  Direct inversion of deflectometry data, such as proton radiographs and
shadowgraphs, is a well-posed problem with a unique solution for the transverse
deflection of each particle or ray if their trajectories do not cross. When
trajectories cross, there exists an infinite set of solutions. In proton
radiography direct inversion determines the line-integrated transverse Lorentz
force. We have tested five publicly available direct inversion routines with a
view to analyzing proton radiographs of cylindrical implosions on the OMEGA
laser; four Monge-Amp\`{e}re solvers [github.com/flash-center/PRaLine,
github.com/flash-center/PROBLEM,
github.com/mfkasim1/invert-shadowgraphy/tree/fast-inverse,
github.com/OxfordHED/proton-radiography-no-source], and a power-diagram method
[github.com/mfkasim1/invert-shadowgraphy]. Test problems were generated using
four field profiles, three cylindrical and one spherical, with varying field
amplitudes in proton-tracing routines. Two Monge-Amp\`{e}re solvers did not
run, the other two failed to reproduce radiographs when trajectories crossed,
although for one field profile the solutions only diverged from the original at
the boundary. The power-diagram method was successful even when proton
trajectories crossed, giving a solution that minimized proton deflection, but
failed for profiles that produced a single, sufficiently sharp peak. For cases
where trajectories do not cross, the Monge-Amp\`{e}re solvers have the
advantage of being considerably faster than the power-diagram routine, up to
$1000$ times in our tests. The test problems are provided in the supplementary
information in pradformat [github.com/phyzicist/pradformat].
