Cardinal H N, Fenster A
Department of Medical Biophysics, University of Western Ontario, London, Canada.
Med Phys. 1990 May-Jun;17(3):327-41. doi: 10.1118/1.596512.
We propose the use of conic and cubic surface equations (surfaces of second and third order) to directly approximate the dual-energy equations (the integral equations for the dual-energy log-signal functions, i.e., the negative logarithms of the relative detector signals, considered as functions of the basis-material component thicknesses of the object) and especially their inverses. These types of surface equations require a minimum number of calibration points, and their solutions are smooth, monotonic functions with the correct linear asymptotic behavior. The accuracy of this method is investigated and compared to that of conventional polynomial approximations, both for simulated and real calibration data, taken from two split-detector systems. These systems provide a more stringent test of our method than comparable dual-kVp systems, due to the greater nonlinearity of their log-signal and inverse functions. For these systems, we show that direct approximation of the inverse dual-energy equations using the simple eight-term rational form of the conic surface equation provides an extremely fast decomposition algorithm, which is accurate, robust in the presence of noise, and which can be calibrated with as few as 9 calibration points, or robustly calibrated, with a built-in accuracy check, using only 16 calibration points. Also, we show that extreme accuracy of approximation (to within less than 10(-6) in log-signal and 1 micron in material thickness) is theoretically attainable using the eighteen-term form of the cubic surface equation, which has a closed-form analytic solution. Finally, we consider the effects of noise on calibration accuracy, and derive simple formulas which relate the true and apparent root-mean-square (rms) accuracies. These formulas then allow the comparison of the true rms calibration accuracies of various surface approximations, considered as functions of the total calibration heat loading of the x-ray tube.
我们建议使用二次和三次曲面方程（二阶和三阶曲面）来直接近似双能方程（双能对数信号函数的积分方程，即相对探测器信号的负对数，被视为物体基础材料成分厚度的函数），尤其是它们的逆方程。这类曲面方程所需的校准点数量最少，其解是具有正确线性渐近行为的光滑单调函数。我们研究了该方法的准确性，并将其与传统多项式近似方法的准确性进行了比较，比较对象包括取自两个分离探测器系统的模拟校准数据和实际校准数据。由于其对数信号和逆函数的非线性更强，这些系统对我们的方法提供了比类似双千伏系统更严格的测试。对于这些系统，我们表明，使用二次曲面方程的简单八项有理形式直接近似双能逆方程，可提供一种极其快速的分解算法，该算法准确、在存在噪声时稳健，并且可以用少至9个校准点进行校准，或者通过内置的准确性检查，仅使用16个校准点就可进行稳健校准。此外，我们表明，使用具有封闭形式解析解的十八项三次曲面方程形式，理论上可实现极高的近似精度（对数信号误差小于10^(-6)，材料厚度误差小于1微米）。最后，我们考虑了噪声对校准精度的影响，并推导了将真实和表观均方根（rms）精度相关联的简单公式。这些公式随后允许比较各种曲面近似的真实均方根校准精度，将其视为X射线管总校准热负荷的函数。
