Bias-Reduction in Variational Regularization

Eva-Maria Brinkmann (1) , Martin Burger (1) , Julian Rasch (1) , Camille Sutour (2, 1)

1 Institut für Numerische und Angewandte Mathematik
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

The aim of this paper is to introduce and study a two-step debiasing method for variational reg-ularization. After solving the standard varia-tional problem, the key idea is to add a consecutive debiasing step minimizing the data fidelity on an appropriate space, the so-called model manifold. The latter is defined by Bregman distances or infimal convolutions thereof, using the subgradient appearing in the optimality condition of the variational method. For particular settings, such as anisotropic 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. Besides visual and PSNR-based evaluations, different notions of bias and variance decomposi-tions 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.

Domaines: Traitement des images [eess.IV]

Eva-Maria Brinkmann, Martin Burger, Julian Rasch, Camille Sutour. Bias-Reduction in Variational Regularization. 2016. ⟨hal-01450731⟩