Analytic Inversion of a Conical Radon Transform Arising in Application of Compton Cameras on the Cylinder

Abstract

Single photon emission computed tomography (SPECT) is a well-established clinical tool for functional imaging. A limitation of current SPECT systems is the use of mechanical collimation, where only a small fraction of the emitted photons are actually used for image reconstruction. This results in a large noise level and finally in a limited spatial resolution. In order to decrease the noise level and to increase the imaging resolution, Compton cameras have been proposed as an alternative to mechanical collimators. Image reconstruction in SPECT with Compton cameras yields the problem of recovering a marker distribution from integrals over conical surfaces. Due to this and other applications, such conical Radon transforms recently got significant attention. In the current paper we consider the case where the cones of integration have vertices on a circular cylinder and axis pointing to the symmetry axis of the cylinder. Our setup does not use all emitted photons but a much larger fraction than systems based on mechanical collimation. Further, it may be simpler to be fabricated than a Compton camera system collecting full five-dimensional data. As main theoretical results in this paper we derive analytic reconstruction methods for the considered transform. We also investigate the V-line transform with vertices on a circle and symmetry axis orthogonal to the circle, which arises in the special case where the absorber distribution is located in a horizontal plane.