  Solving inverse problems is central to a variety of important applications,
such as biomedical image reconstruction and non-destructive testing. These
problems are characterized by the sensitivity of direct solution methods with
respect to data perturbations. To stabilize the reconstruction process,
regularization methods have to be employed. Well-known regularization methods
are based on frame expansions, such as the wavelet-vaguelette (WVD)
decomposition, which are well adapted to the underlying signal class and the
forward model and furthermore allow efficient implementation. However, it is
well known that the lack of translational invariance of wavelets and related
systems leads to specific artifacts in the reconstruction. To overcome this
problem, in this paper we introduce and analyze the translation invariant
diagonal frame decomposition (TI-DFD) of linear operators as a novel concept
generalizing the SVD. We characterize ill-posedness via the TI-DFD and prove
that a TI-DFD combined with a regularizing filter leads to a convergent
regularization method with optimal convergence rates. As illustrative example,
we construct a wavelet-based TI-DFD for one-dimensional integration, where we
also investigate our approach numerically. The results indicate that filtered
TI-DFDs eliminate the typical wavelet artifacts when using standard wavelets
and provide a fast, accurate, and stable solution scheme for inverse problems.
