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Thu, 02 Feb 2012 05:02:50 +0000
Thu, 02 Feb 2012 05:02:50 +0000

Homogeneous Penalizers and Constraints in Convex Image Restoration
https://kluedo.ub.unikl.de/frontdoor/index/index/docId/2866
Recently convex optimization models were successfully applied for solving various problems in image analysis and restoration. In this paper, we are interested in relations between convex constrained optimization problems of the form \(min\{\Phi(x)\) subject to \(\Psi(x)\le\tau\}\) and their nonconstrained, penalized counterparts \(min\{\Phi(x)+\lambda\Psi(x)\}\). We start with general considerations of the topic and provide a novel proof which ensures that a solution of the constrained problem with given \(\tau\) is also a solution of the onconstrained problem for a certain \(\lambda\). Then we deal with the special setting that \(\Psi\) is a seminorm and \(\Phi=\phi(Hx)\), where \(H\) is a linear, not necessarily invertible operator and \(\phi\) is essentially smooth and strictly convex. In this case we can prove via the dual problems that there exists a bijective function which maps \(\tau\) from a certain interval to \(\lambda\) such that the solutions of the constrained problem coincide with those of the nonconstrained problem if and only if \(\tau\) and \(\lambda\) are in the graph of this function. We illustrate the relation between \(\tau\) and \(\lambda\) by various problems arising in image processing. In particular, we demonstrate the performance of the constrained model in restoration tasks of images corrupted by Poisson noise and in inpainting models with constrained nuclear norm. Such models can be useful if we have a priori knowledge on the image rather than on the noise level.
RenĂ© Ciak; Behrang Shafei; Gabriele Steidl
preprint
https://kluedo.ub.unikl.de/frontdoor/index/index/docId/2866
Thu, 02 Feb 2012 05:02:50 +0000