Abelian and Non-Abelian X-ray transforms are examples of integral-geometric transforms with applications to X-ray Computerized Tomography and the imaging of magnetic fields inside of materials (Polarimetric Neutron Tomography).
(1). We will first discuss recent results on a sharp description of the mapping properties of the X-ray transform (and its associated normal operator I*I) on the Euclidean disk, associated with a special L2 topology on its co-domain.
(2). We will then focus on how to use this framework to show that attenuated X-ray transforms (with skew-hermitian attenuation matrix), more specifically their normal operators, satisfy similar mapping properties.
(3). Finally, I will discuss an important application of these results to the Bayesian inversion of the problem of reconstructing an attenuation matrix (or Higgs field) from its scattering data corrupted with additive Gaussian noise. Specifically, I will discuss a Bernstein-VonMises theorem on the ‘local asymptotic normality’ of the posterior distribution as the number of measurement points tends to infinity, useful for uncertainty quantification purposes. Numerical illustrations will be given.