X-ray transform

Summary

The X-ray transform sends a function $f$ on $\R^n$ to the function that gives the integral of $f$ along each line. In the plane it is the Radon transform, up to the parametrization of lines. In higher dimensions lines are smaller than hyperplanes, and the two transforms differ. It models what X-ray tomography measures in three dimensions, where the data are line integrals. It is injective on integrable functions, and in dimension $n \ge 3$ the data are overdetermined, since the set of lines has dimension $2n - 2 > n$.