  We consider the problem of reconstructing the features of a weak anisotropic
background potential by the trajectories of vortex dipoles in a nonlinear
Gross-Pitaevskii equation. At leading order, the dynamics of vortex dipoles are
given by a Hamiltonian system. If the background potential is sufficiently
smooth and flat, the background can be reconstructed using ideas from the
boundary and the lens rigidity problems. We prove that reconstructions are
unique, derive an approximate reconstruction formula, and present numerical
examples.
