  We explain the main concepts centered around Sharafutdinov's ray transform,
its kernel, and the extent to which it can be inverted. It is shown how the ray
transform emerges naturally in any attempt to reconstruct optical and stress
tensors within a photoelastic medium from measurements on the state of
polarization of light beams passing through the strained medium. The problem of
reconstruction of stress tensors is crucially related to the fact that the ray
transform has a nontrivial kernel; the latter is described by a theorem for
which we provide a new proof which is simpler and shorter as in Sharafutdinov's
original work, as we limit our scope to tensors which are relevant to
Photoelasticity. We explain how the kernel of the ray transform is related to
the decomposition of tensor fields into longitudinal and transverse components.
The merits of the ray transform as a tool for tensor reconstruction are studied
by walking through an explicit example of reconstructing the
$\sigma_{33}$-component of the stress tensor in a cylindrical photoelastic
specimen. In order to make the paper self-contained we provide a derivation of
the basic equations of Integrated Photoelasticity which describe how the
presence of stress within a photoelastic medium influences the passage of
polarized light through the material.
