No abstract

We propose a sparse reconstruction framework for solving inverse problems. Opposed to existing sparse regularization techniques that are based on frame representations, we train an encoder-decoder network by including an`I-penalty. We demonstrate that the trained decoder network allows sparse signal reconstruction using thresholded encoded coefficients without losing much quality of the original image. Using the sparse synthesis prior, we propose minimizing the`I-Tikhonov functional, which is the sum of a data fitting term and the`I-norm of the synthesis coefficients, and show that it provides a regularization method.

No abstract

No abstract

Recently, a large number of efficient deep learning methods for solving inverse problems have been developed and show outstanding numerical performance. For these deep learning methods, however, a solid theoretical foundation in the form of reconstruction guarantees is missing. In contrast, for classical reconstruction methods, such as convex variational and frame-based regularization, theoretical convergence and convergence rate results are well established. In this paper, we introduce deep synthesis networks for regularizing inverse problems (DESYRE) using neural networks as nonlinear synthesis operator bridging the gap between these two worlds. The proposed method allows to exploit the deep learning benefits of being well adjustable to available training data and on the other hand comes with a solid mathematical foundation. We present a complete convergence analysis with convergence rates for the proposed deep synthesis regularization. We present a strategy for constructing a synthesis network as part of an analysis–synthesis sequence together with an appropriate training strategy. Numerical results show the plausibility of our approach.