Spot size and detector unsharpness determination for numerical measurement uncertainty determination

As the experimental measurement uncertainty evaluation for computed tomography is currently an expensive and timeconsuming task, numerical uncertainty evaluation (according to GUM Supplement 1) using a virtual CT is a promising alternative. For this digital twin to simulate accurate measurement results, the virtual CT system needs to be modelled as reallistically as possible using adequate input parameters for the simulation. This paper describes a method for the separated determination of two significant input parameters: focal spot size and detector unsharpness. The method is tested using a wide range of measurement parameters on a Zeiss Metrotom 1500 with different test specimen. It could be shown that the method yields reasonable results for most parameters. In addition, there is a dependence of the focal spot size on the ratio of tube voltage and tube current at constant power. This contradicts the wide-spread rule of thumb that the spot size depends on the tube power only. Indications of a changing focal spot shape at high powers could be detected.


Introduction
The importance of computed tomography (CT) for dimensional metrology has increased over the recent years.There are some advantages of CT metrology compared to other established dimensional metrology procedures such as constant measurement machine time for any arbitrary number of measurands.Furthermore, the precise non-destructive and contactless testing of inner geometries is a unique property in dimensional metrology.This represents a huge advantage for rising additive manufacturing technologies [1].Unfortunately, the expensive and time-consuming task specific uncertainty determination according to VDI/VDE 2630 Part 2.1 [2] constitutes a disadvantage of this comparatively young technology.One approach to adressing this problem is the numerical uncertainty evaluation using a virtual metrological CT (VMCT) (compare [3,4,5]) according to GUM supplement 1 [6] which would potentially consume less machine time.This approach is only feasible with a realistic simulation and adequate input parameters are necessary to obtain a realistic simulation.Since focal spot size and detector unsharpness are both influence factors that lead to unsharpness of the projection data, the separated determination of these parameters is nontrivial but necessary for a faithful simulation allowing e.g.tube power variation.In DIN EN 12543-1 [7], the focal spot size is described as "the X-ray emitting surface on the target of the X-ray tube from the perspective of the measuring device".In prior studies, focal spot size and detector unsharpness were determined by evaluating projections of sharp copper edges and modelling the unsharpness as a convolution of two Gaussian curves [3].There are other methods determining the focal spot size from the projection data by evaluating the projection of sharp edges, slits and pinholes (compare [8,9,10,11]) as well as using micro resolution charts with different slit widths, e.g. the JIMA RT RC-02 sample.The main disadvantage of these methods is that only the combined unsharpness due to both effects is evaluated.In other applications, this is unproblematic due to the use of very high magnifications that render the detector unsharpness negligible but are unacessible in usual CT setups.Furthermore, the quoted methods require expensive samples or an expensive experimental setup.Thus, the goal of this work is to present an inexpensive and easy to implement method that can be applied to any industrial CT setup in a straightforward fashion.This work extends the prior studies [3] by testing a wider parameter set.Most of this work was done during a Bachelor's thesis at FMT [12].The evaluated measurements show that it is possible to detect changes of the focal spot shape which arise at high tube power.Further, the influence of the ratio of tube power and tube voltage on the resulting spot size is discussed.

Theory and unsharpness determination
The approach has been introduced before in [3] and [13].While the effect of the focal spot size on the projection unsharpness depends on the geometric magnification, the influence of the detector unsharpness on the projection is assumed to be magnification independent.In consequence, it is possible to separate these influences by evaluating the combined unsharpness in the projection data for different geometric magnifications.In the mathematical model of the experiment, the edge is assumed to be geometrically ideal.This assumption is discussed in Chapter 3 based on a comparison of different real specimen.Figure 1 shows the mathematical model for the superposition of focal spot size and detector unsharpness with a sharp edge.
Figure 1: Superposition of focal spotsize and detector unsharpness with the resulting projection data (from [13]) The resulting projection can be modelled using Fourier optics.In the following, normalisation constants are omitted to enhance readability.The sharp edge is modelled by a step function (equation ( 1)).The effect of the focal spot size on the projection data is assumed to be a convolution with a Gaussian (equation ( 2)).The Gaussian's effective standard deviation scales with the geometric magnification .The detector unsharpness, which is assumed independent of the magnification, is also modelled by a Gaussian function (equation ( 3)).The resulting projection intensity along the edge on the detector, , is then described as the convolution of these functions in equation (4).

∝
(1) Here is the standard deviation of the Gaussian spot size, is the standard deviation of the Gaussian detector unsharpness and is the geometric magnification.Straightforward algebra shows that the resulting convolution kernel is still a Gaussian function (compare equation ( 5 is the resulting combined standard deviation.For the following analysis, the detector unsharpness and spot size are assumed to be isotropic and symmetrical.This assumption can be tested and holds true for the experimental setup at the Institute of Manufacturing Metrology (compare Chapter 4).The intensity profile along the edge is, according to equation ( 1) -( 5), given by: The derivative of this intensity profile again yields a Gaussian: ∝exp is given by (compare equation ( 5)): and can thus be calculated by the evaluation of intensity profiles at different magnifications and a subsequent fit of the resulting values.Assuming the spot size scales linearly with the tube power , ; ⋅ .In a first step, the grey values of 17 points on the projected edge are automatically selected (compare red lines in Figure 2).At each step, 30 adjacent pixel rows are averaged to supress single pixel noise and to obtain one averaged edge transition as grey value curve.As depicted in Figure 2, the derivative of the transition from high to low grey values has the shape of a Gaussian curve which can be fitted according to equation ( 7) to obtain as parameter for the transition.This fit is performed for each of the 17 averaged transitions.The 17 resulting values of are again averaged to yield one average value which is taken as the resulting edge unsharpness for this projection.In the next step, many projections are evaluated at different magnifications with the same tube power.The experimentally determined values of are then plotted against • as shown in Figure 3.By fitting a curve according to equation (8) to the experimental data, information about the detector unsharpness and the spot size can be gained separately.As the quality of the fit depends on the number of magnifications that are used, an adequate number of datapoints must be obtained.

Specimen
The measurements are generally, unless stated otherwise, performed using metallic sheets of 10 mm length and width with a thickness of 1 mm.The measured edge is finely grinded flat to prevent additional unsharpness.
Possible influence factors have to be tested to ensure that the variations of the grey value profiles is caused only by detector unsharpness and spot size.The effect of prefiltering has been tested using a 0.5 mm copper plate but it is negligibly small.The influence of the specimen material and thickness on the results has been determined through dedicated measurements.In these, specimen from copper and tungsten are compared to two steel specimen of different thicknesses (1 mm, 0.5 mm).The choosen material seems to only have a secondary influence on the resulting projection unsharpness.Also a variation of the sheet thickness only introduced a small effect.The dependence of the material on the detected unsharpness is shown in Figure 4.At higher magnification, the geometrical quality of the edge becomes increasingly important.In order to gauge the influence, the steel specimen, which is finely grinded flat at 0°, is compared to a non-post-processed edge as well as to edges grinded at defined angles.A less sharp jump between material and air produces a wider unsharpness in the resulting projection data.Figure 5 shows the resulting unsharpness depending on the edge quality.As expected, a non-flat edge (45°) leads to a wider transition.Small deviations from the perfect edge form (the edge grinded at 2° or the untreated edge) still have a small broadening effect compared to the finely grinded edge.Therefore, only measurements using a grinded edge are used.The quality of the edges has been tested by optical measurements using the coordinate measurement machine WerthVideo-Check-IP 250.Since the form deviations of the grinded edge as shown in Figure 6 are below the theoretical resolution limit (considering an overlaying detector unsharpness of roughly 3 pixels) of the Metrotom at the magnifications used, an error caused by the form deviations can be excluded.When comparing the results of the 0° grinded edge to a 2° grinded edge, there is almost no difference noticeable.

Experimental procedure and data evaluation
In order to obtain an appropriate number of data points, the measurement is carried out on a wide range of tube powers as well as on different current -voltage ratios.Experimental determination of the parameters is carried out over the possible power range of the Metrotom from 5 W to 500 W. The images were taken as uint16 grey value images using the Zeiss Metrotom 1500 at FMT with the Metrotom OS software (version 3.2.4.17214).The specimen is positioned on a fixture in the center of the beam path so that its surface becomes perpendicular to the X-rays.Perpendicularity is assessed by comparing projection data at 9 ° rotation, centering of the fixture is ensured visually by positioning a cylindrical specimen and rotating it by 360°.Gain and integration time of the detector were adjusted so that the detector remains in its linear response.To exclude systematic deviations of the detector and to compensate geometrical effects, a flat field correction is performed by manual acquisition of five bright images at active tube and free beam path and three dark images without active tube.The projections were then corrected pixelwise with the pixel-wise bright image mean and the dark image mean according to: Here, is the uncorrected grey value of the pixel and the corrected one.In this way, systematic errors of the detector can be excluded and grey values are normalised for further processing.The resulting projection data is superimposed with noise that increases with the applied gain.Even though the data has a good signal to noise ratio of about 20 dB for all parameter sets, it is necessary to smooth the data to get an adequate Gaussian fit for the derivative of the grey values of the transition (compare Chapter 2).This is done by applying a Gaussian filter to the data.As more intense smoothing yields a larger transition, a compromise between smoothing the noise and keeping the form of the transition must be found (compare diferent window lengths in figure 7).For all measurements presented in this paper, a Gaussian filter with 10 px width was used.All data analysis and image processing is carried out according to the method presented in Chapter 2 using MATLAB.
Figure 7: Influence of the applied Gaussian filter to the resulting grey value profile As mentioned in Chapter 2, the hypothesis of isotropic and symmetrical detector unsharpness and spot size needed to be tested.To gain information about the detector unsharpness, a circular steel sheet (radius 10 mm, thickness 1 mm) as depicted in Figure 8 (left) was tested at low power and low magnification to surpress unsharpness caused by the spot.When evaluating the grey value transition around the circle, there is almost no deviation detectable.To gain information about the spot, the edge is projected at high magnification and high power (500 W) to get the influence of the detector negligibly small.By rotating the edge in different angles (experimental setup in Figure 8; right), the resulting unsharpness depending on the applied angle can be determined.No significant deviations could be detected.During the experimental investigations, clear deviations could be observed between different measurement days, especially if between those the CT had a regular servicing with tube maintenance.Only consistent data of the same servicing period is shown in each single plot in this paper.This means that for the virtual simulation of a CT, new determination of the exact spot size and profile might be necessary after each servicing.

Results
Measurements for a wide range of tube powers were conducted using the steel specimen with 1 mm thickness.The evaluation of the detector unsharpness for different tube settings yields the assumption that the detector unsharpness is independent of the tube power (compare Figure 9).The determined standard deviation of the spot intensity, which is shown in Figure 10, is only approximately linearly proportional to the applied tube power.This shows a clear deviation from the assumption that the spotsize is proportional to the tube power only.If the spot size is quantified with the full width at half maximum, this corresponds to a spot size of ca. 1 µm/W tube power.Between the 300 W and 350 W measurements, the power dependence shows a clear change.This correlates with a change in tube acceleration voltage -while the measurements up to 300 W are conducted with 100 kV acceleration voltage, the voltage needed to be increased to 200 kV to obtain 350 W. This observation was a reason to further analyse the influence of voltage-current-ratio.To analyse the dependence of the measured edge transition unsharpness on the current-voltage-ratio, measurements using the tungsten specimen were performed at 200 W with varying ratio between tube voltage and tube current.The results are shown in Figure 11.There seems to be a ratio at which the focal spot size becomes minimal.Particulary at high voltage or high current, the spot size increases.Without further methods to inspect the interior of the X-ray tube, this effect cannot be analysed, but it is possible that it is due to the way the electron optics of the X-ray tube are regulated depending on tube parameters.At high tube powers, deviations from the assumed model (presented in Chapter 2) were detected.Figure 12 shows measurements using the steel specimen with 1 mm thickness for 200 W and 500 W tube power.While the Gaussian fit is a reasonable approximation for moderate tube powers, at higher power the deviations become prominent in the form of side maxima.The reason for this deviation might be extra focal radiation or intended defocusing of the electron optics to prevent heat damage of the target.Again, a direct analysis of the effect is out of the scope of this method.
This symmetric three-term Gaussian has been fitted to the derivative data (compare figure 13).While this clearly constitutes a better approximation for the curve, the deviations are still large and the model is based on an unproven hypothesis.A better model should either yield a better fit to the curve or be motivated by more insight into the actual focal spot shape.
Figure 13: three-term gaussian fit

Conclusion
It has been shown that the method presented can be applied for moderate power and moderate magnifications.The assumptions of a symmetrical and isotropic focal spot could be confirmed by measurements.In addition, the influence of the material and the edge quality was checked for the results.For high powers and high magnifications, deviations of the spot influence from a Gaussian become apparent.There, an alternative possibility of fitting the data has been shown.This alternative fit yields better resuls but is not satisfying.Here, a better spot size model based on more knowledge about the tube might be helpful.It could also be shown that the ratio of voltage and current at constant power influences the focal spot.For the measurements at constant power, there is a ratio at which the focal spot size becomes minimal.This information could be useful for measurements in which the penumbra effect is the limiting factor.The preliminary study presented in this paper will be continued in the EMPIR project AdvanCT to achieve systematic parameter determination for virtual metrological CT systems ('digital twins').

Figure 2 :
Figure 2: Determination of the combined unsharpness from spot size and detector unsharpness quantified with

Figure 3 :
Figure 3: Experimentally determined values of with varying at constant tube power of 100 W and fitted according to equation (8)

Figure 5 :
Figure 5: Influence of the edge quality for the steel specimen (180 W tube power, 200 kV voltage, 900 mA current)

Figure 6 :
Figure 6: Perpendicular view of the 0° grinded edge using a WerthVideo-Check-IP 250 (middle) in comparison to the unfinished edge (left) and the 45° angeled edge in side view (right)

Figure 8 :
Figure 8: Projection of circular specimen (left) and experimental setup for testing the hypothesis of spot isotropy and symmetry (right)

Figure 11 :
Figure 11: Experimentally determined dependence of on the current -voltage ratios and magnifications at same power (200 W)