SINGULARITIES OF THE X-RAY TRANSFORM AND LIMITED DATA TOMOGRAPHY IN R(2) AND 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 R3 is also analyzed.