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

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-fanbeam 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.

Cone-Beam Pair-Wise Data Consistency Conditions in Helical CT

“…Those zeroth order DCC, also referred to as integral invariants, have been known for quite some time and derived in many ways: in the Fourier domain in [5], from some homogeneous function and its symmetric group in [13] or from John's equation in [15]. In [24], they were all proven mathematically equivalent and it was later proven that if these pair-wise fan-beam DCC are verified, then the Grangeat-based pair-wise DCC will also be verified [25]. Diagnostic CT scanners generally have an x-ray source that follows a helical trajectory coupled with a cylindrical detector.…”

Automatic alignment for three-dimensional tomographic reconstruction

“…Local consistency conditions can also be defined in some cases, e.g. based on John's equation for cone-beam illumination [38]. Although these conditions can be evaluated in a region-of-interest tomography setting, they instead rely on the assumption that the angular sampling is sufficient to allow for the numerical computation of derivatives that approximate a continuum of projection angles.…”

Optimization based beam-hardening correction in CT under data integral invariant constraint

“…As demonstrated in our previous work (Tang et al 2011), the correction of bone-induced beam-hardening artifacts in CT can be accomplished by turning the calibration-oriented iterative operation into solving an optim ization problem, in which a quadratic objective function based on the Helgasson-Ludwig consistency constraint (HLCC) (Natterer 1986, Li and Wang 1995, Basu and Bresler 2000, Wang and Sze 2001, Yu and Wang 2007, Xu et al 2010, Clackdoyle 2013, Clackdoyle and Desbat 2015 is defined. In principle, the optimization based beam-hardening correction (BHC) using data consistency constraint (DCC) (John 1938, Patch 2002a, 2002b, Defrise et al 2003, Chen and Leng 2005, Leng et al 2007, Tang et al 2012 is sustained by projection data, and thus, its performance can be maintained over all scenarios while the tedious calibration process that is prone to errors can be avoided. However, the original HLCC is defined under parallel beam geometry whereas the projection data are acquired under fan beam geometry.…”