Cone-beam consistency conditions for planar trajectories with parallel and perpendicular detectors

Abstract

Cone-beam (CB) projections provide a first-order model for x-ray imaging with an area detector. CB consistency conditions (CBCCs), also known as range conditions for the 3D divergent x-ray transform, are equations that express the redundant information in a collection of CB projections. For applications purposes, CBCCs are most suitably expressed in terms of detector coordinates. CBCCs are only known for a few geometrical configurations, which depend on the source and detector trajectories. Here we only consider source trajectories that lie in a plane, and detector orientations that are parallel to the trajectory plane, or perpendicular to it. The parallel detector is stationary, but the vertical detector rotates around the center of the circular trajectory. We unify and generalize the existing known CBCCs for planar trajectories, by creating an intermediate geometry consisting of a parallel, rotating detector, and we develop new CBCCs for this geometry. Our main result is a theorem on CBCCs for a perpendicular detector, which must necessarily move in response to movement of the source. We also provide a theorem for the more difficult situation of a perpendicular detector but without the restriction that the target object be on one side or the other of the trajectory plane. We present a simple numerical simulations for a toy calibration problem to provide an example application of the new CBCCs.