X-ray source tracking to compensate focal spot drifts for dimensional CT measurements

Accurate CT measurements rely on a stable X-ray focal spot. Because focal spot drift deteriorates CT data, especially at high resolutions, drift compensation is crucial. Here, we present an optical X-ray tube tracking system that does not require fiducial markers and/or radiographic image registration. To assess its performance, a calibrated, small volume (< 4 mm) multi-sphere standard was CT scanned. The deviations of unidirectional dimensional measurements were reduced by almost a factor of two when compensating for the X-ray tube drift.


Introduction
A variety of error sources influence the accuracy of dimensional computed tomography (CT) measurements [1].Drift of the Xray focal spot causes blurring of CT data and can be detrimental for dimensional measurements [2,3], especially at high resolutions [4].To mitigate this effect, tracking highly absorbing, stationary fiducial markers was suggested [5,6].However, this method relies on a stable mount for the fiducial markers, consumes detector space, thus limiting the maximum resolution, and is not applicable for arbitrary magnifications.
Here, we investigated the accuracy of an X-ray tube tracking system, based on CMOS imaging sensors attached to the X-ray tube close to the target [7].The employed high-resolution transmission type X-ray tube has a JIMA resolution between 0.6 µm and 10 µm, depending on the operating parameters.The investigation includes improving the accuracy of the electron-beam centring procedure, measuring the X-ray tube drift, and evaluating the influence of the drift compensation on CT measurements.

Implementation of the X-ray tube tracking system
The X-ray tube tracking system is part of the METAS-CT geometry measurement system (figure 1a) [7].The tracking system consists of two CMOS imaging sensors attached to the tube head close to the target (figure 1b).Two optical fibres attached to the metrology frame illuminate the imaging sensors with a near Gaussian laser beam profile.Its intensity-weighted average is used to determine the position of the X-ray tube relative to a metrology frame with an accuracy below 0.1 µm [8].The physical implementation of the system is shown in figure 2. The attachment of the CMOS sensors to the X-ray tube is beneficial, since their heat is dissipated through the X-ray tube water-cooling system.Attaching the optical fibres to the metrology frame via numerically temperature compensated aluminium brackets, avoids heat flux into it.The tube tracking system measures the position of the outer steel body of the X-ray tube with respect to the metrology frame.The position of the actual focal spot on the target with respect to the X-ray-tube body is determined by the focal spot centring procedure.Prior to a measurement the electron beam, which is accelerated onto a metal target to create X-rays, is centred relative to the centre of an aperture.This aperture provides a reference location with respect to the outer steel body of the X-ray tube, which is measured by the above presented tube tracking system.The centring procedure is thus critical for the absolute measurement of the focal spot position and for long CT scans, where re-centring might be required due to electron beam drift.The target accuracy of the method should be a fraction of a voxel, e.g.0.1 µm for 1 µm voxels.The standard method to centre the electron-beam is implemented in the X-ray tube controller ("standard").It was improved by adjusting the parameters, e.g.increasing the averaging time ("improved").Eventually, we implemented a method that fits a double-sigmoid curve to the target current vs. centring-coil current data ("fit"), according to 1 with the centring-coil current being modulated and the target current being measured.is an offset and the amplitude of the target current, and the centre and width of the aperture by means of the centring coil current, and and the slopes of the two transitions, which correlate with the width of the electron beam.Figure 3a shows an example of the recorded target current, which drops when the aperture blocks the electron beam.
To determine the position of the X-ray focal spot independently, radiographs of a steel sphere (Ø 1.5 mm) glued to a quartz rod were analysed (figure 4a).Therefore, the sphere centre was determined by thresholding the images and determining the sphere centre using the particle analysis module in FIJI [9]. Figure 3b shows the standard deviation over five repeated centring cycles for the different methods.The "standard", "improved" and "fit" methods had a maximum deviation of, respectively, 2.1 µm, 1.1 µm and 0.25 µm.Whereas all methods are sufficient for CT measurements with voxel sizes of several micrometres, highresolution scans, i.e. voxel sizes ~ 1 µm, require sub-micrometre re-centring.This demand is only met by the method that employs a fit to the data to determine the centre of the aperture.So far, we could not identify the reason for the difference in accuracy for the y and z direction.

Compensation of X-ray focal spot drift through measurement of the X-ray tube position
The position of the X-ray focal spot was derived from radiographs of a steel sphere (Ø 1.5 mm) as described in section 3.1 (figure 4a).The physical positions of the X-ray tube and the steel sphere were simultaneously tracked using the METAS-CT metrology system [7].Whereas, the y-and z-coordinate were derived from the sphere position in the radiograph, the x-coordinate along the magnification axis was calculated from the change in sphere diameter.Since this is a less sensitive measure, it is more prone to noise.Figure 4b and c shows the actual focal spot position derived from radiographs with and without drift compensation using the X-ray tube tracking system.Along the magnification axis (x-coordinate) the thermal expansion of the X-ray tube head is visible.It is mainly caused by the power dissipated in the focussing coils of the X-ray tube.After compensation, the stability was within ±1 µm over 6 h.Sub-micrometre focal spot stability was achieved parallel to the detector plane (y-and z-coordinate) after a 15 min warm-up period.The oscillations observed in z-direction are due to temperature fluctuations in the actively cooled X-ray tube mounting structure.The residual focal spot instabilities are explained by electron beam fluctuations that are not registered by the X-ray tube tracking system.Such fluctuations can arise from subtle instabilities in the coil currents or local temperature variations.
Figure 4: (a) The X-ray focal spot drift was independently measured by tracking a steel sphere (blue curves).The X-ray tube tracking system was then used to compensate for the drift (red curves): (b) Focal spot drift in the central X-ray beam direction, i.e.X-ray tube expansion (x; see figure 1a for coordinate system), and (c) parallel to the detector plane (y and z; see figure 2a for coordinate system).

Influence of X-ray tube maintenance on X-ray focal spot position
Usually our CT system requires re-calibration after opening the cathode chamber for maintenance (see figure 5).Since the Xray tube measurement system remains stationary, it can also track the positional shift of the tube head after opening and closing the cathode chamber.Table 1 shows the positional shifts after opening the cathode chamber over the past months.The deviations are quite considerable, especially when compared to the drifts observed during a scan (figure 4).CT scans are very prone to yshifts, which are equal to an offset of the rotary axis that causes radial blur in the data.Displacements in the x-direction, i.e. parallel to the magnification axis, furthermore introduce scaling errors.In future, we plan to use the X-ray tube tracking system to make re-calibration after tube maintenance unnecessary.Table 1: Relative X-ray tube displacement after opening and closing the cathode chamber for maintenance (coordinate system in figure 1a).

Validation by dimensional CT measurements
Because X-ray focal spot drift is in particular detrimental at high magnifications, a small reference object was CT scanned and dimensionally measured.A multi-sphere standard consisting of 22 ruby spheres with a diameter of 0.3 mm and a maximum measurement length of 3.6 mm was thus employed (ZEISS MTCnano [10]).The positions, diameters and form errors of the spheres were calibrated on an ultra-precise tactile coordinate-measuring machine (METAS µCMM [11]) with an extended measurement uncertainty of 0.08 µm.The CT scan parameters are provided in table 2 and the X-ray tube position recorded during the scan is shown in figure 6a.After acquisition, the projections were corrected for gain, intensity and distortion [12].Projection data were reconstructed using the FDK algorithm in CERA 5.1.The uncompensated reconstruction was performed assuming a static CT geometry.The only correction applied was a horizontal rotary axis offset determined from the projection data.For the compensated reconstruction, the CT geometry was parametrised for each projection.Therefore, projection matrices were calculated based on the measured X-ray source position and a static rotary axis and detector position and supplied to the CERA reconstruction.The volume data was loaded into VG Studio MAX where the surface was determined using the local gradient method (advanced surface determination).The tactile probing points were mapped onto the CT data as shown in figure 6b.These points were used to fit spherical primitives to the CT data to obtain the position and diameters of all spheres.Subsequently, all 231 sphere centre-to-centre distances were calculated for the tactile reference and the CT data.Figure 7 shows the scale corrected sphere centre-to-centre distance deviations between the tactile reference measurement and the CT scan.By compensating the X-ray source drift, the standard deviation was reduced by nearly a factor of two.The major influence was thermal expansion of the X-ray tube, leading to a dynamic scale error that cannot be corrected for by adjusting the scale globally.This emphasises the importance of such a correction for high-resolution CT scans.

Conclusions and outlook
We evaluated an optical, image sensor based, X-ray tube tracking system for its performance to compensate for focal spot drifts.After a warm-up phase of 15 min, the system, which measures the positon of the tube target, did account for focal spot drifts within ±0.5 µm parallel to the detector surface and ±1 µm parallel to the magnification axis.Uncompensated for, the X-ray tube's thermal expansion was on the order of 10 µm, introducing significant scaling errors.It is emphasised that the stability of the focal spot relative to the X-ray tube is limiting the accuracy of the method.Subsequently, the X-ray tube tracking system's performance to compensate dimensional CT scans was evaluated.At high magnifications the improvements for unidirectional measurements, i.e. sphere centre-to-centre distances, were nearly a factor of two.In future, the system could also be used to track the displacement of the X-ray tube after maintenance, which would render re-calibration of the CT system unnecessary.

Figure 1 :
Figure 1: (a) Schematic of the METAS-CT geometry measurement system: The source position xs is determined using the X-ray tube tracking system, whereas the rotary axis and detector positions, xr and xd, are measured with laser interferometers.(b) X-ray tube tracking system (dashed box in (a)): A laser beam is directed onto a CMOS image sensor, attached to the X-ray tube, to measure its position.

Figure 2 :
Figure 2: Photograph of the X-ray tube tracking system: (a) front view of the X-ray tube with (b) image sensors attached.(c) The fibre is attached to the metrology frame by a bracket.The coordinate system used in the entire publication is shown in (a).

3 X-ray focal spot centring, tracking, and drift compensation 3 . 1
Improving the focal spot centring procedure

Figure 3 :
Figure 3: Improving the electron-beam centring procedure: (a) Target current in dependence of the centring-coil current.At the edges an aperture blocks the electron beam, causing the target current to drop.A double sigmoid (see equation (1)) is fitted to the data to determine the aperture centre.(b) Standard deviation (bar) and maximum deviation (error bar) over five repeats of the tested centring methods (see figure 2a for coordinate system).

Figure 5 :
Figure 5: The X-ray tube cathode chamber is opened (a) and closed (b) for maintenance, e.g.filament replacement, on a regular basis.

Figure 6 :
Figure6: (a) X-ray source drift measured during the CT scan of the small multi-sphere standard using the tracking system.(b) 3D rendering of the CT data of the small multi-sphere standard (Ø 3.9 mm, spheres Ø 0.3 mm); the tactile probing points are indicated on each sphere.

Figure 7 :
Figure 7: Scale corrected sphere centre-to-centre distances for an uncompensated ( = 0.20 µm) and an X-ray source drift compensated ( = 0.11 µm) CT scan; voxel size 2.1 µm.The equivalent Gaussian distributions are shown on the right side.

Table 2 :
CT scan parameters for the small multi-sphere standard