Software-based compensation of computed tomography instrument misalignments – experimental study

A newly developed software-based compensation of geometrical misalignments is applied to several computed tomography (CT) instruments. The actual geometry of each CT instrument is measured using a dedicated procedure. Then, given the measured geometries, software-based compensation is applied. Dimensional measurements on a validation object show a consistent reduction in observed errors as a result of the compensation. Robustness of the compensation method is confirmed also on a CT instrument with severe misalignments.


Introduction
Currently, X-ray computed tomography (CT) measuring instruments almost exclusively employ the standard circular trajectory for data acquisition in combination with the Feldkamp-Davis-Kress (FDK) reconstruction algorithm.FDK reconstruction is sensitive to the geometrical alignment of the CT instrument: relative motions between X-ray source, detector, and object during data acquisition must be consistent with the corresponding trajectory in the reconstruction algorithm.In practice, however, this prerequisite is not fulfilled as misalignments of the CT instrument are inevitable.These geometrical misalignments, if not considered, result in errors of CT measurements.
There is currently no standardized or easily implementable method to compensate geometrical misalignments of the CT instrument [1].In many cases, the procedure of mechanical adjustment of the CT instrument is time-consuming and impractical.Even then, sizable deviations from ideal alignment can still remain.These remaining deviations, albeit smaller than the original geometrical misalignments, can still present significant artifacts in the reconstructed volume and systematic errors in dimensional measurements [2].In [3], we introduced a modified FDK reconstruction method with embedded misalignment compensation.The method performance was confirmed in several simulation studies.In this paper, we apply the modified FDK method to several lab-based CT instruments.The actual geometrical parameters of the CT instruments are measured by the dedicated procedure presented in [4,5].Then, given the measured geometries, software-based compensation is applied.Dimensional measurements on a validation object show a consistent reduction in observed errors.The robustness of the compensation technique is confirmed on a CT instrument with severe misalignments.

CT instrument geometry
The geometry of a cone-beam CT instrument is described as the relative position and orientation of the three major instrument components: X-ray focal spot, axis of object rotation, and flat-panel detector.The geometrical parameterization of a typical conebeam CT instrument with a circular scanning trajectory is illustrated in figure 1.
A global right-handed Cartesian coordinate frame (GCF) is defined by three mutually orthogonal axes X, Y, and Z.The Y axis is parallel to the rotation axis, the Z axis passes through the focal spot and is perpendicular to the rotation axis, and the X axis is given by cross product of Y and Z, i.e.X = (Y×Z).The coordinate frame origin coincides with the X-ray source focal spot.The detector orientation is defined by two orthonormal vectors eu and ev, while the detector position is given by the location of its geometrical center d = (xd, yd, zd) in the GCF.The normal to the detector plane en is given by the cross product of eu and ev, i.e. en = (eu×ev).The set of three vectors eu, ev, en forms an orthonormal basis and can be explicitly parameterized using three extrinsic rotations in a local coordinate frame with origin at d.These rotations are in the sequence (1) η (about a local Z' axis, parallel to the global Z axis), ( 2) φ (about a local Y' axis, parallel to the global Y axis), and (3) θ (about a local X' axis, parallel to the global X axis).The position of the axis of rotation is given by r = (0, 0, zr).For the given parameterization, the ideal alignment of a cone-beam CT instrument with a circular scanning trajectory can be expressed by the following set of conditions.
Consequently, the geometry of an ideally aligned CT instrument can be fully described by the parameters zd and zr, where zd is typically referred to as source-to-detector distance (SDD) and zr is the source-to-rotation axis distance (SRD).

Geometrical adjustment of a CT instrument
The quality of CT dimensional measurements is typically dependent on the ideal alignment of the instrument.A conventional procedure for compensation of misalignments consists of 3 major steps, summarized below and illustrated in figure 2. (1) After identifying misalignments by way of a geometrical calibration procedure [4,5], the instrument is mechanically adjusted .Adjustment of the CT instrument includes not only alignment of the source, the rotation axis and the detector plane with respect to each other to satisfy criteria discussed in the previous section, but also error mapping and compensation of kinematic positioning errors of the sample rotation stage, e.g.erroneous zr.Often, adjustment of the CT instrument is proprietary to the instrument manufacturers and is not easily implementable by users.
(2) Radiographs of a workpiece are acquired and reconstructed with conventional FDK reconstruction algorithm, often implemented in commercial software.Some tomographic reconstruction software packages perform what is known as 'center of rotation estimation' prior to filtering and backprojection the radiographic data.This technique estimates and compensates any discrepancy between the projection of the axis of rotation on the detector plane and the detector central pixel column (approximately equivalent to the parameters η, xd, and yd, in the parameterization presented in this paper).In an ideally aligned CT instrument, the projection of the axis of rotation should coincide with the detector central pixel column (or the edge between two central columns in a detector with an even number of pixel columns).
(3) In [6,7], the authors showed that positioning of the sample stage has poor repeatability even after error mapping and compensation by the CT instrument manufacturer.Therefore, a procedure known as 'scale calibration' or 'scale compensation' is performed to compensate an error in magnification factor [8,9].Typically, a reference object consisting of several spheres in a particular arrangement with known center-to-center distances, e.g.measured by coordinate measuring machine (CMM), are CT measured at the same position of the sample stage as the workpiece.Since kinematic errors in the manipulator can result in non-repeatable positioning of the sample stage, the sample stage should not be moved between steps (2) and (3).CT-measured center-to-center distances are compared to the known values and a scaling error between the two datasets is used to calculate a voxel scaling factor, which is applied to rescale the dataset obtained in step (2) [8,9].Steps ( 2) and ( 3) can be performed in the reverse order or simultaneously in one scan.

Software-based compensation of CT geometrical misalignments
In [3], the conventional FDK method was modified to account for non-ideal geometry of a CT instrument.We briefly revisit the modification procedure here.
The conventional FDK algorithm consists of three steps: 1. weighting of acquired radiographs, 2. convolution of the weighted radiographs with a Ramp filter, and 3. weighted back-projection of the filtered radiographs along the original rays.
The weighting coefficient for an individual pixel in the first step is the cosine of the angle between the ray from the X-ray source to the pixel and the ray from the source intersecting the detector plane orthogonally.The weighting coefficient depends on the position and orientation of the detector with respect to the source and has to be re-calculated to agree with the actual geometry of the CT instrument.
A ramp filter is applied to the radiographs to reduce blurring in the reconstructionan effect inherent to back-projection.In the conventional FDK algorithm, the filtration line must be parallel to the X axis.This condition is violated if the detector has angular misalignments φ and/or η.However, we showed in [3] that violation of this condition does not have any impact on the quality of reconstruction and consecutive measurement results of simulated datasets.
Finally, the geometry of the back-projection step is modified to correspond to the actual CT instrument geometry, i.e. instead of back projecting from an ideally-aligned CT instrument, we use the measured geometrical parameters of the CT instrument to calculate trajectories of the rays from the X-ray source through the voxels of the measurement volume and to the detector pixels

Data acquisition
The ball plate presented in [5] and shown in figure 3 was CT measured with three separate instruments: the Twinned Orthogonal Adjustable Tomograph (TORATOM) instrument at the Center of Excellence Telč (CET) [10], a Nikon XT H 225 ST, and a Nikon XT H 450; the Nikon CT instruments are both at KU Leuven.The ball plate consists of 15 grade 20 chrome steel 2.5 mm diameter spheres arranged on a carbon fiber plate in a dedicated pattern.Sphere positions on the workpiece were measured by CMM with a maximum permissible error (MPE) of 2 + L/400 μm, where L is the measured length in mm.Prior to CT measuring the ball plate, the geometrical estimation procedure described in [4,5] was applied to measure the instrument geometry at the same sample stage position as the ball plate.Scan settings were selected to guarantee X-ray penetration at all rotation positions and maximize the acquired gray value range for all individual radiographs.Radiographic acquisition parameters for each instrument are listed in table 1.
Two datasets were acquired on the TORATOM.Dataset D1 corresponds to the initial TORATOM instrument alignment upon our arrival to CET.The geometry of the TORATOM instrument was subsequently modified to have severe misalignments (see [5] for the procedure for modification of the TORATOM instrument geometry); this new geometry corresponds to dataset D2.Datasets D3 and D4 were acquired on the Nikon XT H 225 ST and Nikon XT H 450 instruments, respectively; the measured geometries of the commercial CT instruments at KU Leuven did not include any intentional misalignments.Geometrical parameter values for acquisitions D1-D4 estimated using the procedure from [4,5] are shown in table 2.   1), henceforth 'modified reconstruction'.Conventional reconstruction includes center-of-rotation estimation (discussed in section 3).For conventional reconstruction of datasets D1-D2, we set SRD and SDD to equal the estimated zr and zd parameters (table 2), respectively.For conventional reconstruction of datasets D3 and D4, SRD and SDD were provided by the acquisition software.

Results
Sphere center-to-center distances and sphere radii were measured on the conventional reconstruction and modified reconstruction datasets of the ball plate using VG Studio MAX (Volume Graphics, GmbH).Strong form deviations were observed in the area where sphere surface was in contact with glue and the carbon fiber surface due to erroneous segmentation.Therefore the hemisphere contacting the carbon fiber support and glue was excluded from dimensional measurements.
For all datasets except D2 we plot results of both conventional and modified reconstructions.Conventional reconstruction of dataset D2, corresponding to the severe misalignments, contained strong artifacts (see figure 4) despite center of rotation estimation.In table 3 we report mean absolute error (MAE) and total absolute error (TAE) in center-to-center distance segments compared to CMM values.For datasets D1-D4, there is significant reduction in MAE and TAE for modified reconstruction.However, both MAE and TAE values for modified reconstruction of datasets D3 and D4 are almost twice as large than equivalent metrics for D1 and D2.In [4,5] 'coupling'' of estimated zr and zd were observed, which might introduce the observed increase in MAE and TAE for D3 and D4.
Errors in measured radii relative to their manufacturing specification are shown in figure 6 for each sphere in the reconstructed datasets.Overall, the modified FDK algorithm reduces the sphere radius errors observed the conventional reconstruction, which is also confirmed by the data presented in table 4. For datasets D1-D2, errors in radius after the modified reconstruction follow roughly the same pattern.D3 and D4 demonstrate different error distributions, which most probably is related to different scan settings and environmental conditions.

Conclusion
The FDK algorithm was modified to the actual CT instrument geometry as estimated using a state-of-the-art estimation procedure and was applied to several CT instruments to compensate observed geometrical misalignments.The experimental study was performed on three CT instruments: from the TORATOM instrument at CET, a Nikon XT H 225 ST, and a Nikon XT H 450. Three CT acquisitions were performed without changing initial geometry of the CT instruments.For one acquisition, we intentionally misaligned the experimental CT instrument to check for robustness of the method.Results showed significant reductions of observed measurement errors.For example, center-to-center distance measurement errors in one of the experimental acquisitions of the workpiece were reduced from a maximum of approximately 250 μm under conventional reconstruction to below 20 μm using the modified reconstruction for a 140 mm length segment.Radius measurements were also reduced in the modified reconstruction.

Figure 1 .
Figure 1.Parameterization of the CT instrument geometry.

Figure 2 .
Figure 2. Conventional procedure for compensation of CT geometrical misalignments.

Figure 4 .
Figure 4. Conventional reconstruction results for dataset D2. (Left) Magnified and cropped gray-value slice of sphere 1. (Right) Threedimensional rendering of segmented surface for sphere 1. Center-to-center distance results for datasets D1-D4 are shown in figure 5.In total, 105 unique distances between pairs of spheres in the 15-sphere ball plate were measured and compared to CMM values.For datasets D1 and D2, modified FDK clearly outperforms conventional reconstruction.For datasets D3 and D4, the error in measurements of center-to-center distance increases with distance, indicating scale error, i.e. erroneous SRD and/or SDD.Datasets D1 and D4 also exhibit a larger dispersion of errors in conventional reconstruction compared to datasets D2-D3, which is related to the presence of relatively high detector out-of-plane misalignment θ.

Figure 5 .
Center-to-center distance measurement errors.In total, 105 unique distance between sphere pairs in the 15-sphere ball plate were measured and compared to CMM values.Conventional reconstruction results are shown as squares, whereas modified FDK results are shown as circles.

Figure 6 .
Errors in measured radii relative to their manufacturing specification.Conventional reconstruction results are shown as squares, whereas modified FDK results are shown as circles.

Table 2 .
Measured instrument geometry for each acquired dataset of the ball plate.

Table 3 .
Mean absolute error (MAE) and total absolute error (TAE) in center-to-center distance segments compared to CMM values.Statistical information is calculated based on 105 unique distances measured in dataset.

Table 4 .
Mean absolute error (MAE) and total absolute error (TAE) in measured sphere radii compared to manufacturing specification.Statistical information is calculated based on radius error measurements for all spheres.