  We propose a novel learning-based framework for image reconstruction
particularly designed for training without ground truth data, which has three
major building blocks: energy-based learning, a patch-based Wasserstein loss
functional, and shared prior learning. In energy-based learning, the parameters
of an energy functional composed of a learned data fidelity term and a
data-driven regularizer are computed in a mean-field optimal control problem.
In the absence of ground truth data, we change the loss functional to a
patch-based Wasserstein functional, in which local statistics of the output
images are compared to uncorrupted reference patches. Finally, in shared prior
learning, both aforementioned optimal control problems are optimized
simultaneously with shared learned parameters of the regularizer to further
enhance unsupervised image reconstruction. We derive several time
discretization schemes of the gradient flow and verify their consistency in
terms of Mosco convergence. In numerous numerical experiments, we demonstrate
that the proposed method generates state-of-the-art results for various image
reconstruction applications--even if no ground truth images are available for
training.
