Physics-informed compressed sensing: reconstruction of magnetic resonance velocimetry signals as an inverse Navier-Stokes problem

Magnetic resonance velocimetry (MRV) can measure all three components of a time varying velocity field but as the spatial resolution is increased the measurements become increasingly noisy. To acquire velocity images of acceptable signal-to-noise ratio, repeated scans are required, leading to long acquisition times. We present an algorithm that is capable of reconstructing magnetic resonance velocimetry signals from a single scan, by formulating a Bayesian inverse Navier--Stokes problem for the unknown velocity field. In this way we can infer the most likely boundaries of the flow, the boundary conditions, the viscosity, and the reconstructed velocity field, and estimate the uncertainty in the prediction. Our physics-based approach does not only provide a way to reconstruct the MRV signal, but it can furthermore infer hidden flow quantities such as the pressure and the wall shear stress, which are otherwise difficult to measure.