  Reconstructing medical images from partial measurements is an important
inverse problem in Computed Tomography (CT) and Magnetic Resonance Imaging
(MRI). Existing solutions based on machine learning typically train a model to
directly map measurements to medical images, leveraging a training dataset of
paired images and measurements. These measurements are typically synthesized
from images using a fixed physical model of the measurement process, which
hinders the generalization capability of models to unknown measurement
processes. To address this issue, we propose a fully unsupervised technique for
inverse problem solving, leveraging the recently introduced score-based
generative models. Specifically, we first train a score-based generative model
on medical images to capture their prior distribution. Given measurements and a
physical model of the measurement process at test time, we introduce a sampling
method to reconstruct an image consistent with both the prior and the observed
measurements. Our method does not assume a fixed measurement process during
training, and can thus be flexibly adapted to different measurement processes
at test time. Empirically, we observe comparable or better performance to
supervised learning techniques in several medical imaging tasks in CT and MRI,
while demonstrating significantly better generalization to unknown measurement
processes.
