Two-step regularization methods for linear inverse problems

Abstract

In this paper we investigate reconstruction methods for the treatment of ill-posed inverse problems. These methods are based on a data estimation operator S λ followed by a classical regularization operator R α T α,λ = R α S λ . As a particular example of such a two-step regularization method we investigate in detail the combination of a wavelet shrinkage operator S λ followed by Tikhonov regularization R α . The nonlinear shrinkage operator is applied to noisy data and partially recovers the smoothness properties of the exact data. We prove order optimality for the proposed scheme and confirm the theoretical results with an example from medical imaging.