Beam hardening correction and its influence on the measurement accuracy and repeatability for CT dimensional metrology applications
Ye Tan1,2, Kim Kiekens1,2, Frank Welkenhuyzen2, Jean-Pierre Kruth2, Wim Dewulf1,2
1Group T-International University College Leuven, KU Leuven Association
Andreas Vesaliusstraat 13, 3000 Leuven, Belgium, e-mail:
2Katholieke University Leuven, Department of Mechanical Engineering, PMA DivisionCelestijnenlaan 300B, 3001 Heverlee, Belgium, e-mail:
Abstract
The beam hardening effect is one of the major artifacts of X-ray computed tomography. It not only complicates medical inspection and material analysis, but also influences the accuracy and repeatability of dimensional measurements.Therefore, many efforts have been devoted to develop beam hardening correction methods since the early 1970s. In practice, the combination of hardware filters (pre-filtration) and linearization algorithms is frequently used.In general, this combination can largely eliminate the cupping effect, resulting in more homogeneous gray values throughout the same material and improving the surface appearance. However, while correcting for the beam hardening effect, other problems are generated,including noise magnification and contrast reduction. Moreover, experiments reveal that in some cases,beam hardening correction methods induce edge offsets of internal structures that are largely dependent on the amount of surroundingmaterial. This worsens themeasurement uncertainty. Therefore, this paper investigates the influence of beam hardening correction on the measurement accuracy and uncertainty for CT metrology applications.
Keywords: CT,dimensional metrology, beam hardening correction
1 Introduction
With the fast development of X-ray sources (smaller focal spot size), better detectors, more advanced reconstruction algorithms and analyzing softwares, using industrial CT machine as a measuring tool becomes possible [1-3].For CT dimensional metrology applications, image quality and volume artifacts are considered unimportant unless they harm accurate surface determination. Systematic errors and constant edge offsets are troublesome but can be well compensated by dedicated calibration objects and by applying proper edge correction terms [4]. However, random and non-constant edge offset errors are most undesirable, because they increase the overall measurement uncertainty.
1.1 Beam hardening effect
Most industrial X-ray sources produce X-ray beams with polychromatic spectrum. Since the X-ray attenuation is energy dependent, as the beam penetrates a workpiece, low energy (soft) X-rays are more rapidly attenuated than high energy (hard) X-rays. Consequently, the number of photons hitting an X-ray detector (hence the grey value of the corresponding pixel) is not strictly linearly related to the penetrated material thickness. On the other hand, most reconstruction algorithms presume linear attenuation, causing the outer edge to color differently from inner material, known as the cupping effect (Fig. 1). Furthermore, streaks and dark bands often appear between dense objects (Fig. 2). [5]
Figure 1.Reconstructed slices of a steel cylinder (left) and corresponding grey valueprofile along the red line (right). [5]
Figure 2. (a) 2D X-ray image of an aluminium profile with four steel spheres; (b) streak artifacts visible in a reconstructed CT slice of the red section in (a); and (c) 3D CT voxel model of the objects. [5]
1.2Beam hardening correction methods
Beam hardening correction has been a research topic for decades. Except for hardware filtering, a broad varityof artifact reduction algrothims have been developed. Daviset al. suggest to model X-ray generation, transmission, detection and to use step wedge transmission measurementsfor beam hardening correction [6]. Similiarly, Kachelrieß et al. use an empirically determinedprecorrection function of polynomial form to correct for cupping artifact [7]. Amirkhanovet al. propose a projection based metal-artifact reduction method, in which the segmented metal parts are reprojected as void and interpolated for a second reconstruction [8]. Dual energy methods were also developed, in which the energy-dependence of the attenuation coefficients is modelled as a linear combination of two basic functions representing the separate contributions of the photo-electric effect and the scattering [9-11].Van Gompel et al. suggest several iterative algorithms based on minimizing the difference between the measured sinogram data and a simulated polychromatic sinogram[12]. More recently, a referenceless beam hardening correction technique for multi-material objects has been developed [13].
1.3 Data processing (hardware and software)
The measurement hardware and software used for all measurements reported in this paper are listed in Table 1. The 2D projection images are reconstructed using a filtered back projection algorithm. A linearization technique based on pre-defined polynomial curves of maximum fourth order has been applied for beam hardening correction:
Y = a( b + cX + dX2 + eX3 + fX4)
Where X represents the initial grey value of a pixel in an X-ray image, Y represents the corrected (linearized) grey value, and a through f represent coefficients that can be fine-tuned in order to obtain cupplingfree images. Six beam hardening correction presets have been applied in this paper (Table 2).
Table 1.Measurement equipment and software
CT device / XT-H225 with Tungsten targetReconstruction and beamhardening correction / CTPro XT 2.2 SP2 with linearizationa
Thresholding and dimensional measurements / VGStudio MAX 2.1.2
a CTPro also supports beam hardening correction based on look-up tables.
For surface determination, a local adaptive thresholding method has been applied [14]. Two methods have been used to calculate the rescaling factor for voxel size calibration: using the calibrated distance between centre positions of two adjacent steel spheres; using the centre position of two ruby-spheres mounted at a fixed distance from each other and calibrated using a CMM.
Table 2. Parameters used for the six beam hardening correction presets
Parameters / presets1 / 2 / 3 / 4 / 5 / 6
a / 1 / 1 / 1 / 1 / 1 / 1
b / 0 / 0 / 0 / 0 / 0 / 0
c / 1 / 0.75 / 0.5 / 0.2 / 0.1 / 0
d / 0 / 0.25 / 0.5 / 0.8 / 0.9 / 0.2
e / 0 / 0 / 0 / 0 / 0 / 0.8
f / 0 / 0 / 0 / 0 / 0 / 0
2 Measurements and results
As mentioned before, the ability to measure internal structures is the major asset of CT metrology. Thus, this section investigates whether the above mentioned beam hardening correction algorithm(linearization technique based on pre-defined polynomial curves) influences the dependence of the measurement result on the amount and type of surrounding materials. This has beentested using several object set-ups, including both single and multi-material cases.
2.1 Preliminary notes on the experimental results
Before discussing the influence of beam hardening correction on the measurement uncertainty, it is necessary to clearify one common characteristic ofthe reported measurement results. All results reveal one obvious trend: the objects’ top parts appear larger than their lower parts. As shown in Figure 3, two steel cylinders (4mm and 5mm, dimensional tolerance 1m) are scanned together in one set-up. After reconstruction and surface determination, their diameters are measured at ca. 70 different cross sections (equallyspaced slices) from top to bottom. The resulting plot clearly shows that the dimensions for both cylinders are decreasing from top to bottom. The top diameter can be upto 5m larger than the bottom diameter. This error is most probably caused by a misalignment of the rotation axis and the X-ray detector at the time of performing the experiments. Though annoying, this trend does not influence the conclusions related to the impact of the beam hardening correction methods, which will be characterized by sudden discontinuities rather than by steady decreases.
2.2Single material case
Steel parts are very often seen in industrial assemblies, and are troublesome in many CT metrology applications due to beam hardening artifacts. Therefore, this section investigates the influence of beam hardening correction on the measurement uncertainty of steel parts.
Figure 4 demonstrates the measurement result of a 6mm steel cylinder (tolerance 1m) partly surrounded by a hollow steel cylinder near the bottom. The CT scan has been processed using beam hardening correction presets 1 and 2 (see Table 2). All surfaces have been determined with local adaptive thresholding method [14]. Subsequently, the diameter of the inner steel cylinder is measured on a series of equidistance slices from top to bottom. Figure 4 shows the CT measurement error as a function of slice number. The above mentioned error due to the tilt/detector angle is observedonce more. More importantly, the cylinder dimension experiences a significant jump due to the changing surrounding (inner cylinder enters the hollow cylinder towards bottom). This jump is around 2m in of the absence of beam hardening correction and increases to 6m when applying slight beam hardening correction.
The above mentioned result is confirmed with another setup shown in Figure 5. A 3mm steel cylinder (tolerance 1m) is partly surrounded with a steel stepped hollow cylinder. The scan data was processed with beam hardening correction presets 1 to 5 (see Table2, starting from no correction upto 2nd order polynomial correction curve). After surface determination, the diameter of the innersteel cylinder is measured on a series of equidistance slices from top to bottom. Sudden dimensional changes are again present at the height where the surrounding situation alters. By comparing the measurement results of 5 beam hardening correction presets, it can be concluded that the sudden dimensional variation increaseswhen increasing the correction level (from 4m to 8m). Starting from slice Nr.30, the dimensional variations (2m) are mainly due to increased noise (lower signal to noise ratio) for the central cylinder.
Figure 5. Ø3mm steel cylinder (±1m tolerence) is placed inside another hollow stepped steel cylinder. After reconstruction and local thresholding, its diameter is measured slice by slice from top towards bottom. Different polynomial curves for beam hardening correction have been tested. Significant dimensional variation is detected, while beam hardening correction tends to magnify such variation.
2.3Simulation verification
As mentioned in the previous sections, the decreasing trend on cylinder diameter over the height has no influence on the beam hardening correction induced dimensional changes. However, it is difficult to eliminate such misalignment error due to its various potential causes: X-ray source, rotation axis and detector deformation etc. Thus, in order to exclude the misalignment error and to verify the previous conclusions, a similar setup as in Figure 4has been simulated under the same scanning conditions (X-ray power, filter, magnification etc.). Figure 6 demonstrates the geometry of the simulated setup and the evolution of the gray value profile while increasing the order of the polynomial curve for beam hardening correction. Over-correction is observed with beam hardening correction preset 3, while preset 2 can effectively eliminate the cupping effect. The dimensional measurement results are shown
Figure 6. (a) Dimension of the steel cylinders. All dimensions are in milimeter. (b) The gray value profile along the red line with (top-left) and without (bottom-right) surrounding material, using different beam hardening correction presets (see table 2).
in Figure 7. The decreasing trend is eliminated due to the “perfect” alignment in the simulated machine configuration. However, sudden discontinuities can be observed at the location where the surrounding situation changes (in this case, when the middle cylinder enters the hollow cylinder). Moreover, this sudden dimensional variation increases with increasing beam hardening correction level. This is similar to what has been observed with real CT measurements. It needs to be mentioned that the specifications of the X-ray detector used in this simulation varies from the one used in the experiments.
Figure 7.CT measurement simulation results: the influence of beam hardening correction on the measured diameter of a 6mm steel cylinder partly surrounded by a hollow steel cylinder.
2.4Multi-material case
Figure 8. Influence of beam hardening correction on the measured diameter of a 4mm steel cylinder (tolerance 1m) partly surrounded by a hollow aluminium cone. The diameter of the reference steel cylinder is measured on different slices from top (left) to bottom (right).
Many applications in CT metrology require scanning multi-material assemblies. Because steel and aluminium parts are often encountered in such applications, this section investigates the influence of beam hardening correction on the measurement uncertainty of combined steel-aluminiumparts.The results are demonstrated in Figures 8 and 9. The inner steel cylinder is the same, but there is a slight difference between these two setups. The aluminium cylinder in Figure 8 has a sharp top; hence the amount of material surrounding the central cylinder is changing gradually. In contrast, the top of the aluminium cylinder in Figure 9 already has a certain thickness. Thus, for the middle steel cylinder, its surrounding changes suddenly at that point. By observing the measurement results from Figure 8 and 9, it can be concluded that for steel-aluminium combinations, applying beam hardening correction can result in dimensional changes when the surrounding situation alters. For these experiments, the magnitude of such change is around 5-10m when using beam hardening correction preset Nr.2 and more than 20m if more severe corrections are applied. Moreover, both experiments show that the local dimensional variations increase when the surrounding material increases.Less sudden dimensional jump is not found in Figure 6 due to the gradual increasing material thickness of the aluminium surrounding.
Figure 9. Influence of beam hardening correction on the measured diameter of a 4mm steel cylinder (tolerance 1m) partly surrounded by a hollow aluminium cone. The diameter of the reference steel cylinder is measured on different slices from top (left) to bottom (right) [5].
4 Discussion and conclusion
Beam hardening correction has been a major research topic for decades. This paper investigates whether CT dimensional metrology applications also benefit from such correction. On the one hand, beam hardening correction brings in benefits such as enchanced image quality and homogeneous gray value for the single material parts. On the other hand, it is shown that beam hardening correction can introduce a surrounding material dependent dimensional error, thus enlarging the measurement uncertainty. However, many existing beam hardening correction methods have not been tested, thus further research is necessary to investigate to which extent the reported results applies. Furthermore, other material combinations and geometries should also be investigated.
Acknowledgements
The authors thank Andrew Ramsey for his valuable advice. Furthermore we acknowledge the support of the Research Foundation Flanders (FWO) via project G.0711.11 N and G.0618.10.
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