  The characteristic feature of inverse problems is their instability with
respect to data perturbations. In order to stabilize the inversion process,
regularization methods have to be developed and applied. In this work we
introduce and analyze the concept of filtered diagonal frame decomposition
which extends the standard filtered singular value decomposition to the frame
case. Frames as generalized singular system allows to better adapt to a given
class of potential solutions. In this paper, we show that filtered diagonal
frame decomposition yield a convergent regularization method. Moreover, we
derive convergence rates under source type conditions and prove order
optimality under the assumption that the considered frame is a Riesz-basis.
