Article

Computational Optimization and Applications

, Volume 54, Issue 2, pp 371-398

A class of quasi-variational inequalities for adaptive image denoising and decomposition

  • Frank LenzenAffiliated withHeidelberg Collaboratory for Image Processing & Image and Pattern Analysis Group, University of Heidelberg Email author 
  • , Florian BeckerAffiliated withHeidelberg Collaboratory for Image Processing & Image and Pattern Analysis Group, University of Heidelberg
  • , Jan LellmannAffiliated withHeidelberg Collaboratory for Image Processing & Image and Pattern Analysis Group, University of Heidelberg
  • , Stefania PetraAffiliated withHeidelberg Collaboratory for Image Processing & Image and Pattern Analysis Group, University of Heidelberg
  • , Christoph SchnörrAffiliated withHeidelberg Collaboratory for Image Processing & Image and Pattern Analysis Group, University of Heidelberg

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Abstract

We introduce a class of adaptive non-smooth convex variational problems for image denoising in terms of a common data fitting term and a support functional as regularizer. Adaptivity is modeled by a set-valued mapping with closed, compact and convex values, that defines and steers the regularizer depending on the variational solution. This extension gives rise to a class of quasi-variational inequalities. We provide sufficient conditions for the existence of fixed points as solutions, and an algorithm based on solving a sequence of variational problems. Denoising experiments with spatial and spatio-temporal image data and an adaptive total variation regularizer illustrate our approach.

Keywords

Quasi-variational inequalities Adaptive image denoising Total variation regularization Solution-dependent adaptivity