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hal-00722743, version 1

## Smooth objectives composed of asymptotically affine data-fidelity and regularization. Bounds for the minimizers and parameter choice

F. Baus 1, Mila Nikolova (, ) 2, Gabriele Steidl () 1

(2012-07-19)

Abstract: We examine properties of the minimizer u* of a class of differentiable functionals where both the data-term and the regularization term are symmetric and nearly affine beyond a small neighborhood of the origin. Customarily, such functions are used to regularize a quadratic data-fidelity term in order to produce solutions where edges are preserved. The functionals we consider in this paper behave quite differently. They were recently successfully applied to provide a strict order for the pixels of digital (quantized) images f thus enabling exact histogram specification. We give upper and lower bounds for the error $\|u* - f\|_\infty$, where the upper bound is independent of the input image f. Interestingly, in the numerical experiments with natural digital images f, the estimated upper bound is easily reached up to a small error. To explain this phenomenon we give simple statistical estimates for the behavior of neighboring pixels. We apply our estimates to specify the parameters of the model.

• 1:  University of Kaiserslautern
• University of Kaiserslautern
• 2:  Centre de Mathématiques et de Leurs Applications (CMLA)
• CNRS : UMR8536 – École normale supérieure de Cachan - ENS Cachan

• hal-00722743, version 1
• oai:hal.archives-ouvertes.fr:hal-00722743
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• Submitted on: Sunday, 5 August 2012 11:11:42
• Updated on: Sunday, 5 August 2012 21:15:44