  Optimal transport problems pose many challenges when considering their
numerical treatment. We investigate the solution of a PDE-constrained
optimisation problem subject to a particular transport equation arising from
the modelling of image metamorphosis. We present the nonlinear optimisation
problem, and discuss the discretisation and treatment of the nonlinearity via a
Gauss--Newton scheme. We then derive preconditioners that can be used to solve
the linear systems at the heart of the (Gauss--)Newton method. With the optical
flow in mind, we further propose the reduction of dimensionality by choosing a
radial basis function discretisation that uses the centres of superpixels as
the collocation points. Again, we derive suitable preconditioners that can be
used for this formulation.

Numerical Analysis