The mathematical equivalence of consistency conditions in the divergent-beam computed tomography

In this paper, we discuss the mathematical equivalence among four consistency conditions in the divergent-beam computed tomography (CT). The first is the consistency condition derived by Levine et al. by degenerating the John's equation; the second is the integral invariant derived by Wei et al. using the symmetric group theory; the third is the so-called parallel-fan-beam Hilbert projection equality derived by Hamaker et al.; and the fourth is the fan-beam data consistency condition (FDCC) derived by Chen et al. using the complex analysis theory. Historically, most of these consistency conditions were derived by their corresponding authors using complicated mathematical strategies, which are usually not easy to be precisely understood by researchers with only a general engineering mathematical background. In this paper, we symmetrically re-derive all these consistency conditions using a friendly mathematical language. Based on theoretical derivation, it has been found that all these consistency conditions can be viewed as a necessary condition for the specific solution to John's equation. From the physical point of view, all these consistency conditions have been essentially expressed as a similar constraint on the projection data acquired with arbitrary two x-ray source points. Numerical simulations have been carried out to experimentally evaluate and verify their merits.