The Microlocal Properties of the Local 3-D SPECT Operator

Abstract

We prove microlocal properties of a generalized Radon transform that integrates over lines in ${{\mathbb R}^3}$ with directions parallel to a fairly arbitrary curve on the sphere. This transform is the model for problems in slant-hole SPECT and conical-tilt electron microscopy, and our results characterize the microlocal mapping properties of the SPECT reconstruction operator developed and tested by Quinto, Bakhos, and Chung. We show that, in general, the added singularities (or artifacts) are increased as much as the singularities of the function we want to image. Using our microlocal results, we construct a differential operator such that the added singularities are, relatively, less strong than the singularities we want to image.