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

翻译：端到端（E2E）展开式优化框架在磁共振（MR）图像恢复中展现出巨大潜力，但训练过程中面临内存占用过高的问题。此外，这类确定性方法无法提供从后验分布中采样的能力。本文提出了一种内存高效的后验分布端到端学习方法。我们将该分布表示为数据一致性驱动的似然项与先验能量模型的组合，其中先验能量模型通过卷积神经网络（CNN）进行参数化。CNN权重通过最大似然优化以端到端方式从训练数据中学习。基于该学习模型，可通过最大后验（MAP）优化从欠采样测量值中恢复图像。此外，后验模型支持采样以生成重建结果的不确定性图。并行MR图像重建实验表明，我们的方法性能与内存密集型的端到端展开式算法相当，优于其内存高效的对应方法，并能提供不确定性图。本框架为三维及更高维度的MR图像重建铺平了道路。