Affiliations: Department of Radiology and Imaging Sciences, Emory
University School of Medicine, Atlanta, GA, USA | Institute of Image Processing and Pattern Recognition,
Xi'an Jiaotong University, Xi'an, Shaanxi, China | School of Automation, Xi'an University of Posts and
Telecommunications, Xi'an, Shaanxi, China
Note: [] Corresponding author: Shaojie Tang, Department of Radiology and
Imaging Sciences, Emory University School of Medicine, Atlanta, GA 30322, USA.
E-mail: shaojie.tang@emory.edu
Abstract: 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.
