Technical note: Error analysis of material-decomposition-based effective atomic number quantification method

The effective atomic number (Z_eff ) is widely applied to the identification of unknown materials. One method to determine the Z_eff is material-decomposition-based spectral X-ray imaging. The method relies on certain approximations of the X-ray interaction cross-sections such as empirical model coefficients. The impact of such approximations on the accuracy of Z_eff quantification has not been fully investigated.

To perform an error analysis of the material-decomposition-based Z_eff quantification method and propose a coefficient calibration-in-groups method to improve the modeling accuracy and reduce the Z_eff quantification error.

The model of the material-decomposition-based Z_eff quantification method relies on the dependence of the interaction cross-sections  (σ_PE ) on the atomic number Z and corresponding coefficient, that is, $$ . In this work, all the data is from the National Institute of Standards and Technology (NIST) website. First, the coefficient m is calibrated through a logarithmic fitting method to quickly determine the m values for any certain energy and Z_eff ranges. Then materials including elements and common compounds with Z_eff ranging from 6-20 are selected as the objects whose effective atomic numbers are to be quantified. Different combinations of basis materials are applied to decompose the object materials and their quantification errors are analyzed. With the help of error analysis, the object materials are divided into high-error and low-error groups based on the decomposition coefficient ratio $$ , which is found to have a strong correlation with error, and their coefficients are calibrated in groups. Finally, the average errors of three m selection strategies: (1) using an empirical m value of 3.94, which is also considered a standard method; (2) using a single m value, which is calibrated through the logarithmic fitting method; (3) using different m values calibrated in groups, are calculated to test the effectiveness of our method.

The approximation of the X-ray interaction cross-section leads to certain errors in Z_eff quantification and the error distributions for different basis materials are different. The average errors for most basis material combinations (C(6)/Ca(20), C(6)/Al(13), Al(13)/Ca(20), C(6)/Ne(10), Na(11)/P(15)) are lower than 0.5, maintaining good average accuracy. While the average error for S(16)/Ca(20) is up to 0.8461, leading to more misjudgments on atomic number. Meanwhile, the error distribution regularity can be observed. The Pearson's correlation coefficients of absolute errors and decomposition coefficient ratios are 0.743, 0.8432 and 0.7126 for basis material combinations C(6)/Ca(20), C(6)/Al(13) and Al(13)/Ca(20), indicating a good correlation. The method using either empirical m value of 3.94 or single calibrated m value of 4.619 has relatively high average errors. The proposed method using different m values calibrated in groups has the lowest average errors 0.254, 0.203 and 0.169, which are reduced by 21.6%(0.07), 3.8%(0.008) and 62.9%(0.286) respectively compared with the standard method.

The error analysis demonstrates that the approximation of X-ray interaction cross-sections leads to inevitable errors, while also revealing certain error distribution regularity. The coefficient calibrated-in-groups method has better modeling accuracy and has effectively reduced the error compared with the standard method using a single empirical m value of 3.94.