Effective filtering and interpolation of 2D discrete velocity fields with Navier–Stokes equations

Abstract

We introduce a new variational technique to interpolate and filter a two-dimensional velocity vector field which is discretely sampled in a region of and sampled only once at a time, on a small time-interval . The main idea is to find a solution of the Navier–Stokes equations that is closest to a prescribed field in the sense that it minimizes the l2 norm of the difference between this solution and the target field. The minimization is performed on the initial vorticity by expanding it into radial basis functions of Gaussian type, with a fixed size expressed by a parameter ϵ. In addition, a penalty term with parameter ke is added to the minimizing functional in order to select a solution with a small kinetic energy. This additional term makes the minimizing functional strongly convex, and therefore ensures that the minimization problem is well-posed. The interplay between the parameters ke and ϵ effectively contributes to smoothing the discrete velocity field, as demonstrated by the numerical experiments on synthetic and real data.