Velocimetry and the aperture problem for 2D incompressible flows*

The inference of velocity fields from 2D movies evolving conserved scalars (optical flow) is fundamentally ambiguous due to the well-known “aperture problem”: velocities along isocontours of the scalar are not visible. This may even corrupt the inference of velocity fields averaged at scales longer than the typical length scale of features in the scalar field, as in the barber-pole effect. However, for divergence-free flows, a stream-function formulation allows us to show that the "invisible velocity" vanishes in the surface average over any closed scalar isocontour. This error-free averaged velocity may be used as an “anchor” for a more reliable inference of the larger-scale velocity field, or to test model-based optical-flow schemes. We have also used the stream-function formulation to derive a new method of optical flow for divergence-free flows. We discuss the new algorithm, including details of discretization, boundary conditions, and image preprocessing that can significantly affect its performance. A simple implementation of the new method is shown to work well for a number of synthetic movies, and is also applied to a GPI movie of edge turbulence in NSTX.