The Non-Abelian X-Ray Transform on Asymptotically Hyperbolic Spaces

The inverse problem of medical CT scans is to recover an image of the X-ray absorption coefficient ϕ inside a patient from absorption data collected after irradiating the patient with X-rays. We consider a generalization of this problem that turns the scalar X-ray absorption equation into a coupled linear system (where the coefficient ϕ is now a square matrix) which asks whether the coefficient ϕ is still recoverable. This finds application in a form of imaging called polarimetric neutron tomography. In addition, we consider the mentioned problem on a class of unbounded geometries called asymptotically hyperbolic spaces. We demonstrate that under certain regularity conditions the coefficient ϕ in this case is also recoverable. If the coefficient is of the form ϕ + A where A is a matrix that depends linearly on velocity (e.g. comes from a connection), then the problem isn’t solvable but the pair (ϕ, A) can be recovered up to a gauge.