ENSURE: A General Approach for Unsupervised Training of Deep Image Reconstruction Algorithms

Image reconstruction using deep learning algorithms offers improved reconstruction quality and lower reconstruction time than classical compressed sensing and model-based algorithms. Unfortunately, clean and fully sampled ground-truth data to train the deep networks is often not available in several applications, restricting the applicability of the above methods. This work introduces the ENsemble Stein's Unbiased Risk Estimate (ENSURE) framework as a general approach to train deep image reconstruction algorithms without fully sampled and noise-free images. The proposed framework is the generalization of the classical SURE and GSURE formulation to the setting where the images are sampled by different measurement operators, chosen randomly from a set. We show that the ENSURE loss function, which only uses the measurement data, is an unbiased estimate for the true mean-square error. Our experiments show that the networks trained with this loss function can offer reconstructions comparable to the supervised setting. While we demonstrate this framework in the context of MR image recovery, the ENSURE framework is generally applicable to arbitrary inverse problems.