  Most existing methods for Magnetic Resonance Imaging (MRI) reconstruction
with deep learning use fully supervised training, which assumes that a high
signal-to-noise ratio (SNR), fully sampled dataset is available for training.
In many circumstances, however, such a dataset is highly impractical or even
technically infeasible to acquire. Recently, a number of self-supervised
methods for MR reconstruction have been proposed, which use sub-sampled data
only. However, the majority of such methods, such as Self-Supervised Learning
via Data Undersampling (SSDU), are susceptible to reconstruction errors arising
from noise in the measured data. In response, we propose Robust SSDU, which
provably recovers clean images from noisy, sub-sampled training data by
simultaneously estimating missing k-space samples and denoising the available
samples. Robust SSDU trains the reconstruction network to map from a further
noisy and sub-sampled version of the data to the original, singly noisy and
sub-sampled data, and applies an additive Noisier2Noise correction term at
inference. We also present a related method, Noiser2Full, that recovers clean
images when noisy, fully sampled data is available for training. Both proposed
methods are applicable to any network architecture, straight-forward to
implement and have similar computational cost to standard training. We evaluate
our methods on the multi-coil fastMRI brain dataset with a novel
denoising-specific architecture and find that it performs competitively with a
benchmark trained on clean, fully sampled data.
