  In this paper we present a novel method for the numerical solution of linear
transport equations, which is based on ridgelets. Such equations arise for
instance in radiative transfer or in phase contrast imaging. Due to the fact
that ridgelet systems are well adapted to the structure of linear transport
operators, it can be shown that our scheme operates in optimal complexity, even
if line singularities are present in the solution.
  The key to this is showing that the system matrix (with diagonal
preconditioning) is uniformly well-conditioned and compressible -- the proof
for the latter represents the main part of the paper. We conclude with some
numerical experiments about $N$-term approximations and how they are recovered
by the solver, as well as localisation of singularities in the ridgelet frame.
