Analytic approximation of the log-signal and log-variance functions of x-ray imaging systems, with application to...

In the analysis of x-ray system performance, the log-signal function, or negative logarithm of the relative detector signal, and the analogously defined log-variance function, are of central importance. These are smooth, monotonic functions of object thickness, which are nonlinear for nonmonoenergetic x-ray source spectra. If we assume a dual-energy decomposition of the object into two basis materials, then they can be written as analytic functions f(x,y) and f*(x,y), respectively, of the component thicknesses (x,y) of the object. In this paper, we analytically develop the Taylor series of these functions, prove that they converge everywhere, and parametrize their coefficients via suitable central spectral moments of the basis-material attenuation coefficients. We then show how the lower-order moments can be used to construct, in closed form, smooth, monotonic, second-order (conic) surface functions which closely approximate f(x,y) and f*(x,y) over the entire feasible domain. A simplified construction, based on using appropriate asymptotic values of the basis-material attenuation coefficients to match the asymptotic behavior of these functions, is also given. The inclusion of image components with K-edge absorption spectra, such as iodine, is done without effort. Extension of the results to the construction of similar (virtually exact) third-order (cubic) surface approximations is straightforward. As an illustration of the broad applicability of this approach, we extend our analysis to the construction of similar approximations to the inverse (decomposition) functions for an arbitrary dual-energy system, and investigate their numerical accuracy for a model dual-kVp system. We conclude that this extended analysis provides an accurate description of the system behavior in terms of a small number of physically meaningful parameters. This parametrization permits greater physical insight into the system behavior, while at the same time simplifying its mathematical description, and similarly facilitates the analysis of various measures of imaging performance via either analytic or numerical methods.