Introduction

Since the 1970s, computed tomography (CT) has been developed into a powerful tool for industrial non-destructive testing (NDT), in which the structural integrity of assembled devices is assessed and the quality of manufactured components is inspected. The main focus of NDT applications is qualitative feature recognition (or visualization), such as flaw detection, morphological characterization of internal structures, and material wear inspection. With the development of state-of-the-art precision engineering (e.g., additive manufacturing [1, 2]), there is an increasing demand for accurate dimensional measurements to assure manufacturing quality (e.g., wall thickness, pore size, and geometry tolerance verification) without destroying the part. The CT technique plays a significant role in dimensional metrology [3-6], especially for the dimensional determination of the internal or hidden structures of a component.

In industrial CT systems, X-rays are generally produced by the so-called bremsstrahlung, which is a process in which high-energy electrons are braked by an anode and a continuous spectrum is produced. Because the X-ray absorption of a material is energy dependent, beam-hardening artifacts occur when a polychromatic spectrum is used for CT reconstruction. Owing to the influence of beam-hardening artifacts, serious errors occur in dimensional measurements [7-11], which hamper accurate quantitative analysis and inspection. Since the invention of CT techniques, several methods have been developed to correct this influence. However, those correction algorithms have their own limitations, such as the reduction of photon flux by pre-filtering, the requirement of prior knowledge of the imaging materials or spectrum [8, 10, 12-22], two scans at preferably non-overlapping spectra [23, 24], and computational complexity [21, 22, 25-28]. To resolve the beam-hardening effect fundamentally, it is necessary to develop a monochromatic X-ray source; in particular, a monochromatic gamma-ray source with strong penetration power for industrial high-Z material imaging that is characterized by high beam quality, a suitable footprint, and a moderate cost.

With the development of high-brightness electron beams and high-power lasers, the inverse Compton scattering (ICS, also called Thomson scattering in the low-energy region [29, 30]) gamma-ray source has been developed as an excellent light source for advanced gamma-ray imaging because it can provide quasi-monochromatic, continuously energy-tunable, small-focal-spot, and high-brightness gamma rays [31-33]. Furthermore, the footprint of this type of light source is at the room or container scale, making it flexible for clinical or industrial NDT applications. Therefore, ICS light sources have recently attracted significant attention in the field of advanced radiation imaging, including monochromatic and spectral imaging [34-37], phase contrast imaging [38-42], polarization-based imaging [43, 44], and nuclear resonance fluorescence imaging [45-49]. For an ICS light source, the gamma-ray photon energy Eγ in a head-on interaction geometry between the relativistic electron and laser, and considering the relativistic approximation (, , and ), can be described as [50] (1)where E_l is the laser photon energy, γ is the relativistic Lorentz factor, a₀ is the magnitude of the normalized laser vector potential, which can be neglected in the linear scattering process (), and θ is the detection angle between the electron moving direction and gamma-ray observation direction. The gamma-ray energy is correlated with the detection angle. The detection angle must be confined by a collimator to obtain quasi-monochromatic gamma rays. Therefore, the field of view (FOV) for imaging is very small, especially in the high-energy region where γ is very large and Eγ is more sensitive to changes in θ. For example, for gamma rays with a peak energy of 2 MeV generated using a laser with a wavelength of 800 nm, the typical FOV within which the gamma-ray bandwidth is determined only by the beam parameters (intrinsic bandwidth, not influenced by the energy-angle correlation) is ~5 mm or or less at 10 m downstream of the interaction point (IP) between the relativistic electron and laser. Hence, the translation + rotation scan scheme of first-generation CT must be used to realize CT for centimeter-scale objects, which has been proven to be time consuming.

To improve the imaging efficiency of gamma-ray CT based on ICS light sources, the imaging FOV must be increased without reducing the beam intensity, which means that special efforts must be made to correct the influence of the intrinsic energy-angle correlation spectrum of this type of light source on CT reconstruction. In this study, a dual-energy scan scheme is proposed to realize large-FOV gamma-ray CT by taking full advantage of the straightforward energy tunability of ICS light sources. The feasibility of this scheme was investigated based on a typical ICS light source using Monte Carlo simulations.

Methods

2.1

Principle of gamma-ray dual-energy scan scheme

According to the interaction mechanism between X-ray photons and materials, the linear attenuation coefficient μ(E) of a material can be decomposed into three parts: (2)where f_PE(E), f_CS(E), and f_PP(E) are the energy E dependencies of the photoelectric, Compton scattering, and pair production effects, respectively. The decomposition coefficients a_PE, a_CS, and a_PP are related with the material properties: (3a) (3b) (3c)where ρ, Z, and A denote the mass density, atomic number, and atomic weight, respectively, K₁, K₂, and K₃ are constants, and .

In the low-energy region (E<1.022 MeV), the pair production process cannot occur. Hence, only the photoelectric and Compton scattering terms in Eq. (2) contribute to the linear attenuation coefficient μ(E). For the photoelectric term, f_PE(E) is empirically considered as 1/E³ when no electron-shell discontinuity occurs in the given energy region, which restricts the model to application to very-high-Z materials. For the Compton scattering term, f_CS(E) is explicitly described by the well-known Klein–Nishina formula, which assumes the unbound electron and at-rest conditions, neglecting the incoherent scattering function and Doppler broadening, respectively. Using dual-energy CT scans, the projections of a_PE and a_CS can be calculated, based on which the spatial distribution of a_PE and a_CS can be reconstructed. Thus, a monochromatic CT image of the imaging object can be obtained using Eq. (2) by neglecting the pair production term. This is the principle of the dual-energy method for beam-hardening correction in the diagnostic X-ray energy region [23, 24], which has been applied to energy-angle correlation correction for monochromatic CT based on an ICS light source in the keV energy region [35].

In the high-energy region (e.g., hundreds of keV to several MeVs), the contribution of the photoelectric effect to the linear attenuation coefficient μ(E) is usually negligible compared with those of the Compton scattering and pair production terms. In principle, the spatial distributions of a_CS and a_PP for an imaging object can also be reconstructed using a dual-energy scan. However, no explicit expression for f_PP(E) exists owing to the complex energy dependence of the pair production effect. In this case, the basis material decomposition model is often adopted [51].

In the basis material decomposition model, an arbitrary material can be considered as a mixture of two basis materials and its linear attenuation coefficient μ(E) can be decomposed as (4)where μ_BM,1(E) and μ_BM,2(E) are the linear attenuation coefficients of the two basis materials, whose values at any gamma-ray energy are known, and c₁ and c₂ are the decomposition coefficients. When a dual-energy scan is performed, the projections of the linear attenuation coefficient at both high and low gamma-ray energies can be obtained as follows: (5a) (5b)where P_H and P_L are the measured projections at the high-energy spectrum (SpecH) and low-energy spectrum (SpecL), respectively, denotes a position vector in Euclidean space, and S(E) is the intensity-normalized energy spectrum [ and ], which can be approximated by a Gaussian distribution at any local position of the gamma-ray beam profile for an ICS light source. Because the local bandwidth of an ICS light source is very narrow (the typical rms value is several percent), compared with the energy dependence of μ(E) in the high-energy region, S(E) can be treated as a Dirac function δ(E). Thus, Eq. (5) can be further simplified as (6a) (6b)where E_H and E_L are the peak energies in the high- and low-energy spectra, respectively. By combining Eqs. (4) and (6), the following linear equation system can be established: (7)with (8a) (8b)The projections C₁ and C₂ can be calculated by solving the linear equation system of Eq. (7). Thus, the projection P(E) of the linear attenuation coefficient at an arbitrary energy E can be obtained as follows: (9)and the monochromatic CT image of the imaging object at energy E can be obtained by reconstructing P(E).

The gamma-ray energy of an ICS light source can be easily adjusted by changing either the electron energy or interaction angle between the electron and laser. The influence of the energy-angle correlation spectrum encountered in a large-FOV imaging geometry using ICS light sources on CT reconstruction can be easily resolved using a dual-energy scan. Although an energy-angle correlation exists in the scan of each gamma-ray peak energy, the local quasi-monochromaticity [δ(E) approximation] at an arbitrary detection angle of the FOV (or an arbitrary detection position within the gamma-ray beam profile) can be satisfied (see Sect. 2.2 for the simulated gamma-ray spectra). Hence, two projections at different gamma-ray energies (E_H and E_L) can be obtained at any detection angle of the FOV. The two gamma-ray energies E_H and E_L can be calculated using Eq. (1) when the gamma-ray peak energies (E_H,max and E_L,max) are determined, based on which the values of μ_BM,1 and μ_BM,2 can be obtained at the two gamma-ray energies E_H and E_L. By solving Eq. (7) and using the composition relation in Eq. (9), the projections of the imaging object at any detection angle of the FOV can be corrected to the same gamma-ray energy, as illustrated in Fig. 1. Therefore, a monochromatic CT image of the imaging object can be reconstructed.

Fig. 1Full-screen View

(Color online) Dual-energy scan scheme for obtaining monochromatic projections of the imaging object at different detector pixels Di and Dj

2.2

Monte Carlo simulation

Monte Carlo (MC) simulations were performed using the Geant4 toolkit [52] to demonstrate the feasibility of the large-FOV gamma-ray CT scheme. The imaging geometry was modeled based on the imaging system constructed for the very compact ICS gamma-ray source (VIGAS) [53-55] that is under construction at Tsinghua University. A fan-beam geometry was adopted, and the imaging layout is illustrated in Fig. 2. Gamma-ray photons were generated from the IP (gamma-ray spot size 10 μm, rms) and propagated R₁ = 10 m to the imaging object. The spectrum of the gamma-ray photons, as shown in Fig. 3, was generated from CAIN [56], which is the most commonly used MC code for ICS simulations, by considering the practical beam parameters of the VIGAS. The gamma-ray peak energies E_L,max and E_H,max of the dual-energy scan were 2 and 4 MeV, respectively. The imaging object was scanned separately by the two spectra. At the VIGAS, gamma rays with peak energies of 2 and 4 MeV were generated by the interaction of 290 MeV electrons and lasers with wavelengths of 800 and 400 nm, respectively. The CAIN simulation results of the gamma-ray spectrum were loaded into Geant4 for further imaging simulations. Both the energy-angle correlation and energy spread of the spectrum were considered in the MC simulation. To acquire the projection data of the imaging object, an ideal transmission detector with a pixel size of 0.2 mm was placed R₂ = 0.5 m downstream of the imaging object. The photon intensity of the generated gamma rays of an ICS light source decreases with the detection angle, as shown in Fig. 3(a). To guarantee the necessary photon intensity at the boundary of the gamma-ray beam profile, the gamma-ray collection angle θ_c was selected as 1/γ (~1.76 mrad), which corresponds to an FOV of ~3.5 cm at the position of the imaging object. Compared with the quasi-monochromatic case (typical FOV of ~5 mm or less at 2 MeV), the FOV increases more than seven-fold in the dual-energy scan scheme.

Fig. 2Full-screen View

(Color online) Imaging layout for large-FOV gamma-ray CT based on an ICS light source (not to scale)

Fig. 3Full-screen View

(Color online) Simulation spectra of the VIGAS using the CAIN software: (a) angular distribution of the generated gamma rays, (b) local rms energy spreads at different detection angles (or different detection positions within the gamma-ray beam profile), and normalized gamma-ray spectra at different collection angles θ_c for MeV (c) and 4 MeV (d), respectively. The pink dotted lines in (a) are the theoretical results obtained from Eq. (1). Although the gamma-ray energy spread increases with the detection angle, the spectrum is still sufficiently narrow that it can be considered to be quasi-monochromatic at different detection angles

The imaging object was an aluminum (Al) cylinder with a diameter of 3.0 cm, inside which there were four cylindrical columns with a diameter of 8.0 mm. The four inner columns were made of iron (Fe), copper (Cu), water (H₂O), and silicon (Si), respectively. For the CT scan of the imaging object, the rotation axis was the central axis of the Al cylinder, located at the center of the gamma-ray beam.

In the MC simulation, 360 projections that were evenly distributed in the angular range of 0–360° were acquired for each CT scan. In each projection, 9×10⁷ gamma-ray photons were simulated to balance the statistical error and time cost.

Results and discussion

The basis materials selected for the CT reconstruction of the imaging object were iron (Fe) and carbon (C), the linear attenuation coefficients of which can be found in the National Institute of Standards and Technology (NIST) [57]. The CT reconstruction results of the imaging object using the original energy-angle correlation spectrum with a gamma-ray peak energy of 2 MeV and the monochromatic CT reconstruction results at a gamma-ray energy of 2 MeV using the dual-energy scan scheme are illustrated in Figs. 4(a) and (b), respectively. To avoid the mosaic phenomenon in the CT reconstruction image caused by undersampling and to obtain a smooth reconstruction result, 512 × 512 pixels were selected in the CT reconstruction region, which led to a virtual resolution that was much higher than the practical resolution of the transmission detector. The horizontal and vertical center profiles of the reconstruction results are shown in Figs. 4(c) and (d), respectively, to provide a quantitative comparison. Similar CT reconstruction results at a gamma-ray energy of 4 MeV are illustrated in Fig. 5. Owing to the influence of the energy-angle correlation spectra, obvious cupping artifacts appeared in the reconstructed images. Using the dual-energy scan scheme described in Sect. 2.1, the cupping artifacts could be perfectly corrected, and the reconstructed linear attenuation coefficient of the imaging object agreed well with its theoretical value, as shown in Figs. 4(c), 4(d), 5(c), and 5(d). Considering that gamma-ray energy is correlated with the detection angle, different parts of the imaging object are irradiated by gamma rays with different energies. The energy-modified linear attenuation coefficient of the imaging object, which yields the theoretical μ(E) of the imaging object with the gamma-ray energy E calculated based on the energy-angle correlation relation in Eq. (1), is also illustrated in Figs. 4(c), 4(d), 5(c), and 5(d) using black dotted lines. The energy-modified linear attenuation coefficient of the imaging object exhibited a similar variation tendency to the reconstruction result using the original energy-angle correlation spectrum, in which the linear attenuation coefficient for the same material increased with the radius. However, a quantitative comparison shows that the two results do not agree well, particularly around the center of the imaging object. Therefore, the single-spectrum correction using the energy-angle correlation information in Eq. (1) cannot reflect the practical reconstruction results.

Fig. 4Full-screen View

(Color online) CT reconstruction results of the imaging object: (a) direct reconstruction using the original energy-angle correlation spectrum at a gamma-ray peak energy of 2 MeV; (b) monochromatic reconstruction at a gamma-ray energy of 2 MeV using the dual-energy scan scheme; and (c) and (d) horizontal and vertical center profiles [white dotted lines in (a) and (b)] of the reconstruction images, respectively

Fig. 5Full-screen View

(Color online) CT reconstruction results of the imaging object: (a) direct reconstruction using the original energy-angle correlation spectrum at a gamma-ray peak energy of 4 MeV; (b) monochromatic reconstruction at a gamma-ray energy of 4 MeV using the dual-energy scan scheme; and (c) and (d) horizontal and vertical center profiles [white dotted lines in (a) and (b)] of the reconstruction images, respectively

The electron density and effective atomic number of the imaging object reconstructed using the pre- and post-processing methods are shown in Figs. 6 and 7, respectively. For both reconstruction methods, the reconstruction quality of the electron density was much better than that of the effective atomic number, which can be attributed to the division operation in Eq. (12b). Compared with the pre-processing method, there were serious artifacts around the boundaries of H₂O, Fe, and Cu in the Z_eff image reconstructed using the post-processing method.

Fig. 6Full-screen View

(Color online) Electron density and effective atomic number of the imaging object reconstructed using the pre-processing method: (a) ρ_e image; (b) and (c)horizontal and vertical center profiles of (a), respectively; (d) Z_eff image; and (e) and (f) horizontal and vertical center profiles of (d), respectively. The white dotted squares in (a) and (d) are ROIs selected for quantitative analysis of the reconstruction results

Fig. 7Full-screen View

(Color online) Electron density and effective atomic number of the imaging object reconstructed using the post-processing method: (a) ρ_e image; (b) and (c) horizontal and vertical center profiles of (a), respectively; (d) Z_eff image; and (e) and (f) horizontal and vertical center profiles of (d), respectively. The white dotted squares in (a) and (d) are ROIs selected for quantitative analysis of the reconstruction results

To evaluate the reconstruction quality of the two methods quantitatively, five regions of interest (ROIs) were selected, as indicated by the white dotted squares in Figs. 6(a), 6(d), 7(a), and 7(d). The ρ_e and Z_eff reconstruction results for the five materials are presented in Tables 1 and 2, respectively. Both the mean value and its standard error calculated over the ROIs are provided for the reconstructed values in Tables 1 and 2. In terms of ρ_e, the precision (standard error) of the five materials reconstructed using the post-processing method was much higher than the value using the pre-processing method. Meanwhile, the accuracy (relative error) of ρ_e reconstructed using the post-processing method was slightly higher than the value using the pre-processing method for all materials, except for Cu. In terms of Z_eff, the reconstruction precision using the post-processing method was slightly lower than that using the pre-processing method for all materials except for H₂O, and the reconstruction accuracy using the post-processing method was similar to that using the pre-processing method for all materials except for H₂O. Therefore, the post-processing method is more suitable for ρ_e reconstruction, whereas the pre-processing method is more suitable for Z_eff reconstruction. For moderate-Z materials (e.g., Fe and Cu), excellent reconstruction was obtained, with accuracies of less than 3% and 1.5% for ρ_e and Z_eff, respectively. For Fe, the relative errors of ρ_e and Z_eff were the smallest, which was expected because Fe is a basis material. For relatively low-Z materials (e.g., Al, Si, and H₂O), the lower ρ_e reconstruction accuracy may be attributed to the less accurate basis material decomposition of the linear attenuation coefficient. Basis material decomposition models with higher accuracy must be developed in future studies to improve the reconstruction accuracy of ρ_e. In addition, the gamma-ray energy selected for the dual-energy scan may be a reason for the high relative errors of the ρ_e and Z_eff reconstruction for these low-Z materials. Because gamma-ray energies of 2 and 4 MeV are too high for low-Z materials in the imaging object, the statistical error of the monochromatic CT reconstruction was relatively high, as shown in Figs. 4(d) and 5(d), which resulted in a high relative error in the ρ_e and Z_eff reconstruction. To reduce the relative errors of ρ_e and Z_eff caused by the selection of gamma-ray energy for the dual-energy scan, the photon number used for the MC simulation should be increased or the gamma-ray energy should be reduced. Although the pre-processing method for Z_eff reconstruction has advantages over the post-processing method in terms of boundary artifact suppression and high-precision reconstruction, the reconstruction precision of Z_eff owing to the division operation in Eq. (12b) was significantly lower than that of ρ_e. Therefore, effective reconstruction methods must be developed to improve the reconstruction precision of Z_eff.

Because the FOV using the dual-energy scan scheme is increased by more than seven-fold compared with the quasi-monochromatic scan (translation + rotation scheme), the imaging time using the dual-energy scan scheme (only rotation is required) when CT scans using different gamma-ray energies are carried out separately can be reduced by at least 3.5 times. However, dual-color gamma rays can be easily produced simultaneously for an ICS light source [58-60] using one of the following three schemes: (i) dual-color lasers interacting with the same electron beam, (ii) a single-color laser interacting with the same electron beam at different interaction angles by beam splitting, and (iii) dual-color electron beams interacting with the same laser. Using dual-color gamma rays, a dual-energy scan can be realized simultaneously by combining it with a layered transmission detector (the front layer for low-energy gamma-ray detection and the rear layer for high-energy gamma-ray detection, as in the case of early dual-energy CT [61, 62]). Therefore, the imaging time can be reduced further by 2 times. Another advantage of the dual-energy scan scheme compared with the quasi-monochromatic scan is that the ρ_e and Z_eff of the imaging object can be obtained simultaneously.

Conclusion

Quasi-monochromatic, continuously energy-tunable, and high-brightness gamma rays produced by ICS light sources can fundamentally resolve beam-hardening artifacts in CT reconstruction, which can improve the accuracy of quantitative analysis and inspection in CT-based dimensional metrology. An effective method for large-FOV imaging was developed to improve the imaging efficiency of gamma-ray CT using this type of light source. Using a dual-energy scan scheme, the influence of the intrinsic energy-angle correlation spectrum of ICS light sources on CT reconstruction was resolved, and a monochromatic CT image of the imaging object was accurately reconstructed. Furthermore, the electron density ρ_e and effective atomic number Z_eff of the imaging object were obtained. For ρ_e reconstruction, the post-processing method has an obvious advantage over the pre-processing method; however, for Z_eff reconstruction, the pre-processing method is preferred.