D-symmetric density functions in tomographic image reconstruction
By: Rolf Clackdoyle, Laboratoire TIMC-IMAG (Techniques de l’Ingénierie Médicale
et de la Complexité - Informatique, Mathématiques et Applications), Université Grenoble Alpes, France
A three-dimensional density function (or "object function") generates 2D parallel projection images when irradiated by an x-ray source far from the object ("at infinity"). These projections are parametrized by the relative orientation of the x-ray source with respect to the object. Each orientation, i.e. each point on the unit sphere, gives rise to a different projection.
On the other hand, if the x-ray sources lie relatively close to the object, the resulting projections are called cone-beam (CB) projections rather than parallel projections. The CB projections are parametrized by the location of the x-ray source in space; each source location gives a different CB projection.
Twenty-five years ago, Edholm and co-workers demonstrated that for x-ray sources lying on a plane, the CB projections of a given object will all perfectly match the parallel projections of another suitably-defined object. The correspondence of the projections is determined by mapping each source location on the source plane to a specific orientation of the parallel projections.
This result appears to be nothing more than an amusing geometrical fact about parallel and CB projections. At the time, Edholm wrote, "it is unlikely that it will provide us with fundamentally new insights." However, image reconstruction theory from parallel projections is much simpler than from CB projections, and recently there has been substantial progress on data consistency conditions for CB (and fan-beam) projections heavily based upon this link between CB and parallel projections.
Here, we discuss the operation of D-symmetrization, which produces special "D-symmetric" density functions. These D-symmetric objects have the magical property that their parallel projections are the same as their CB projections. The parallel and CB projections match perfectly when viewed on a (possibly virtual) detector that is a distance D from the plane of the x-ray sources.