Data Consistency Conditions for Cone-Beam Projections on a Circular Trajectory

摘要

The well-known Helgason-Ludwig data consistency conditions (DCCs) for parallel projections of a two-dimensional function take the form of homogeneous polynomials in cos φ and sin φ, where φ is the angle of the parallel projection. In this note, we establish necessary DCCs for cone-beam (CB) projections of a three-dimensional function taken along a circular trajectory. Our DCCs take the form of homogeneous polynomials in cos λ and sin λ, where λ is the angular position of the CB projection. This trigonometric polynomial format for the DCCs is particularly convenient for medical imaging applications, and these new DCCs for the standard CB geometry will potentially lead to new DCC applications in X-ray computed tomography (CT) and pinhole single photon emission computed tomography (SPECT) where CB projections are measured. If, as is usually the case, the object intersects the trajectory plane, then singularities appear in the DCC formulas. We describe how to interpret and handle these singularities as generalized functions. Numerical simulations are provided for illustration.