  In this paper we analyse the stability of the system of partial differential
equations modelling scalar image velocimetry. We first revisit a successful
numerical technique to reconstruct velocity vectors ${u}$ from images of a
passive scalar field $\psi$ by minimising a cost functional, that penalises the
difference between the reconstructed scalar field $\phi$ and the measured
scalar field $\psi$, under the constraint that $\phi$ is advected by the
reconstructed velocity field ${u}$, which again is governed by the
Navier-Stokes equations. We investigate the stability of the reconstruction by
applying this method to synthetic scalar fields in two-dimensional turbulence,
that are generated by numerical simulation. Then we present a mathematical
analysis of the nonlinear coupled problem and prove that, in the two
dimensional case, smooth solutions of the Navier-Stokes equations are uniquely
determined by the measured scalar field. We also prove a conditional stability
estimate showing that the map from the measured scalar field $\psi$ to the
reconstructed velocity field $u$, on any interior subset, is H\"older
continuous.
