Denoising and super-resolution of Flow Data by Physics-Informed Markov Random Fields

This presentation will discuss the denoising and super-resolution of fluid flows using a Bayesian inverse formulation for recovering a flow field from sparse and noisy observations. We assume the noisy flow data follows a distribution modelled by a Markov random field biased towards satisfying the discrete approximations of the continuity and vorticity equations, corrupted by additive Gaussian white noise. We show that the resulting maximum a posteriori estimation requires solving a sparse nonlinear regularized least-squares problem, introduced as Optimizing a Discrete Loss (ODIL), whose solution is the super-resolved and denoised image. We derive a simple expression involving sparse finite-difference differentiation matrices for the Jacobian of the problem and solve it with the Gauss-Newton method. Contrary to physics-informed neural network based approaches, this method does not require training data. We present results on a variety of vascular flows motivated by denoising and super-resolution of MRI velocimetry. Within this framework, we showcase how the flow field and pressure recovery errors are influenced by the bias toward continuity and vorticity equations.