Cardinal H N, Fenster A
John P. Robarts Research Institute, Imaging Research Laboratories, London, Ontario, Canada.
Med Phys. 1991 Sep-Oct;18(5):867-79. doi: 10.1118/1.596744.
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.
在X射线系统性能分析中，对数信号函数，即相对探测器信号的负对数，以及类似定义的对数方差函数，至关重要。它们是物体厚度的平滑单调函数，对于非单能X射线源光谱而言是非线性的。如果我们假设将物体进行双能分解为两种基础材料，那么它们可以分别写为物体组成厚度(x,y)的解析函数f(x,y)和f*(x,y)。在本文中，我们通过解析方法展开这些函数的泰勒级数，证明它们在各处都收敛，并通过基础材料衰减系数的适当中心光谱矩对其系数进行参数化。然后我们展示了如何利用低阶矩以封闭形式构造平滑、单调的二阶（圆锥）表面函数，该函数在整个可行域内紧密逼近f(x,y)和f*(x,y)。还给出了一种基于使用基础材料衰减系数的适当渐近值来匹配这些函数渐近行为的简化构造方法。包含具有K边吸收光谱的图像成分，如碘，不费吹灰之力。将结果扩展到构造类似的（几乎精确的）三阶（立方）表面近似很简单。作为这种方法广泛适用性的一个例证，我们将分析扩展到为任意双能系统构造类似的逆（分解）函数近似，并研究其对于模型双千伏系统的数值精度。我们得出结论，这种扩展分析通过少量具有物理意义的参数提供了对系统行为的准确描述。这种参数化允许对系统行为有更深入的物理洞察同时简化其数学描述，并且类似地便于通过解析或数值方法分析各种成像性能度量。
