End-to-end reconstruction meets data-driven regularization for inverse problems

We propose an unsupervised approach for learning end-to-end reconstruction operators for ill-posed inverse problems. The proposed method combines the classical variational framework with iterative unrolling, which essentially seeks to minimize a weighted combination of the expected distortion in the measurement space and the Wasserstein-1 distance between the distributions of the reconstruction and ground-truth. More specifically, the regularizer in the variational setting is parametrized by a deep neural network and learned simultaneously with the unrolled reconstruction operator. The variational problem is then initialized with the reconstruction of the unrolled operator and solved iteratively till convergence. Notably, it takes significantly fewer iterations to converge, thanks to the excellent initialization obtained via the unrolled operator. The resulting approach combines the computational efficiency of end-to-end unrolled reconstruction with the well-posedness and noise-stability guarantees of the variational setting. Moreover, we demonstrate with the example of X-ray computed tomography (CT) that our approach outperforms state-of-the-art unsupervised methods, and that it outperforms or is on par with state-of-the-art supervised learned reconstruction approaches.