Automated Regularization Parameter Selection in Multi-Scale Total Variation Models for Image Restoration

Abstract

A multi-scale total variation model for image restoration is introduced. The model utilizes a spatially dependent regularization parameter in order to enhance image regions containing details while still sufficiently smoothing homogeneous features. The fully automated adjustment strategy of the regularization parameter is based on local variance estimators. For robustness reasons, the decision on the acceptance or rejection of a local parameter value relies on a confidence interval technique based on the expected maximal local variance estimate. In order to speed-up the performance of the update scheme a generalized hierarchical decomposition of the restored image is used. The corresponding subproblems are solved by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques. The paper ends by a report on numerical tests, a qualitative study of the proposed adjustment scheme and a comparison with popular total variation based restoration methods.

Homogeneous Penalizers and Constraints in Convex Image Restoration

“…For 'sparsity' models the reader may again consider [61,40]. Further modifications for spatially adapted regularization parameter selection as in [15,22] are also of interest. However, one of the main topics is still the estimation of the noise statistics.…”

Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials

“…This example is useful for spatially or temporally varying "tube" constraints, which arise in the regularization of inverse problems subject to variable noise levels [12]. The indicator function of temporally variable constraints also appears in Moreau's sweeping process, which is a model for several phenomena from nonsmooth mechanics such as elastoplasticity [28].…”

Investigating the Influence of Box-Constraints on the Solution of a Total Variation Model via an Efficient Primal-Dual Method

“…We remark, that a scalar regularization parameter might not be the best choice for every image restoration problem, since images usually consist of large uniform areas and parts with fine details, see for example [36,38]. It has been demonstrated, for example in [36,38,40,41] and references therein, that with the help of spatially varying regularization parameters one might be able to restore images visually better than with scalar parameters.…”

“…For minimizing problem (4) we derive a semi-smooth Newton method, which should serve us as a good method for quickly computing rather exact solutions. Second order methods have been already proposed and used in image reconstruction; see [21,[36][37][38][39]. However, to the best of our knowledge till now semi-smooth Newton methods have not been presented for box-constrained total variation minimization.…”

“…It has been demonstrated, for example in [36,38,40,41] and references therein, that with the help of spatially varying regularization parameters one might be able to restore images visually better than with scalar parameters. In this vein we also consider…”