A Radon-type transform arising in Photoacoustic Tomography with circular detectors

Abstract

PAT is the best-known example of a hybrid imaging method. In this article, we define a Radon-type transform arising in a version of PAT that uses integrating circle detectors and describe how the Radon transform integrating over all circles with a fixed radius is determined from this Radon-type transform. Also, the inversion of the Radon transform integrating over all circles is provided. Various configuration of detectors have their own geometry problems. Here we consider three situations: when the centers of the detectors are located on a cylinder, on a plan, and on a sphere. This transform is similar to a toroidal transform, which maps a given function to its integrals over a set of tori. We also study this mathematically similar object.