Mapping properties of X-ray transforms near convex boundaries (Joint with IP seminar)

On a Riemannian manifold with boundary, the X-ray transform
integrates a function or a tensor field
along all geodesics through the manifold. The reconstruction of the
integrand of interest from its X-ray transform
is the basis of important inverse problems with applications to
seismology and medical imaging.

The inversion of the X-ray transform is often done by inverting the
normal operator (composition of the X-ray
transform and its adjoint, the "backprojection" operator). This
requires the design of appropriate Sobolev scales or
Frechet spaces over the manifold and the manifold of its geodesics,
compatible with an appropriate formulation of
forward and backward mapping properties for the X-ray transform, the
backprojection operator, and their composites.
For example, a landmark result by Pestov and Uhlmann, paving the way
toward their 2005 result on the boundary rigidity
of simple surfaces, was the design of a good Frechet setting where the
backprojection operator is surjective.

In this talk, I will review recent results attempting to shed
additional light on the (forward and backward) mapping
properties of the X-ray transform and its normal operator(s) on
convex, non-trapping manifolds. I will discuss
recent joint works with Gabriel Paternain and Richard Nickl; Rafe
Mazzeo; Rohit Mishra and Joey Zou.