An indirect estimation of x-ray spectrum via convolutional neural network and transmission measurement.

Objective.In this work, we aim to propose an accurate and robust spectrum estimation method by synergistically combining x-ray imaging physics with a convolutional neural network (CNN).Approach.The approach relies on transmission measurements, and the estimated spectrum is formulated as a convolutional summation of a few model spectra generated using Monte Carlo simulation. The difference between the actual and estimated projections is utilized as the loss function to train the network. We contrasted this approach with the weighted sums of model spectra approach previously proposed. Comprehensive studies were performed to demonstrate the robustness and accuracy of the proposed approach in various scenarios.Main results.The results show the desirable accuracy of the CNN-based method for spectrum estimation. The ME and NRMSE were -0.021 keV and 3.04% for 80 kVp, and 0.006 keV and 4.44% for 100 kVp, superior to the previous approach. The robustness test and experimental study also demonstrated superior performances. The CNN-based approach yielded remarkably consistent results in phantoms with various material combinations, and the CNN-based approach was robust concerning spectrum generators and calibration phantoms.Significance. We proposed a method for estimating the real spectrum by integrating a deep learning model with real imaging physics. The results demonstrated that this method was accurate and robust in estimating the spectrum, and it is potentially helpful for broad x-ray imaging tasks.

Technical note: SpekPy Web-online x-ray spectrum calculations using an interface to the SpekPy toolkit.

Knowledge of the photon spectrum emitted from an x-ray tube is frequently needed in imaging and dosimetry contexts. As the spectrum characteristics are influenced by several parameters and routine measurement of a spectrum is often impractical, a variety of software programs have been developed over the decades for convenient calculations. SpekPy is a state-of-the-art software package containing several spectrum models, and was created to estimate photon spectra originating from x-ray tubes using a small set of input parameters (e.g., anode material, anode angle, tube potential, filtration, etc.). SpekPy is distributed as a Python toolkit and is available free of charge. The toolkit does, however, lack a graphical user interface and a user is required to write a Python script to make use of it. In this work this limitation is addressed by introducing a web application called SpekPy Web: a graphical user interface together with an application programmable interface (API). These developments both make the SpekPy spectrum models accessible to a broader set of users and increases the ease of use for existing users. SpekPy Web is hosted at: https://spekpy.smile.ki.se. The functionality of the software is demonstrated, using its API, by estimating first half-value layers (HVLs) for 15 standard beam qualities from the International Bureau of Weights and Measures (BIPM). The estimated HVLs were found to all be within 3.5% agreement when compared to experimental values, with an average calculation time of 2.5 s per spectrum. half-value-layer, software, x-ray spectrum.

Worst-Case X-ray Photon Energies in Ultrashort Pulse Laser Processing.

Ultrashort pulse laser processing can result in the secondary generation of unwanted X-rays if a critical laser irradiance of about 1013 W cm-2 is exceeded. Spectral X-ray emissions were investigated during the processing of tungsten and steel using three complementary spectrometers (based on CdTe and silicon drift detectors) simultaneously for the identification of a worst-case spectral scenario. Therefore, maximum X-ray photon energies were determined, and corresponding dose equivalent rates were calculated. An ultrashort pulse laser workstation with a pulse duration of 274 fs, a center wavelength of 1030 nm, pulse repetition rates between 50 kHz and 200 kHz, and a Gaussian laser beam focused to a spot diameter of 33 µm was employed in a single pulse and burst laser operation mode. Different combinations of laser pulse energy and repetition rate were utilized, keeping the average laser power constant close to the maximum power of 20 W. Peak irradiances I0 ranging from 7.3 × 1013 W cm-2 up to 3.0 × 1014 W cm-2 were used. The X-ray dose equivalent rate increases for lower repetition rates and higher pulse energy if a constant average power is used. Laser processing with burst mode significantly increases the dose rates and the X-ray photon energies. A maximum X-ray photon energy of about 40 keV was observed for burst mode processing of tungsten with a repetition rate of 50 kHz and a peak irradiance of 3 × 1014 W cm-2.

Low-energy X-ray irradiation: A novel non-thermal microbial inactivation technology.

Over the last several decades, food irradiation technology has been proven neither to reduce the nutritional value of foods more than other preservation technologies, nor to make foods radioactive or dangerous to eat. Furthermore, food irradiation is a non-thermal food processing technology that helps preserve more heat sensitive nutrients than those found in thermally processed foods. Conventional food irradiation technologies, including γ-ray, electron beam and high energy X-ray, have certain limitations and drawbacks, such as involving radioactive isotopes, low penetration ability, and economical unfeasibility, respectively. Owing to the recent developments in instrumentation technology, more compact and cheaper tabletop low-energy X-ray sources have become available. The generation of low-energy X-ray, unlike γ-ray, does not involve radioactive isotopes and the cost is lower than high energy X-ray. Furthermore, low-energy X-ray possesses unique advantages, i.e., high linear energy transfer (LET) value and high relative biological effect (RBE) value. The advantages allow low-energy X-ray irradiation to provide a higher microbial inactivation efficacy than γ-ray and high energy X-ray irradiation. In the last few years, various applications reported in the literature indicate that low-energy X-ray irradiation has a great potential to become an alternative food preservation technique. This chapter discusses the technical advances of low-energy X-ray irradiation, microbial inactivation mechanism, factors influencing its efficiency, current applications, consumer acceptance, and limitations.

A simple method for estimating the potential of the pyroelectric crystal surface.

The potential of the pyroelectric crystal surface has been estimated using experimental data. The temperature of the pyroelectric crystal, the electron current from the crystal surface to the target, and the X-ray spectrum were simultaneously measured. The potential calculated from the temperature and the electron current was compared with the experimental endpoint energy of the X-ray spectrum. The calculated potential reasonably agreed with the experimental endpoint energy.

X-ray Emission Hazards from Ultrashort Pulsed Laser Material Processing in an Industrial Setting.

Interactions between ultrashort laser pulses with intensities larger than 1013 W/cm2 and solids during material processing can lead to the emission of X-rays with photon energies above 5 keV, causing radiation hazards to operators. A framework for inspecting X-ray emission hazards during laser material processing has yet to be developed. One requirement for conducting radiation protection inspections is using a reference scenario, i.e., laser settings and process parameters that will lead to an almost constant and high level of X-ray emissions. To study the feasibility of setting up a reference scenario in practice, ambient dose rates and photon energies were measured using traceable measurement equipment in an industrial setting at SCHOTT AG. Ultrashort pulsed (USP) lasers with a maximum average power of 220 W provided the opportunity to measure X-ray emissions at laser peak intensities of up to 3.3 × 1015 W/cm2 at pulse durations of ~1 ps. The results indicate that increasing the laser peak intensity is insufficient to generate high dose rates. The investigations were affected by various constraints which prevented measuring high ambient dose rates. In this work, a list of issues which may be encountered when performing measurements at USP-laser machines in industrial settings is identified.

Mathematical estimation of half-value layer thicknesses.

OBJECTIVE: The objective of this article is to introduce a simplified and swift method to satisfactorily estimate the half-value layers (HVL), quarter-value layer (QVL), and tenth-value layer (TVL) from the x-ray spectra emitted by any diagnostic radiology or kV radiotherapy x-ray tubes. METHODS: A CdTe x-ray and Gamma detector (X-123 CdTe, AmpTek Inc.) is used to measure the x-ray spectra at four different x-ray energies (low, mid, high energy x-rays) with different external filtering. The software "SpekCalc GUI" (Developed in McGill University, Montreal, Canada) is also used to obtain the simulated x-ray spectra. Both measured and simulated spectra are used to compute the HVL thicknesses of Aluminum by a mathematical method presented in this article. Next, the HVL thicknesses for corresponding tube potentials are also measured by calibrated ionization chamber and varying thicknesses of aluminum plates. Finally, the computed and measured HVL, QVL, and TVL thicknesses are compared to evaluate the efficacy of the presented method. RESULTS: The results show acceptable concordance between computed and measured quantities. The disagreement rates between measured HVL and the values derived mathematically from the x-ray spectra are 10 to 90 micrometers of Aluminum at tube potentials of 31 kV to 120 kV. As it is shown, a negligible discrepancy is observed between the analytical estimation and the experimental assessments. CONCLUSION: The HVL is an essential component in the evaluation of the quality of an x-ray beam. However, its measurement could occasionally be challenging, time-consuming, or uncertain due to some technical difficulties. Although the scope of this study is not to undermine the value of conventional and widely accepted practice to determine the HVL thickness, the introduced method provides the fast, more convenient, and comparably reliable technique to estimate the HVL, QVL, and TVL by employing the given x-ray spectrum.

X-ray Dose Rate and Spectral Measurements during Ultrafast Laser Machining Using a Calibrated (High-Sensitivity) Novel X-ray Detector.

Ultrashort pulse laser machining is subject to increase the processing speeds by scaling average power and pulse repetition rate, accompanied with higher dose rates of X-ray emission generated during laser-matter interaction. In particular, the X-ray energy range below 10 keV is rarely studied in a quantitative approach. We present measurements with a novel calibrated X-ray detector in the detection range of 2-20 keV and show the dependence of X-ray radiation dose rates and the spectral emissions for different laser parameters from frequently used metals, alloys, and ceramics for ultrafast laser machining. Our investigations include the dose rate dependence on various laser parameters available in ultrafast laser laboratories as well as on industrial laser systems. The measured X-ray dose rates for high repetition rate lasers with different materials definitely exceed the legal limitations in the absence of radiation shielding.

Estimating the Product of the X-ray Spectrum and Quantum Detection Efficiency of a CT System and Its Application to Beam Hardening Correction.

Lab-based X-ray computed tomography (XCT) systems use X-ray sources that emit a polychromatic X-ray spectrum and detectors that do not detect all X-ray photons with the same efficiency. A consequence of using a polychromatic X-ray source is that beam hardening artefacts may be present in the reconstructed data, and the presence of such artefacts can degrade XCT image quality and affect quantitative analysis. If the product of the X-ray spectrum and the quantum detection efficiency (QDE) of the detector are known, alongside the material of the scanned object, then beam hardening artefacts can be corrected algorithmically. In this work, a method for estimating the product of the X-ray spectrum and the detector's QDE is offered. The method approximates the product of the X-ray spectrum and the QDE as a Bézier curve, which requires only eight fitting parameters to be estimated. It is shown experimentally and through simulation that Bézier curves can be used to accurately simulate polychromatic attenuation and hence be used to correct beam hardening artefacts. The proposed method is tested using measured attenuation data and then used to calculate a beam hardening correction for an aluminium workpiece; the beam hardening correction leads to an increase in the contrast-to-noise ratio of the XCT data by 41% and the removal of cupping artefacts. Deriving beam hardening corrections in this manner is more versatile than using conventional material-specific step wedges.

An X-ray spectrum estimation method from transmission measurement combined with scatter correction.

PURPOSE: Conventional x-ray spectrum estimation methods from transmission measurement often lead to inaccurate results when extensive x-ray scatter is present in the measured projection. This study aims to apply the weighted L1-norm scatter correction algorithm in spectrum estimation for reducing residual differences between the estimated and true spectrum. METHOD: The scatter correction algorithm is based on a simple radiographic scattering model where the intensity of scattered x-ray is directly estimated from a transmission measurement. Then, the scatter-corrected measurement is used for the spectrum estimation method that consists of deciding the weights of predefined spectra and representing the spectrum as a linear combination of the predefined spectra with the weights. The performances of the estimation method combined with scatter correction are evaluated on both simulated and experimental data. RESULTS: The results show that the estimated spectra using the scatter-corrected projection nearly match the true spectra. The normalized-root-mean-square-error and the mean energy difference between the estimated spectra and corresponding true spectra are reduced from 5.8% and 1.33 keV without the scatter correction to 3.2% and 0.73 keV with the scatter correction for both simulation and experimental data, respectively. CONCLUSIONS: The proposed method is more accurate for the acquisition of x-ray spectrum than the estimation method without scatter correction and the spectrum can be successfully estimated even the materials of the filters and their thicknesses are unknown. The proposed method has the potential to be used in several diagnostic x-ray imaging applications.

Radiological aspects of CO2 peripheral DSA: Preliminary analysis on the dedicated protocols.

OBJECTIVES: Thanks to its lack of allergic reactions and renal toxicity, CO2 represents an alternative to iodine as a contrast medium for peripheral subtraction angiography. Since CO2 has a lower and negative contrast than iodine, postprocessing DSA and stacking are mandatory. So, it seems that higher doses than traditional iodine angiography are required. We addressed the dosimetric aspects of CO2 angiography for two different commercial DSA-apparatus. MATERIALS AND METHODS: Two different radiological suites were analyzed by recreating the same setup on all the apparatuses: we used a PMMA slabs phantom with a MPD Barracuda dosimeter on its side to collect all radiological parameters. RESULTS: Results show that the irradiation parameters were left completely unchanged between the traditional and CO2 angiographic programs. CONCLUSIONS: This leads to thinking that these CO2 protocols do not operate on the X-ray emission, but only differ on image manipulation. The possibility of improvements by changing radiological parameters are still not explored and really promising.

Xpektrin, un simulador de espectros fácil de usar y ampliamente distribuible para radiología general / Xpektrin, an easy to use and highly distributable X-Ray spectra simulator in general radiography

Resumen: Se presenta una aplicación basada en Microsoft Excel llamada Xpektrin para el cálculo de dosis en radiología general. La aplicación permite simular espectros de rayos X en radiología general utilizando el modelo TASMICS a partir de mediciones del kerma en aire (Kair) y de la capa Hemirreductora (HVL). Tiene implementado el cálculo de magnitudes radiométricas y dosimétricas, como el kerma en aire en la superficie de entrada (Ke) y la dosis en piel (Dskin), en función de la elección arbitraria de los factores de exposición, el tipo y grosor de filtro, la distancia foco-piel y el tamaño de campo. Xpektrin fue validado con la herramienta computacional SPEKTR 3.0, utilizando mediciones de dosis y de HVL de tubos de rayos X de tres recintos hospitalarios. Se encontró buena correlación en ambas aplicaciones entre las mediciones experimentales y los valores calculados de HVL y con coeficientes de Pearson R² ≥ 0.99 en todos los casos. Sin embargo, se obtuvo mejor concordancia con los valores experimentales de HVL con Xpektrin (mediana de diferencias -0.43%, -0.04% y 0.01%) que con SPEKTR 3.0 (mediana de diferencias -3.31%, 0.10% y -7.85%), en particular para el tubo con mayor filtración. Xpektrin está optimizada para ser utilizada en los departamentos de radiología para la determinación de dosis de pacientes individuales en función de los parámetros utilizados durante la exposición, por lo que puede ser utilizada como parte de un sistema de registro dosimétrico o como apoyo para el establecimiento de niveles de referencia para diagnóstico (NRD), siendo particularmente útil en servicios con equipos sin registros automáticos de dosis. Además, debido a sus características de simulador, puede ser útil como herramienta pedagógica. El uso de Excel permite que sea altamente distribuible y fácil de usar, sin necesidad de conocimientos de programación.

Abstract: Xpektrin, an easy to use and highly distributable X-Ray Spectra Simulator in General Radiography. An application based on Microsoft Excel called Xpektrin is presented for dose calculation in general radiology. The application was developed to simulate X-ray spectra in general radiography using the TASMICS model. Using as inputs air kerma (Kair) and Half-value layer (HVL) measurements, Xpektrin allows the calculation of several radiometric and dosimetric quantities, such as the entrance surface air kerma (Ke) and the skin dose (Dskin), depending on the exposure factors, filter material type, filter thickness, focus-skin distance and field size. Xpektrin was validated against the Matlab toolkit SPEKTR 3.0, using dose and HVL measurements of X-ray tubes from three different hospitals. It was found good correlation in both applications between the experimental measurements and the calculated HVL and Kair values with Pearson coefficients R² ≥ 0.99 in all cases. However, experimental and calculated HVL have better agreement with Xpektrin (median percent difference -0.43%, -0.04% and 0.01%) than SPEKTR 3.0 (median percent difference -3.31%, 0.10% and -7.85%), particularly for the tube with greater filtration thickness. Xpektrin is optimized to be used in radiology departments for patient dose determination depending on the exposure parameters and may be used as part of a dosimetric record system or as a support for the determination of Diagnostic Reference Levels, which may be useful when no automatic dose records are available. In addition, due to its simulator characteristics, it can be useful as a pedagogical tool. Using Excel allows Xpektrin to be highly distributable and easy to use, without the need for programming skills.

Estimating the spectrum in computed tomography via Kullback-Leibler divergence constrained optimization.

PURPOSE: We study the problem of spectrum estimation from transmission data of a known phantom. The goal is to reconstruct an x-ray spectrum that can accurately model the x-ray transmission curves and reflects a realistic shape of the typical energy spectra of the CT system. METHODS: Spectrum estimation is posed as an optimization problem with x-ray spectrum as unknown variables, and a Kullback-Leibler (KL)-divergence constraint is employed to incorporate prior knowledge of the spectrum and enhance numerical stability of the estimation process. The formulated constrained optimization problem is convex and can be solved efficiently by use of the exponentiated-gradient (EG) algorithm. We demonstrate the effectiveness of the proposed approach on the simulated and experimental data. The comparison to the expectation-maximization (EM) method is also discussed. RESULTS: In simulations, the proposed algorithm is seen to yield x-ray spectra that closely match the ground truth and represent the attenuation process of x-ray photons in materials, both included and not included in the estimation process. In experiments, the calculated transmission curve is in good agreement with the measured transmission curve, and the estimated spectra exhibits physically realistic looking shapes. The results further show the comparable performance between the proposed optimization-based approach and EM. CONCLUSIONS: Our formulation of a constrained optimization provides an interpretable and flexible framework for spectrum estimation. Moreover, a KL-divergence constraint can include a prior spectrum and appears to capture important features of x-ray spectrum, allowing accurate and robust estimation of x-ray spectrum in CT imaging.

Counting-loss correction for X-ray spectra using the pulse-repairing method.

Facing the technical problem of pulse distortion caused by frequent resetting in the latest high-performance silicon drift detectors, which work under high-counting-rate conditions, a method has been used to remove false peaks in order to obtain a precise X-ray spectrum, the essence of which eliminates distorted pulses. Aiming at solving the problem of counting-loss generated by eliminating distorted pulses, this paper proposes an improved method of pulse repairing. A 238Pu source with activity of 10 mCi was used as the measurement object, and the energy spectrum obtained by the pulse repairing method was compared with that obtained by the pulse elimination method. The ten-measurement results show that the pulse repairing method can correct the counting-loss caused by the pulse elimination method and increase peak area, which is of great significance for obtaining a precise X-ray energy spectrum.

[Measured X-ray Spectra and Absorbed Dose Conversion Factors in a Water Equivalent Phantom Irradiated with Diagnostic X-rays].

The diagnostic X-ray spectra in a water equivalent phantom have been measured. From these measured spectra, the absorbed dose conversion factors of water were derived. The primary X-ray spectra were also measured and the scattered X-ray spectra were calculated by subtraction. The measurements were made at the depths of 0, 5, 10, 15, and 20 cm in a 20 cm-thick phantom and at the X-ray tube voltages 60, 90, and 120 kV by using a small silicon diode detector. The radiation field size was 30×30 cm2 at the phantom surface. In the obtained spectra, the fraction of the scattered photon number is increased with the depth. The X-ray qualities of the spectra in the phantom were near the qualities of primary X-rays when the depth is 0 cm, and became near the qualities of scattered X-rays as the depth increases. The changes of the X-ray qualities due to the depth change were small; photon mean-energy changes were within 4.6 keV. The changes in the absorbed dose conversion factors were also small (within 0.68%). These conversion factors were 0.4-2.3% larger than those obtained from the effective energy of incident X-rays and only -0.3 to 0.5% larger than those obtained from the X-ray spectra calculated from the aluminum half value layer and the tube voltage of incident X-rays. This study shows experimentally that the absorbed dose in a water-like phantom can be calculated with good accuracy by using the absorbed dose conversion factor obtained from the incident X-rays.

Analytical expressions for the reconstructed image of a homogeneous cylindrical sample exhibiting a beam hardening artifact in X-ray computed tomography.

BACKGROUND: Cylindrical phantoms are often imaged by X-ray computed tomography (CT) to evaluate the extent of beam hardening (or cupping artifact) resulting from a polychromatic X-ray source. OBJECTIVE: Our goal was to derive analytical expressions for the reconstructed image of a homogeneous cylindrical phantom exhibiting a cupping artifact, to permit a quantitative comparison with experimental cupping data. METHODS: A filtered backprojection method was employed to obtain the analytical cupping profile for the phantom, assuming that the projection data could be approximated as a power series with respect to the sample penetration thickness. RESULTS: The cupping profile was obtained analytically as a series of functions by employing Ramachandran filtering with an infinite Nyquist wavenumber. The quantitative relationship between the power series of the projection and the nth moment of the linear attenuation coefficient spectrum of the phantom was also determined. Application of the obtained cupping profile to the evaluation of the practical reconstruction filters with a finite Nyquist wavenumber and to the best choice of the contrast agent was demonstrated. CONCLUSIONS: The set of exact solutions derived in this work should be applicable to the analysis of cylindrical phantom experiments intended to evaluate CT systems.

Determination and verification of the x-ray spectrum of a CT scanner.

The accuracy of Monte Carlo (MC) simulations in estimating the computed tomography radiation dose is highly dependent on the proprietary x-ray source information. To address this, this study develops a method to precisely estimate the x-ray spectrum and bowtie (BT) filter thickness of the x-ray source based on physical measurements and calculations. The static x-ray source of the CT localizer radiograph was assessed to measure the total filtration at the isocenter for the x-ray spectrum characterization and the BT profile (air-kerma values as a function of fan angle). With these values, the utilized BT filter in the localizer radiograph was assessed by integrating the measured air kerma in a full 360-deg cycle. The consistency observed between the integrated BT filter profiles and the directly measured profiles pointed to the similarity in the utilized BT filter in terms of thickness and material between the static and rotating x-ray geometries. Subsequently, the measured air kerma was used to calculate the BT filter thickness and was verified using MC simulations by comparing the calculated and measured air-kerma values, where a very good agreement was observed. This would allow a more accurate computed tomography simulation and facilitate the estimation of the dose delivered to the patients.

X-ray spectrum estimation for accurate attenuation simulation.

PURPOSE: To estimate detected x-ray spectra from transmission measurements of known attenuators that allow to accurately simulate the transmission in unknown attenuators. METHODS: Starting from the established spectrum estimation method using the truncated singular value decomposition (TSVD) we extended the algorithm by incorporating prior knowledge about the statistical nature of the transmission data and about high-frequency spectral components like characteristic peaks. Thereby our proposed approach requires only minimal prior knowledge, namely the energy positions of characteristic peaks or k-edges, which are typically well-known. This ensures that the final spectrum is not biased towards a given prior spectrum which is often observed in other methods. The new approach, prior truncated singular value decomposition (PTSVD), is compared to TSVD as well as the expectation-maximization (EM) method in a simulation and a measurement study. The resulting spectra are evaluated according to their ability to reproduce transmission data of attenuators that have not been included into the estimation process. RESULTS: In case of noiseless simulated data, the PTSVD approach outperforms the existing methods in both, estimating the shape of the spectrum as well as providing a spectrum that reproduces the transmission data. Not surprisingly for increasing noise the ability of PTSVD to estimate the spectral shape worsens while it still performs best in reproducing the transmission data. This finding is also confirmed in the measurement study. CONCLUSION: Our new approach allows to estimate detected x-ray spectra that accurately reproduce both transmission measurements that have and have not been included into the estimation process. It is less prone to noise compared to the established TSVD method and potentially leads to smaller transmission errors compared to EM for accurate transmission data while being less biased towards the given prior information.

Precise material identification method based on a photon counting technique with correction of the beam hardening effect in X-ray spectra.

The aim of our study is to develop a novel material identification method based on a photon counting technique, in which the incident and penetrating X-ray spectra are analyzed. Dividing a 40 kV X-ray spectra into two energy regions, the corresponding linear attenuation coefficients are derived. We can identify the materials precisely using the relationship between atomic number and linear attenuation coefficient through the correction of the beam hardening effect of the X-ray spectra.

A simulation-based study on the influence of beam hardening in X-ray computed tomography for dimensional metrology.

X-ray computed tomography (CT) is a radiographic scanning technique for visualising cross-sectional images of an object non-destructively. From these cross-sectional images it is possible to evaluate internal dimensional features of a workpiece which may otherwise be inaccessible to tactile and optical instruments. Beam hardening is a physical process that degrades the quality of CT images and has previously been suggested to influence dimensional measurements. Using a validated simulation tool, the influence of spectrum pre-filtration and beam hardening correction are evaluated for internal and external dimensional measurements. Beam hardening is shown to influence internal and external dimensions in opposition, and to have a greater influence on outer dimensions compared to inner dimensions. The results suggest the combination of spectrum pre-filtration and a local gradient-based surface determination method are able to greatly reduce the influence of beam hardening in X-ray CT for dimensional metrology.