  Deep generative models have gained recent traction in accelerated MRI
reconstruction. Diffusion priors are particularly promising given their
representational fidelity. Instead of the target transformation from
undersampled to fully-sampled data required for MRI reconstruction, common
diffusion priors are trained to learn a task-agnostic transformation from an
asymptotic start-point of Gaussian noise onto the finite end-point of
fully-sampled data. During inference, data-consistency projections are injected
in between reverse diffusion steps to reach a compromise solution within the
span of both the trained diffusion prior and the imaging operator for an
accelerated MRI acquisition. Unfortunately, performance losses can occur due to
the discrepancy between target and learned transformations given the asymptotic
normality assumption in diffusion priors. To address this discrepancy, here we
introduce a novel Fourier-constrained diffusion bridge (FDB) for MRI
reconstruction that transforms between a finite start-point of moderately
undersampled data and an end-point of fully-sampled data. We derive the
theoretical formulation of FDB as a generalized diffusion process based on a
stochastic degradation operator that performs random spatial-frequency removal.
We propose an enhanced sampling algorithm with a learned correction term for
soft dealiasing across reverse diffusion steps. Demonstrations on brain MRI
indicate that FDB outperforms state-of-the-art methods including non-diffusion
and diffusion priors.
