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A direct variational approach to a problem arising in image reconstruction. (English) Zbl 1029.49037

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 \(L^p\).

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

Citations:

Zbl 0981.49024
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