Memory-Efficient Deep End-to-End Posterior Network (Deepen) for Inverse Problems

End-to-End (E2E) unrolled optimization frameworks show promise for Magnetic Resonance (MR) image recovery, but suffer from high memory usage during training. In addition, these deterministic approaches do not offer opportunities for sampling from the posterior distribution. In this paper, we introduce a memory-efficient approach for E2E learning of the posterior distribution. We represent this distribution as the combination of a data-consistency-induced likelihood term and an energy model for the prior, parameterized by a Convolutional Neural Network (CNN). The CNN weights are learned from training data in an E2E fashion using maximum likelihood optimization. The learned model enables the recovery of images from undersampled measurements using the Maximum A Posteriori (MAP) optimization. In addition, the posterior model can be sampled to derive uncertainty maps about the reconstruction. Experiments on parallel MR image reconstruction show that our approach performs comparable to the memory-intensive E2E unrolled algorithm, performs better than its memory-efficient counterpart, and can provide uncertainty maps. Our framework paves the way towards MR image reconstruction in 3D and higher dimensions.