An accurate method for direct dual‐energy calibration and decomposition

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 μm 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.