  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 is actually used for image reconstruction. This results in 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 to 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. As main results 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.
