  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 unavailable in several
applications, restricting the applicability of the above methods. We introduce
a novel metric termed the ENsemble Stein's Unbiased Risk Estimate (ENSURE)
framework, which can be used 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 evaluate the expectation of the GSURE loss functions over the
sampling patterns to obtain the ENSURE loss function. We show that this loss is
an unbiased estimate for the true mean-square error, which offers a better
alternative to GSURE, which only offers an unbiased estimate for the projected
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.
