Error estimates for non-quadratic regularization and the relation to enhancement

In this paper error estimates for non-quadratic regularization of nonlinear ill-posed problems in Banach spaces are derived. Our analysis is based on a few novel features: in comparison with the classical analysis of regularization methods for inverse and ill-posed problems where a Lipschitz continuity for the Frechet derivative is required, we use a differentiability condition with respect to the Bregman distance. Also, a stability result for the regularized solutions in terms of Bregman distances is proven. Moreover, a source-wise representation of the solution as used in standard theory is interpreted in terms of data enhancement. It is also shown that total variation Bregman distance regularization for image analysis, as developed recently, can be considered as a two-step regularization method consisting of a combination of total variation regularization and additional enhancement. This technique can also be applied for filtering.