Bias-Reduction in Variational Regularization

The aim of this paper is to introduce and study a two-step debiasing method for variational regularization. After solving the standard variational problem, the key idea is to add a consecutive debiasing step minimizing the data fidelity on an appropriate set, the so-called model manifold. The latter is defined by Bregman distances or infimal convolutions thereof, using the (uniquely defined) subgradient appearing in the optimality condition of the variational method. For particular settings, such as anisotropic $\ell^1$ and TV-type regularization, previously used debiasing techniques are shown to be special cases. The proposed approach is however easily applicable to a wider range of regularizations. The two-step debiasing is shown to be well-defined and to optimally reduce bias in a certain setting. In addition to visual and PSNR-based evaluations, different notions of bias and variance decompositions are investigated in numerical studies. The improvements offered by the proposed scheme are demonstrated and its performance is shown to be comparable to optimal results obtained with Bregman iterations.