The X-ray transform in 2 dimensions

Abstract: In tomography, one wishes to image an object by gathering data about its cross-sections. I will discuss the mathematics behind the simplest mathematical example, the X-ray transform in 2 dimensions, which, given a function on the plane, associates to every line in the plane the integral of the function over that line. I will discuss a quantitative way to recover a function given its X-ray transform data using the Fourier transform, as well as more qualitative and geometric ways to detect singularities of the function given the singularities of its X-ray transform. Time permitting, I will discuss a phenomenon where artificial lines may be introduced in the reconstructed image, and how one can mathematically account for such artifacts.