inria-00161706, version 4
New algorithm for solving variational problems in $W^{1,p}\SO$ and $BV\SO$: Application to image restoration
N° RR-6245 (2007)
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.
- a – Université de Nice Sophia-Antipolis
- b – INRIA
- 1:
- CNRS : UMR6621 – Université Nice Sophia Antipolis [UNS]
- 2:
- INRIA – Ecole des Ponts ParisTech – Ecole normale supérieure de Paris - ENS Paris
- Domain : Mathematics/Functional Analysis
- Keywords : Calculus of variation – functional analysis – Sobolev spaces – $BV$ – variational approach
- Internal note : RR-6245
- Available versions : v1 (2007-07-11) v2 (2007-07-13) v3 (2007-07-13) v4 (2007-07-13) v5 (2007-07-26)
- inria-00161706, version 4
- http://hal.inria.fr/inria-00161706
- oai:hal.inria.fr:inria-00161706
- From:
- Submitted on: Friday, 13 July 2007 15:19:25
- Updated on: Friday, 13 July 2007 15:19:39
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