  We examine the problem of Bragg-edge elastic strain tomography from energy
resolved neutron transmission imaging. A new approach is developed for
two-dimensional plane-stress and plane-strain systems whereby elastic strain
can be reconstructed from its Longitudinal Ray Transform (LRT) as two parts of
a Helmholtz decomposition based on the concept of an Airy stress potential. The
solenoidal component of this decomposition is reconstructed using an inversion
formula based on a tensor filtered back projection algorithm whereas the
potential part can be recovered using either Hooke's law or a finite element
model of the elastic system. The technique is demonstrated for two-dimensional
plane-stress systems in both simulation, and on real experimental data. We also
demonstrate that application of the standard scalar filtered back projection
algorithm to the LRT in these systems recovers the trace of the solenoidal
component of strain and we provide physical meaning for this quantity in the
case of 2D plane-stress and plane-strain systems.
