No. 1, 63-81 (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 \[ \int|\nabla u|\Biggl(\alpha+ \beta\Biggl|\text{div}{\nabla u\over|\nabla u|}\Biggr|^p\Biggr),\qquad\alpha, \beta> 0,\;p\geq 1, \] 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 \(L^p\). Cited in 1 ReviewCited in 34 Documents MSC: 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 49J45 Methods involving semicontinuity and convergence; relaxation 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 68U10 Computing methodologies for image processing Keywords:image processing; image reconstruction; relaxation; BV; integral varifolds; mean curvature Citations:Zbl 0981.49024 PDFBibTeX XMLCite \textit{L. Ambrosio} and \textit{S. Masnou}, Interfaces Free Bound. 5, No. 1, 63--81 (2003; Zbl 1029.49037) Full Text: DOI