Influencing factors in x-ray spectral estimation of industrial CT using transmission measurements

Abstract The X-ray spectrum provides important information for many industrial CT tasks. Transmission measurement based methods have long been employed in X-ray spectral estimation. There are two advantages of this method. First, no photon counting detector is needed. Second, the detector response is included in the estimation. However, these methods are subject to influencing factors of various sources. This work uses simulation to obtain a thorough insight into these potential influencing factors. The influencing factors are simulated separately and the corresponding impact on the spectral estimation are quantified and compared. This work is intended to serve as a guide for the experimental design of the real world spectral estimation. By quantifying the impact of the potential influencing factors, we can subsequently prioritize them and decide which ones need further action.


Introduction
In industrial CT imaging, multiple important applications rely on the knowledge of the X-ray spectrum, such as dual-energy material decomposition and beam hardening correction [1].The desired spectra for such applications are integrated with the task detector energy absorbing response.The state-of-the-art spectral estimation uses photon counting detectors [2].There are two main drawbacks of this approach.First, it requires specific detectors and experimental setups.Moreover, even if the tube spectrum can be measured with photon counting detectors, the task detector response remains unknown.These drawbacks hinder the usage of photon counting detectors in routine measurements.One practical alternative method involves estimating the spectrum from transmission measurements of a dedicated phantom [3].This method consists of two steps: (1) acquiring the transmission data of a phantom that is comprised of multiple materials with known geometries, and (2) reconstructing the spectrum from the transmission data with algorithms, e.g.truncated singular value decomposition (TSVD) and expectation-maximization (EM).Because the transmission measurements are conducted with the task detector, the estimated spectrum is the product of the tube spectrum and the detector response, which can be applied directly to various industrial applications.Nevertheless, both the phantom preparation and the experimental measurement can be error-prone.In this paper, the various potential influencing factors of the phantom-based transmission measurement method and their influences on measurement accuracy are analysed with the aid of simulation.

Materials and methods 2.1 X-ray transmission model
Transmission measurement based spectral estimation is essentially an inverse problem.As a commonly used CT system has energy integrating detectors, its signal can be modelled with Beer-Lambert law [4] in the case that the spectral and object information are given: where I 0 and I are incident and transmitted photon intensities, respectively.p is the forward transmission intensity, µ(E) is the attenuation coefficient with energy E and l the path length of X-ray through the material.γ(E) is the detector efficiency, S(E) is the equivalent spectrum at the detector.W (E) denotes the weights of the energy intervals of the spectrum.The goal of spectral estimation is to acquire a set of W (E) from a given set of p, µ(E) and l.The outcome is therefore a 9th Conference on Industrial Computed Tomography, Padova, Italy (iCT 2019) normalised spectrum.After discretisation, Equation 1 can be formulated as: where M is the total number of the projection measurements in the set of p, N is the number of sampling intervals of the spectrum, A is a pre-calculated matrix of the product of the attenuation coefficients and path lengths.

EM spectral estimation
The projection measurement p follows a Poisson Distribution.The simple Poisson statistics have been used in emission CT reconstruction [6] with acceptable results.Under the assumption of simple Poisson distribution, the projection measurement subjects to: The log-likelihood objective function of obtaining projection measurement p with the normalised spectrum w can be formulated as: where c is a constant.In order to estimate w, the maximisation of the ln L(p, w) is needed.It can be realised with: The EM approach is applied to solve this maximisation problem.The iterative update equation follows: The projection measurement set p can be produced by a wedge phantom.When the material and geometry information of the phantom is known, A can be calculated using the path length and attenuation coefficients of the material.Only projection images are required.The initial input spectrum is normally acquired from a spectrum simulator with the same tube voltage and target material.

Wedge step arrangement
Two step wedges were designed for this study.The aluminium (Al) alloy wedge consists of 15 steps from 2mm to 30mm in increments of 2mm.The polystyrene (PS) wedge consists of 15 steps from 10mm to 150mm in increments of 10mm.Each phantom produces 15 different grey shades in accordance with the wedge step height.For the parallel beam geometry, the X-ray path length equals the step height at each grey shade.For cone-beam geometry, apart from one step can be roughly aligned with the X-ray source using axial and sagittal lasers, the X-ray path length of the step wedge in relation to the cone-beam geometry is not the real step height of the wedge.Cone-beam path length compensation can be conducted using ray-tracing.An alternative solution is to change the step arrangement of the wedge.As shown in Figure 1, the difference between the true path length and step height increases along the steps away from the centre.To minimise the difference, the highest step are aligned by the centre.and the next highest two steps at opposing sides of the highest step, and so forth (Figure 2).Each grey shade in the phantom image approximately covers a region of 100 × 100 pixels, but the edge pixels of some grey shades contain overlapped grey values.This is because in the cone-beam geometry, some edge pixels traverse more than one step.To avoid error, 50 × 50 pixels from each grey shade centre are acquired and averaged to yield a calibration array of 15 elements.Table 1 shows the summed path length error of each wedge step arrangement.Errors of both wedges reduce to about half of the original.Table 2 shows the difference of a spectral estimation under 120kV using different wedge step arrangements.It is where Ns and N s are the normalised true and estimated spectra, and n is the number of energy bins.A significant improvement with path length compensation can be observed in the conventional wedge.Our improved wedge, to the contrary, does not show that prominent difference.As the error caused by path length is negligible with the improved step arrangement, no path length compensation is needed.

Influencing factors
On a technical level, although the theory of EM method is well developed, spectral estimation procedures based on the Xray transmission measurements are error-prone.There are two main types of influencing factors: model sophistication and experimental preparation.The X-ray transmission model only takes the primary transmission into account.Noise and scattering are not considered.Therefore, a simulated projection image is superimposed with Gaussian noise to evaluate its effects.Scattering is another challenge, as large area detectors are commonly used in industrial CT.Monte Carlo approach can be used to simulate scattering in the projection image.
Regarding phantom preparation, one major concern is the inevitable manufacturing error in the step lengths.As the true step length can be measured with a caliper, and its uncertainty is normally within ±0.1mm, a random manufacturing error is added within this range for all the wedge steps.
Wedge material is another factor worth investigating.In theory, using single wedge material is preferable because it saves time for wedge alignment and scanning.However, double wedges, one aluminium and one plastic, were suggested by [7] for the stability of the estimation algorithm.Therefore, the number of wedge materials is treated as an influencing factor for evaluation.
Another problem is that most accessible metal materials in the workshops are alloys, which composition may deviate from the record.In the simulation experiment, the material of the wedge is Al alloy 2014, whereas the information of alloy 2024 is used to do the spectral estimation.This allows us to assess the effect of alloy composition mismatch.The last influencing factor is the target material of the X-ray tube.When using a simulated spectrum as the initial input of the estimation algorithm, the target material needs be known to determine the location of the characteristic peak.Although it is unlikely that this information is lacking, it is included in the evaluation to study the effects.The influencing factors and the comparison experiments to quantify their effects are summarised in Table 3. Unless explicitly stated in the table, both phantoms are applied in the experiments.Their axial projections are scanned individually.

Experimental setups
Four target spectra were estimated, including 120kV, 160kV, 200kV, and 240kV with an Al filter of 2mm.The initial input spectra of the spectral estimation algorithm were the spectra without a filter at the corresponding energy levels.The initial inputs were therefore significantly softer than the target spectra.The spectrum is first estimated with no influencing factor included.The results are compared in terms of RMSE.The RMSEs of the estimated spectra compared to the true spectra were calculated.In the results, the averaged RMSE of the four spectra under different kVs are presented.Figure 3 shows the spectral estimation of the 120kV spectrum.The energy bin size is 1kV.The simulation was conducted with aRTist software [5].In the subsequent estimations, the influencing factors were included individually.The spectral estimation algorithm was implemented in MATLAB R2015a on a laptop computer with 8GB RAM and a 2.5GHz quad-core processor.The computational time for 3000 iterations was less than 1s.The specifications of the experimental parameters can be found in Table 4.

Simulation studies on the spectral measurement influencing factors
The RMSE with each influencing factor under every energy level is plotted in Figure 4.The averaged RMSE with each influencing factor is shown in Table 5.The percentage change from the influencing factor free estimation is also presented.RMSEs of the wedge geometry, noise, and Al alloy composition are all under 1E − 3, which can be considered as negligible compared to other sources of error.If the characteristic peak location is missing in the initial spectral input, the estimation algorithm will not be able to recover this information.Instead, it yields a harder estimated bremsstrahlung spectrum to compensate the missing characteristic peaks (Figure 5a).Hence, it is pivotal to have the target material information to know the location of the characteristic peaks.From Figure 5b it can be observed that the estimated spectrum with scattering is harder than the true spectrum.Scattering has an effect of increasing the image intensity.Therefore, the algorithm expects a harder spectrum to compensate the increase in the image intensity, which is actually caused by scattering.In addition, as there is a discrepancy between the true primary transmission intensity and scattering contaminated initial intensity, in the worst case the estimated

Figure 1 :
Figure 1: (a) Parallel beam geometry.The X-ray path length is true to the step height.(b) Cone-beam geometry.There is a factor of up to 1/ cos(θ ) increase relative to the parallel beam path length.

Figure 2 :
Figure 2: The projections of the wedges.Both wedges provide 15 different grey shades as they have 15 steps.
(a) 120kV spectrum estimated without characteristic peaks in the initial input.(b) 120kV spectrum estimated with scattering.

Table 1 :
Summed path length errors with different wedge step arrangements.

Table 2 :
RMSE of a 120kV spectrum with different wedge step arrangements.

Table 3 :
Potential influencing factors and the way to take them into account.

Table 4 :
Specifications of the experiments.converge to the right solution at all.Scattering correction is therefore necessary.Using a single Al wedge showed significant effects on the spectral estimation.Therefore, it is recommended to use wedges of two materials with distinct attenuation coefficients to improve the estimation stability.

Table 5 :
Averaged RMSE with different influencing factors.