We review recent progress made by the author on some inverse problems involving
geodesic X-ray transforms on Riemannian surfaces with boundary.
We are concerned with the reconstruction of functions, or more generally, of
symmetric solenoidal tensor fields from knowledge of their X-Ray transform.
Recalling some results known in the simple case (Fredholm equations for functions
and solenoidal vector fields, s-injectivity of the ray transform for tensors of
any order), we then explain how to reconstruct other sections of certain bundles
(k-differentials for k an integer), which in some cases coincide with solenoidal tensor fields,
from knowledge of their ray transform. Such reconstruction formulas take the form
of Fredholm equations when the metric is simple. Furthermore, the error is
proved to be a contraction when the gaussian curvature is small in C^1 norm, in which
case the unknowns can be reconstructed via Neumann series.
Second, we present numerical implementation of these formulas. We observe that,
while the borderline cases where the error operators cease to be contractions are not
well known quantitatively, numerics indicate that, on the examples treated, the Neumann
series converges for a family of metrics that is arbitrarily close to non-simple.