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
1
Group T - International University College Leuven, KU Leuven Association
Andreas Vesaliusstraat 13, 3000 Leuven, Belgium, e-mail: wim.dewulf@groept.be
2
Katholieke University Leuven, Department of Mechanical Engineering, PMA Division
Celestijnenlaan 300B, 3001 Heverlee, Belgium, e-mail: Jean-Pierre.Kruth@mech.kuleuven.be
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 surrounding material. This worsens the
measurement 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 value profile 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]
1.2 Beam hardening correction methods
Beam hardening correction has been a research topic for decades. Except for hardware filtering, a
broad varity of artifact reduction algrothims have been developed. Davis et al. suggest to model X-ray
generation, transmission, detection and to use step wedge transmission measurements for beam
hardening correction [6]. Similiarly, Kachelrieß et al. use an empirically determined precorrection
function of polynomial form to correct for cupping artifact [7]. Amirkhanov et 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
cuppling free images. Six beam hardening correction presets have been applied in this paper (Table 2).
Table 1. Measurement equipment and software
CT device
Reconstruction and beam hardening correction
Thresholding and dimensional measurements
a
XT-H225 with Tungsten target
CTPro XT 2.2 SP2 with linearizationa
VGStudio MAX 2.1.2
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
a
b
c
d
e
f
presets
1
1
0
1
0
0
0
2
1
0
0.75
0.25
0
0
3
1
0
0.5
0.5
0
0
4
1
0
0.2
0.8
0
0
5
1
0
0.1
0.9
0
0
6
1
0
0
0.2
0.8
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 been tested using
several object set-ups, including both single and multi-material cases.
2.1 Preliminary notes on the experimental results
Figure 3. Two steel cylinders
are scanned together in one
set-up; their diameters are
measured at different heights
from top to bottom.
Before discussing the influence of beam hardening correction on the measurement uncertainty, it is
necessary to clearify one common characteristic of the 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 1m) are scanned together in one setup. After reconstruction and surface determination, their diameters are measured at ca. 70 different
cross sections (equally spaced 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 up to 5m
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.
2.2 Single 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 1m) 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 2m in of
the absence of beam hardening correction and increases to 6m when applying slight beam hardening
correction.
Figure 4. Influence of beam
hardening correction on the
measured diameter of a
6mm steel cylinder partly
surrounded by a hollow steel
cylinder.
The above mentioned result is confirmed with another setup shown in Figure 5. A 3mm steel
cylinder (tolerance 1m) 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 inner steel
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 increases when increasing the correction level (from 4m to 8m). Starting from
slice Nr.30, the dimensional variations (2m) are mainly due to increased noise (lower signal to noise
ratio) for the central cylinder.
Figure 5. Ø3mm steel cylinder (±1m 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.
2.3 Simulation 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 4 has been simulated under the same scanning conditions (Xray 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.
2.4 Multi-material case
Figure 8. Influence of beam hardening correction on the measured diameter of a 4mm steel cylinder (tolerance
1m) 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-aluminium parts.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 5-10m when using beam hardening correction preset Nr.2 and
more than 20m 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
1m) 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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