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New algorithm for solving variational problems in $W^{1,p}\SO$ and $BV\SO$: Application to image restoration

Gilles Aubert 1 Pierre Kornprobst 2
2 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique - ENS Paris, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS-PSL - École normale supérieure - Paris, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
Abstract : We propose a new unifying method for solving variational problems defined on the Sobolev spaces $W^{1,p}(\Omega)$ or on the space of functions of bounded variations $BV(\Omega)$ ($\Omega\subset\R^N$). The method is based on a recent new characterization of these spaces by Bourgain, Brezis and Mironescu (2001), where norms can be approximated by a sequence of integral operators involving a differential quotient and a suitable sequence of radial mollifiers. We use this characterization to define a variational formulation, for which existence, uniqueness and convergence of the solution is proved. The proposed approximation is valid for any $p$ and does not depend on the attach term. Implementation details are given and we show examples on the image restoration problem.
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Submitted on : Thursday, July 26, 2007 - 5:26:12 PM
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Gilles Aubert, Pierre Kornprobst. New algorithm for solving variational problems in $W^{1,p}\SO$ and $BV\SO$: Application to image restoration. [Research Report] RR-6245, INRIA. 2007, pp.25. ⟨inria-00161706v5⟩

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