Singularities of the X-Ray Transform and Limited Data Tomography in $\mathbb{R}^2 $ and $\mathbb{R}^3 $

Given a function f, the author specifies the singularities of f that are visible in a stable way from limited X-ray tomographic data. This determines which singularities of f can be stably recovered from limited data and which cannot, no matter how good the inversion algorithm. Microlocal analysis is used to determine the relationship between the singularities of a function f and those of its X-ray transform. The results are applied to determine the singularities that are visible for limited angle tomography and the interior and exterior problems. The author also suggests a practical method to use this relationship to reconstruct singularities of f from limited data $Rf$. The X-ray transform with sources on a curve in $\mathbb{R}^3 $ is also analyzed.