Physics-Constrained Bayesian Inference of Magnetic Resonance Velocimetry Data

Abstract

Magnetic resonance velocimetry provides quantitative three-component three-dimensional velocity data but has poor spatial resolution, poor signal-to-noise ratio, and prohibitively long acquisition times for many applications. When combined with physical knowledge such as conservation of mass, the Navier–Stokes equations, and a constitutive model for the viscosity, however, this data's high information content can generate superresolved fields that include unmeasured variables such as the pressure and wall shear stress. This combination is achieved by asserting a physical model, creating a well-posed Bayesian inverse problem with a small search space, and solving it with gradient-based optimization accelerated with adjoint methods. Different physical models can be proposed and the likelihood of each model can be calculated, given the data. With a similar method, the information content of the data can be calculated, given each model. This careful probabilistic representation of uncertainty allows models to be ranked, data quality to be assessed, and future experiments to be designed. We outline the history and principles of this method, provide two toy problems, and show three applications to magnetic resonance velocimetry data.