On the uncertainty of surface determination in x-ray computed tomography for dimensional metrology

The proposed method for evaluating the uncertainty due to surface determination is evaluated for measured and simulated CT data of a workpiece with various cross-sectional geometries, see figure 2. The dimensional features of the workpiece include both internal and external radii, of which reference measurements are made on a Zeiss PRISMO tactile CMM. The reference measurement uncertainty is evaluated using the substitution method as per ISO15530-3 [4] and a setting ring gauge is used as the measurement standard. The reference measurements are given in table 1.

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The multi cross-section workpiece is scanned with an YXLON Y.FOX CT system. The CT system features a 160 kV x-ray source with a tungsten transmission target. The detector is a Varian PaxScan, with 1480  ×  1848 pixels and a pixel size of 0.127 mm. 720 projections are acquired with the x-ray source voltage and current set to 160 kV and 17.5 µA respectively.

Simulated projections of the workpiece are generated using in-house developed software. The simulation considers the geometric un-sharpness due to the finite size of both the x-ray focal spot and the detector pixels. Additionally, the polychromatic x-ray spectrum and energy dependence of the detector are considered alongside noise in the projections.

Both measured and simulated data sets are reconstructed into 1024³ CT volumes with VGStudio MAX 2.1. The measured data is aligned using a '3-2-1' procedure and CT images extracted in the same planes as the reference measurements, see figure 2.

The edges of objects in CT data are not ideal step edges, but ramp edges, this is due to the point spread function (PSF) that arises from data acquisition and reconstruction [5]. We adopt a discrete 2D ramp edge model based on the work of Rockett [6], who in turn based the model on that of Lyvers & Mitchell [7]. A step edge is parameterised in terms of its sub-pixel displacement from the origin t, its orientation θ and contrast-to-noise ratio (CNR). The intensity of each pixel is calculated based on the relative area of high and low intensity regions, see figure 3.

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To approximate a ramp edge the step edge is convolved with the PSF of the imaging system. An imaging system's response to an edge is the edge response function (ERF), the first derivative of which is an approximation of the line spread function (LSF) which in turn is roughly equivalent to a profile taken perpendicular to a circularly symmetric 2D PSF [8]. ASTM E1695-95 [9] provides instruction on evaluating the ERF and LSF from CT data; the procedure described in the standard is adopted here. Figures 4(a) and (b) show the ERF and LSF evaluated from the inner edge of the CT image shown figure 2(c). A Gaussian function is fitted to the LSF to estimate its spread; due to the rotational symmetry of the Gaussian the PSF is estimated by simply extending the 1D Gaussian to 2D.

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After convolution of the step edge with the PSF the ramp edge is superimposed with white Gaussian noise; the standard deviation of which is defined by evaluating the CNR for the inner edge of the CT image in figure 2(c). The noised ramp edge is then passed to the surface determination algorithm.

The sub-pixel displacement of the edge is varied from 0.1 to 0.4 pixels and the edge orientation from 0 to 45°. The edge symmetry implies all results are periodic for intervals of 45°. For each set of edge parameters 1000 repeated simulations are performed and the mean and standard deviation of the localisation error evaluated, a new noise waveform is generated for each repeat thus forming the Monte Carlo method.

The surface determination method is based on the Canny algorithm [3]. The image is first convolved with a 5  ×  5 Gaussian filter and the image gradient calculated using a 1  ×  5 derivative of Gaussian filter (DOG) applied in the x and y directions. The standard deviation of the Gaussian and DOG filters are both 5/6 pixels. Local maxima are identified via non-maximum suppression (NMS), followed by hysteresis thresholding of the gradient magnitude. Hysteresis thresholding is not necessary when evaluating the edge model since the edge is correctly identified by NMS. However, for the CT data-sets evaluated in section 3.3 hysteresis thresholding is required. The two thresholds are automatically defined by fitting a Rayleigh distribution to a smoothed histogram of the gradient magnitude after NMS. The thresholds are then a function of the distribution peak, as evaluated by Voorhees [10]. Finally, second order polynomials are fitted to the gradient magnitude normal to the edge; the turning point of which gives the sub-pixel surface position.

Figure 5 shows how the localisation error varies as a function of edge displacement t and edge orientation θ in the absence of noise. The localisation error increases with both t and θ and reaches a maximum for t = 0.4 and θ = 35–40°. The localisation error is very small, at worst it is approximately 1/25th of a pixel, however, in the presence of noise the localisation error becomes much larger. Figure 6 shows how the mean localisation error varies as a function of t and θ in the presence of noise. For values of t that are not equal to 0 a bias is observed and the localisation error becomes independent of θ. This is likely due to the combined effect of the PSF and noise blurring the edge into the neighbouring pixels; in CT this is known as the partial fill artefact which arises due to the finite spatial resolution of the imaging system. Mikulastik et al [11] showed that using polynomials for the sub-pixel estimate can introduce a bias in the estimated edge position; the bias seen in figure 6 is in line with the calculations of Mikulastik et al.

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For each run of the Monte Carlo simulation the x and y coordinates of the edge position are estimated, figure 7 shows how the standard deviation of these xy coordinates varies as a function of t and θ. It is found the coordinate uncertainties act in opposition and converge at 45°. Figure 7 also shows for θ = 0° (a vertical edge) the y coordinate uncertainty is minimum whilst the x coordinate uncertainty is a maximum; i.e. the coordinate uncertainty is maximum when perpendicular to the edge and minimum when parallel to the edge. This explains why both the x and y coordinate uncertainties converge at θ = 45°.

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Figure 7 shows the standard deviation of the x and y coordinates vary with both edge orientation and sub-pixel displacement. This is an important result because least-squares fitting is based on the assumption the standard deviation of coordinate error is constant; this property is termed homoscedasticity [12]. Thus for geometric features with varying edge orientation, such as circles, spheres and ellipses this assumption of homoscedasticity is violated. If, however, the uncertainty structure of the coordinate set is known then the weighted least-squares method can be used, whereby each coordinate is weighted by the inverse of its variance.

Figure 7 is particularly useful as it enables the uncertainty of an edge's position to be estimated for a given t and θ. Thus if we know t and θ for each surface point in a CT data-set we can estimate the respective coordinate uncertainties. θ is estimated directly by the surface determination algorithm. Unfortunately t can't be estimated as the actual edge position is unknown. Instead we use the maximum envelope of figure 7; this is shown in figure 8. With figure 8 the uncertainty of an edge's position can be estimated with only knowledge of its orientation θ. Figure 8 forms a LUT that is used in the next section to estimate the uncertainty of surface coordinates from measured and simulated CT data. The individual coordinate uncertainties are then propagated through a least-squares fit to the final measurement result.

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The results of the Monte Carlo simulation are used in this section to estimate the measurement uncertainty due to surface determination for the inner radius of the multi cross-section aluminium workpiece; both measured and simulated data are considered. The graphs in section 3.1 correspond to the simulated CT image in figure 2(c), thus these results are used as a worked example.

Applying the surface determination algorithm to the CT image in figure 2(c) yields a vector of surface coordinates x_i and y_i, alongside a vector of edge directions θ_i, for i = 1 to m. With the aid of figure 8 the uncertainty of each coordinate x_i, y_i, denoted σx_i and σy_i respectively, is estimated. We fit a circle to the coordinates x_i and y_i using weighted least-squares so as to consider the uncertainty of each point. Weighted least-squares minimises

$$\begin{eqnarray}&&F=\underset{i=1}{\overset{m}{\sum}} {{w}_{i}}{{f}_{i}}^{2}\end{eqnarray} \tag{ 1 }$$

where w_i is a vector of weights. The weights are a function of the standard deviation of the localisation error σ_i such that high-quality data points influence the fit more than low-quality data points:

$$\begin{eqnarray}&&{{w}_{i}}=1/\sigma _{i}^{2}.\end{eqnarray} \tag{ 2 }$$

For the linear least-squares circle-fit, ${{f}_{i}}$ in equation (1) is written:

$$\begin{eqnarray}&&{{f}_{i}}=r_{i}^{2}-{{r}^{2}},\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&r_{i}^{2}={{\left({{x}_{i}}-{{x}_{c}}\right)}^{2}}+{{\left({{y}_{i}}-{{y}_{c}}\right)}^{2}},\end{eqnarray} \tag{ 4 }$$

where x_c, y_c represent the centre coordinates of the circle and r its radius. Equation (3) is therefore rewritten:

$$\begin{eqnarray}&&{{f}_{i}}={{\left({{x}_{i}}-{{x}_{c}}\right)}^{2}}+{{\left({{y}_{i}}-{{y}_{c}}\right)}^{2}}-{{r}^{2}}.\end{eqnarray} \tag{ 5 }$$

Expanding:

$$\begin{eqnarray}&&{{f}_{i}}=-2{{x}_{i}}{{x}_{c}}-2{{y}_{i}}{{y}_{c}}+\left(x_{c}^{2}+y_{c}^{2}-{{r}^{2}}\right)+\left(x_{i}^{2}+y_{i}^{2}\right),\end{eqnarray} \tag{ 6 }$$

let

$$\begin{eqnarray}&&\rho =x_{c}^{2}+y_{c}^{2}-{{r}^{2}},\end{eqnarray} \tag{ 7 }$$

which simplifies equation (6) to

$$\begin{eqnarray}&&{{f}_{i}}=-2{{x}_{i}}{{x}_{c}}-2{{y}_{i}}{{y}_{c}}+\rho +\left(x_{i}^{2}+y_{i}^{2}\right).\end{eqnarray} \tag{ 8 }$$

Forming the equations in matrix notation:

$$\begin{eqnarray}\begin{array}{*{35}{l}} W=\left[\begin{array}{c} 1/\sigma _{1}^{2} & \cdots & 1/\sigma _{m}^{2} \\ \end{array}\right],\ \ A=\left[\begin{array}{c} 2{{x}_{1}} & 2{{y}_{2}} & -1 \\ \vdots & \vdots & \vdots \\ 2{{x}_{m}} & 2{{y}_{m}} & -1 \\ \end{array}\right], \\ u=\left[\begin{array}{c} {{x}_{c}} \\ {{y}_{c}} \\ \rho \\ \end{array}\right]~b=\left[\begin{array}{c} x_{1}^{2}+y_{1}^{2} \\ \vdots \\ x_{m}^{2}+y_{m}^{2} \\ \end{array}\right]. \\ \end{array}\end{eqnarray} \tag{ 9 }$$

The u that minimises F is the solution of the linear system of equations

$$\begin{eqnarray}&&({{A}^{T}}WA)u={{A}^{T}}Wb.\end{eqnarray} \tag{ 10 }$$

The radius estimate is recovered by

$$\begin{eqnarray}&&r=\sqrt{x_{c}^{2}+y_{c}^{2}-\rho}.\end{eqnarray} \tag{ 11 }$$

With estimates of x_c, y_c and r the uncertainty of the fitted radius is evaluated via a second Monte Carlo simulation. Pseudo-random numbers are added to each surface coordinate x_i, y_i and a circle fitted to the coordinate set. This is repeated 100 000 times in order to evaluate the standard deviation of the radius estimate. The standard deviations of the pseudo-random numbers added to each coordinate are defined by σx_i and σy_i as evaluated via figure 8. The coordinate set for each run of the Monte Carlo simulation is therefore

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{x}_{i}}={{x}_{i}}+\sigma {{x}_{i}}n \\ {{y}_{i}}={{y}_{i}}+\sigma {{y}_{i}}n \\ \end{array}\end{eqnarray} \tag{ 12 }$$

where n is a random number drawn from the standard normal distribution.

The results of the weighted least-squares circle-fit for the CT image in figure 2(c) are given in table 2 alongside the uncertainty of the radius evaluated via the second Monte Carlo simulation. The results show the radius is measured with sub-pixel uncertainty. Furthermore, the radius uncertainty is significantly lower than the individual coordinate uncertainties shown in figure 8. Such a low uncertainty is unexpected; however; the result can be rationalised based on the following considerations: firstly, the number of surface points used in the geometric fit is large (m = 987); if m is large then the radius uncertainty will be small. Secondly, for the considered measurement task, the dimension of interest r is large compared to the pixel size; r is nominally 22.6 mm whilst the pixel size is 49 µm. Based on this reasoning, the uncertainty due to surface determination will be larger for dimensional features that fill only a portion of a CT image and for features fitted to few surface points.

To verify the result presented in table 2 we make use of the CT simulation tool described in section 2.1; 20 CT data-sets of the cross-section shown in figure 2(c) are generated, each with a different noise waveform. For each data-set surface determination is performed and the radius evaluated; the standard deviation of the repeated radius estimation is calculated and found to be 0.0066 pixels. This is in agreement with the result presented in table 2 and thus verifies our method.

Finally, the developed method is applied to the measured CT image shown in figure 2(d). The steps taken are summarised as follows:

• The PSF of the inner edge is evaluated. Fitting a Gaussian function to the PSF yields a standard deviation of 7.4 pixels.
• The CNR of the inner edge is evaluated and found to be 7.3.
• The above PSF and CNR are input to the Monte Carlo simulation so as to generate a LUT for σx and σy.
• Surface determination is performed giving the surface coordinates x_i and y_i.
• Using the LUT σx_i and σy_i are evaluated.
• The surface coordinates x_i, y_i and their respective uncertainties σx_i and σy_i are input to the second Monte Carlo simulation to estimate uncertainty of the radius due to surface determination.

The results for estimating the uncertainty due to surface determination for the measured CT data are given in table 3. Comparing the results in tables 2 and 3 shows the measurement uncertainty due to surface determination is larger for the measured data than the simulated data. This result is predominantly due to fewer surface points being used when fitting the least-squares circle. For the measured data m = 121 whereas for the simulated data m = 987. Fewer surface points are used because the measured data suffers from additional artefacts, such as ring artefacts, which are caused by defective detector pixels. As a consequence, surface points that deviate from the fitted geometric form by more than one pixel are omitted from the least-squares fit. In general, omitting outliers in this way improves the fit but reduces the number of fit points, which, as can be seen from the results, increases the measurement uncertainty due to surface determination.