Spectral computed tomography (CT) utilizes the absorption characteristics of X-rays of different energies by photon-counting detectors to perform "differential measurements" and obtains the X-ray attenuation characteristics of the object in different energy intervals. It not only allows the identification of materials with similar attenuation coefficients but also the qualitative and quantitative analysis of material properties of the scanned object (e.g., atomic number and electron density). Existing estimation models only consider the main attenuation effect, which is not precise enough for compound materials with complex compositions and makes the calculated equivalent atomic numbers and densities often have an error of more than 10%, thus preventing accurate estimation of compounds with similar equivalent atomic numbers. Since bronzes and their corrosion contain a variety of monomers and structurally complex compounds, and the equivalent atomic numbers of most materials are clustered in the interval of 20-30, the existing methods of estimating atomic numbers and densities could not meet the demand for accurate estimation of bronze materials. To this end, we propose a high-order fitting model and verify it through simulation experiments and actual data experiments, and finally realize the accurate estimation of the equivalent atomic number and density of the materials inside the bronzes.

The existing first-order linear models simplify the relevant physical effects by taking only the photoelectric effect and Compton scattering into consideration, which may not reflect the real physical process precisely. As the actual physical process is very complex, including the photoelectric effect, Compton scattering, and Rayleigh scattering, the relationship between the obtained attenuation coefficient and atomic number may not be a simple linear one. Based on the first-order model, we propose a higher-order fitting model to characterize the complex physical processes. Meanwhile, to verify the feasibility of the model, we design simulation experiments and actual data experiments and analyze the experimental results.

In the simulated experiments, we choose eight metal simulation materials with atomic numbers between 20 and 30 for model fitting (Table 1). Firstly, the body of materials is designed and SpekCalc software is adopted to simulate the energy spectrum from 0 to 3×10⁵V_p to obtain the projections at two energies of 3×10⁵V_p and 1.6×10⁵V_p. Then, the filtered back-projection algorithm is utilized to obtain the reconstructed images of the materials (Fig. 1), with the mean value of the 20×20 part in the center of each material taken as the attenuation coefficient μ. Four of them are leveraged as the base materials in the fitting, and the remaining four materials are for validation. The model estimates the equivalent atomic number of the four validated materials with a maximum error of 1.9% and an average error of 1.2% and estimates the density with a maximum error of 9% and an average error of 8% (Table 2). In the actual data experiments, we select seven major compounds in bronze patina and monomorphic copper totaling eight materials for model fitting (Table 3). By adopting photon counting detector-type spectral CT, one set of data is collected at every interval of 2×10⁴V_p, and a total of seven sets of data are obtained in the range of 1.6×10⁵V_p-2.8×10⁵V_p (Fig. 2). The mean value of the center cut layer of each material is taken as the attenuation coefficient μ. Four of the materials are employed as the base materials, which are validated with the remaining four materials, and the optimal model with the smallest estimation error is finally derived. The maximum error in the estimation of the equivalent atomic number for the four validated materials is 5% with an average error of 4%, and the maximum error in the estimation of the density is 10% with an average error of 4% (Table 4). The results of both simulation experiments and actual data experiments show that the third-order fitting model can estimate the equivalent atomic numbers and densities of the compounds contained in the bronzes relatively and accurately.

We analyze the existing atomic number and density estimation methods, which cannot meet the demand for accurate estimation of bronze materials. To address this problem, we first analyze the optimal method of calculating the equivalent atomic number and then construct a higher-order fitting model based on the data collected by spectral CT in multiple energy ranges to estimate the equivalent atomic number and density of the measured object for the bronze and its corrosion of the main components. Finally, this model is verified by simulation experiments and actual data experiments, and thus the accurate estimation of the equivalent atomic number and density of the materials inside the bronze is realized. In simulation experiments and fitting experiments on the actual data of the main components of bronze corrosion, the results show that the estimation average error of the proposed method is 3.67% for the atomic number and 3.75% for the density.