Uncertainty for uncorrected measurement results in X-ray computed tomography

In a recent paper [1], a mathematical formalism for the uncertainty estimation of dimensional measurements obtained from Xray computed tomography (CT) data was outlined. The formalism includes the treatment of both ‘corrected’—when the final result is corrected for bias—and ‘uncorrected’ measurement results. Such formalism is mainly based on the ISO 15530 series and the VDI/VDE 2630-2.1 guidelines. However, the treatment of uncertainty in the ‘uncorrected’ case—not compensated for bias—is limited to the use of the root-sum-of-squares of standard uncertainties (RSSu) approach. The present paper expands to other possibilities for the uncertainty estimation of ‘uncorrected’ results that could be applied to CT measurements, namely the root-sum-of-squares of expanded uncertainties (RSSU), the algebraic sum of expanded uncertainty with the signed bias (SUMU), the enlargement of the expanded uncertainty by adding the absolute value of the bias (SUMUMAX), and the so-called Uε method that sums the expanded uncertainty with the absolute value of the bias scaled by a factor assigned for a 95% distribution coverage. In addition, the alternative of using a maximum permissible error ( ��) statement—typically specified by the manufacturer of the CT instrument—to get a rough estimate of the expanded uncertainties of CT measurements is considered. Through a concrete example, by using dimensional X-ray CT data extracted from a metallic artifact that has internal features, these possibilities are analysed. From all the possibilities investigated, the RSSu method seems to be the most conservative for the estimation of expanded uncertainties associated with CT dimensional measurement that are not compensated for bias. On the other hand, uncertainty bounds estimated with the ��-based approach vary little from a constant value, and, therefore, risk creating significant underor over-estimation of the uncertainty intervals.


Introduction
As a transition from the classical error analysis to a modern approach of uncertainty analysis in measurement, in 1993 the International Organization for Standardization (ISO) published the first version of the 'Guide to the Expression of Uncertainty in Measurement' (GUM) [2], now managed and published by the Bureau International des Poids et Mesures (BIPM) [3].Within the metrology community and all ISO certified industries, the GUM now requires all measured quantity values to be corrected for all recognized significant systematic effects (and that every effort has been made to identify such effects) before performing any measurement uncertainty analysis.However, in X-ray CT dimensional measurements, a complete correction of all recognized significant systematic errors (known or unknown), as suggested by the GUM in its clause 3.2.4[3], is not always possible due to the complexity of the CT phenomena and the large number of influencing factors [4][5][6][7][8][9][10][11][12].In addition, situations in which a 'gross' estimate of bias (or an expected value for the combined effect of all systematic errors) in a measurement is known or suspected, but in which a specific correction 1 to the final measurement cannot be justified, are relatively common within several metrological contexts, especially when using complex measurement techniques.This is the case of metrology with X-ray CT, mainly due to the numerous sources contributing to the measurement process and analyses of CT data for dimensional measurements.Currently, the uncertainty analysis of CT measurements faces technical and economic challenges when trying to make bias estimates from all sources.An experimental investigation for bias determination on several types of measurands and measurement conditions is time-consuming.Such investigations involve complex multi-parametric measurement otpimization and may be technologically impractical.Therefore, in such cases, subtracting the bias to correct each individual CT measurement result (as the GUM suggests) is not feasible.Unfortunately, the GUM does not provide enough guidelines or recommendations for situations like this, and, therefore, professional judgement is necessary when deciding the appropriate method for reporting uncertainties of measurement results under the presence of uncorrected bias.In such situations, various alternatives to modify and incorporate the influence of a 'gross bias' into the overall uncertainty have been proposed [13][14][15][16][17][18][19][20][21][22].The common feature in all proposed solutions has been to enlarge the reported uncertainty to account for the known or suspected 'gross bias', aiming to at least include the 'true value' of the measurand within the expanded uncertainty interval that would be created around the measurement result.Table 1 lists various alternatives that have been recommended by the metrology community during the last twenty years, including main characteristics and remarks.These observations and justifications for the choice of a particular approach are similarly discussed elsewhere, e.g., see [12,[17][18][19][20][21][22][23][24].
Table 1: Options for modifying the expanded uncertainty (in the presence of bias information) instead of applying a bias correction [12].

Mathematical form
Abbrev.Main characteristics and some remarks for an uncorrected measured value with asymmetric right (upper) and left (lower) uncertainty sides.The uncorrected result is not in the middle of the uncertainty interval and the asymmetry of such interval increases with ||/ .At the expense of its asymmetry, this approach consistently provides a small uncertainty interval with good uncertainty coverage (~95%) for low biases, ||/ < , but for large biases the risk of a reported value lying outside the reported uncertainty interval increases.A limitation of the the asymmetry in this method is that a standard uncertainty cannot be recovered from the + ′ or − ′ values, although it makes the uncertainty interval ± symmetric around the corrected result.
′ = + ||, SUMU MAX Enlarges the expanded uncertainty by adding the absolute value of the bias.This approach considerably overestimates the uncertainty (i.e., provides higher coverage than expected) for low and moderate bias, i.e, when ||/ ≤ , but performs well for large biases.
Adds the absolute value of the bias linearly to the expanded uncertainty with a factor term adjusted to include 95% coverage of the corrected value distribution.This approach provides adjustment factors of coverage for bias values between 0.1 and 2 times the combined uncertainty [19].
In Table 1, the -term used in the mathematical expresions for the enlarged uncertainty ′ accounts for a standard combined uncertainty derived from the following approximation, i.e., the root-sum-of-squares of uncertainties of 'Type A' and 'Type B' (per GUM description) previously presented in [1], which includes the uncertainty of repeatability , the uncertainty associated to mechanical and thermal changes in the measured workpieces (e.g., changes in form or shape, surface texture, thermal expansion, elasticity and plasticity, etc.), and the term to account for the uncertainty of bias (i.e., the uncertainty about the correctness of the bias ).It is worth noting, in Table 1, that the standard uncertainty gets first adjusted or expanded to a certain 'level of confidence' through its multiplication by the coverage factor before being combined with the bias for enlargement.Non-enlarged expanded uncertainties are expressed as = , so the prime symbol ( ′ ) accompanying the in Table 1 denotes 'enlargement'.
Another alternative, when there is an (maximum permissible error) statement associated with the CT measuring system, is to express the expanded uncertainty of a CT measurement as [1,12], where is still the uncertainty from variations associated with mechanical and thermal changes in the measured workpiece.However, although the approach formulated by Eq. ( 2) can work under well defined conditions it is not of general validity.Uncertainty statements in dimensional metrology must be task-specific and should not solely rely on CT system's perfomance information.
Figure 1: Visualization of the NIST Frustum artefact by using a 3D rendering with one half side in solid view and the other half in semitransparent appearance to reveal the artefact's internal geometry.A 2D cross-sectional slice view is also shown.

Uncertainty estimation for X-ray CT measurements
The workpiece for this study is an aluminum prototype of an object, designed and fabricated by NIST, with internal geometric features that are revealed by Figure 1 (and also the left side of Figure 2).The workpiece's external surface is that of a truncated solid cone formed by cutting off the cone's tip section in a plane parallel to its base, and henceforth the workpiece for this study is referred as to the "Frustum artefact" [1].The NIST Frustum artefact was manufactured with two internal features that appear to be cylindrical voids in Figure 1.However, they are actually not cylinders as was revealed by measurements of the diameter dimensions of the top and bottom surfaces (or circles) of each feature, see [1,12]; they are cones.A full uncertainty analysis for dimensional measurements obtained from X-ray CT data, for both 'corrected' and 'uncorrected' results, has previously been presented elsewhere [1].However, the analysis of uncertainty in the 'uncorrected' case was limited to the use of the RSSu approach.This paper builds upon that work to expand such analysis to other possibilities, briefly introduced in Section 1, and apply them to analyze CT dimensional data.

9.35
For illustration, Table 2 shows the numerical results of different uncertainty estimations computed for a measurement of length obtained from X-ray CT data, namely the length "L" highlithed in Figure 1 or, to be clear, the length specified with a nominal value of 10 mm in the design plot that is shown in Figure 2 (on the left side).Figure 2 shows the reference coordinate system used for measurement and extraction of dimensional data.The length "L" was measured as the distance between the "Top plane" surface, which defines the XY plane, and an auxiliary plane constructed on the top of the inner feature that is labelled as F1 (in Figure 2).With the surface data (and volume rendering) obtained from X-ray CT reconstruction, such auxiliary plane was constructed using a grid arrangement of virtual touching probes separated 0.4 mm and evaluated using least squares fitting (and passing low-pass spline filters of 2.5 mm long wavelength cutoff with outlier elimination) [1,12].
Figure 2: Left: design plot for the NIST Frustum artefact.Right: geometrical features defining the reference coordinate system.The Z direction is determined by outward vectors that are perpendicular to the "Top plane" surface defining the XY plane [1].
A total of = 13 measurements were taken with the X-ray CT method, so the Student's distribution safety factor multiplier for a standard coverage level of 68.3% is −, = .68. , = 1.04 (see Table G.2 in the GUM [3]), leading to a standard uncertainty of repeatability of = 1.63 µm for the measurement of length "L" [12].The CT data employed proper dimensional corrections from thermal expansion, with the formula ∆ = − ˚C , using a coefficient of thermal expansion for aluminum of = 23.4 ± 3 µm•m -1 •˚C -1 [25] and knowing that the average temperature during the CT measurements was 21.8 ± 0.5 ˚C.Since no measurements of roughness were reported, the only contributor used for computing the uncertainty in Eq. ( 1) comes from the uncertainty in the value of temperature during measurement and uncertainties of coefficients of thermal expansion associated with values given in engineering references (or specified by different material datasheets, standard reference data or standard reference materials [26,27]).Here, = 6 µm•m -1 •˚C -1 and = 1 ˚C, which lead to a standard uncertainty for mechanical-thermal variations in the workpiece during CT measurement of ≈ = 0.10 µm.The bias , or estimate of uncorrected systematic errors, was computed from the difference between the average of CT measurements and CMM reference data, i.e., ≈ ̅ − , where the reference value was obtained with the NIST M48 CMM system [12,28].Since the differences between CT and CMM reference measurements ∆ = ̅ − were performed in a single experiment, the only contributor to the uncertainty of bias in Eq. ( 1) is the standard uncertainty of the reference value , which can be approximated by ≈ / , with the expanded uncertainty for calibration measurements of the NIST M48 CMM given by [29][30][31][32] where the dimension of length is measured in meters.For the length "L", of nominal value of 10 mm, the standard uncertainty of bias is then ≈ = 0.07 µm and, by combining in quadrature all the standard uncertainties , , and , as indicated by Eq. ( 1), and multiplying the result for the factor = , the expanded uncertainty for the CT measurement would be = 3.26 µm.However, since the bias for such measurement, computed from the difference between CT and CMM measurements ( ≈ ∆ ̅ = ̅ − ) is actually = −3.7 µm, as previously reported elsewhere [1,12], the expanded uncertainty seems to be underestimated.To safeguard an interval around a measured value that includes the operative 'true value' of the measurand-namely the CMM reference-with a high level of confidence, one of the methods presented in Table 1 should be used to enlarge the uncertainty estimate.For example, the RSSu approach gives an expanded uncertainty ′ = 8.09 µm, and in such a case the average result of the CT measurement for the length "L" can be expressed as L = (9.975± 0.008) mm, which in fact includes the operative 'true value' = 9.979 mm but also an overestimated uncertainty value.
Other similar intervals can be created around the (uncorrected) CT measurement with other symmetrical uncertainty enlargement methods, e.g., with the RSSU, SUMU MAX , and U ε approaches, for which uncertainty estimates are presented in Table 2.If the asymmetrical SUMU approach is used instead, an interval of the form ̅ − − ′ , ̅ + + ′ should be created.
Thus, with the uncertainty SUMU upper and lower limits listed in Table 2, the (uncorrected) CT measurement for the length "L" can be expressed as the interval L = (9.975,9.982) mm.
From the data provided in Table 2, note that while the RSSu approach leads to an overestimate of the expanded uncertainty of the uncorrected measurement, with an interval of ±0.008 mm around the ̅ value, the RSSU, SUMU, SUMU MAX , and U ε approaches create smaller uncertainty intervals that still contain the operative 'true value' .This observation is consistent across other sets of measurements as shown in Figure 3, where it can be seen that the uncertainty bounds for the RSSU, SUMU MAX , and U ε approaches are contained within the range of values covered by the RSSu intervals.While the RSSu method creates uncertainty intervals that are at least two times larger than the bias absolute value ||, when using the multiplying factor as equal to 2, the RSSU, SUMU MAX , and U ε approaches generate intervals that are smaller than 2||, but still larger than ||, so they can cover the bias, at least.In reality, the uncertainty intervals generated by the RSSU, SUMU MAX , and U ε methods are similar to each other, see Figure 3.It is however, with the asymmetrical SUMU approach that the smaller uncertainty intervals, still containing the operative 'true values' or CMM reference values, can be generated.The results presented in Figure 3 were computed using CT and CMM data from measurements sets obtained on the NIST Frustum artefact, which were analysed elsewhere [1,12], and correspond to the same data sets containing the measurement of length "L" analyzed above.
Figure 3: Enlarged uncertainty bounds as a function of the bias ( ≈ ̅ − ), for a coverage level of 95% ( ≈ ), when using dimensional measurements obtained for NIST Frustum artefact shown in Figure 2. The uncertainty intervals were computed with the methods for uncertainty enlargement presented in Table 1, and also with the MPE-based method from Eq. ( 2).Symmetrical intervals are drawn around the horizontal axis while the asymmetrical SUMU intervals are drawn around the bias measured value.The inset on the top left corner shows the bias "" values, and also the " " values, that were used to calculate the uncertainty bounds drawn in the main body of the figure for the RSSU, SUMU, SUMU MAX , and U ε approaches.The name of the measurands associated with each bias value are based on Figure 2 labels (and more specifically defined in [1,12]).
In the hypothetical case in which the actual measurements ̅ obtained from the CT machine are considered final, i.e., by ignoring the bias or assuming it did not exist, the " " values presented here would correspond to what would be the standard uncertainties of unbiased (or 'corrected') measurements.However, under such assumption, the expanded uncertainties ≈ that could be associated to the ̅ measurements would be underestimated, as is evident in several cases shown in the inset figure at the top left corner of Figure 3.The " " values would not be sufficiently large enough to cover the references or operative 'true values'.That is precisely the motivation for evaluating and comparing the enlargement uncertainty methods presented in this paper.From the data provided in Figure 3 it is seen that the uncertainty bounds for the RSSU, SUMU MAX , and U ε approaches are contained within the range of values covered by the RSSu intervals.While the RSSu method creates uncertainty intervals that are at least two times larger than the bias absolute value ||, when using a coverage level of about 95% ( = ), the RSSU, SUMU MAX , and U ε approaches generate intervals that are smaller than 2||, but still larger than ||, so they can at least cover the bias.In reality, the uncertainty intervals generated by the RSSU, SUMU MAX , and U ε methods seem to be very similar to each other as is shown in Figure 3.It is however, with the asymmetrical SUMU approach that the smaller uncertainty intervals, still containing the operative 'true values' or CMM reference values, can be generated.Figure 3 also shows the uncertainty bounds generated by the -based method from Eq. (2)-using statements previously defined elsewhere [1].In such a case, the uncertainty limits can be considered essentially constant (with values around ±9.3 µm) in function of the bias.Thus, the -based method creates intervals that overestimate the uncertainties when the bias results from the CT measurements are smaller than 5 µm, but at the same time, it potentially creates intervals that underestimate the uncertainties when the bias results are larger, e.g., larger than 15 µm.

Conclusions
The focus of this paper is to evaluate methods for the estimation of uncertainties that can be applied to the analysis of X-ray CT dimensional measurements for the case of uncorrected measurement results, i.e., under the presence of a known or 'suspected' bias.Two main possibilities were considered: (a) uncertainty 'enlargement' by one of the several methods described in recent literature (RSSu, RSSU, SUMU, SUMU MAX , and U ε ), and (b) uncertainty estimates based on statements as formulated by Eq. (2).A concrete example was presented to illustrate the application of these possibilities.To apply the possibility (a), an estimate of the 'gross bias' is necessary.Knowing such an estimate and not using it to correct the final measurement result sounds somewhat paradoxical.But situations like this are commonly found in attempts to avoid mishandling corrections when the credibility of the 'gross bias' estimate is questionable.In such cases, from the uncertainty 'enlargement' approaches, the RSSu method seems to be the more conservative by producing the largest uncertainties-at least two times larger than the bias absolute value-when = .The second possibility, or possibility (b), produces a rough estimate of uncertainty based on an statement for the CT instrument created in advance, which avoids a direct (or extra) comparison between CT and reference data but is of limited applicability (i.e., not of general validity).By adopting the statements typically reported in the literature of X-ray CT, it is observed that the uncertainty limits produced by Eq. ( 2) vary little from a constant value.Therefore, the -based method presents the risk of creating intervals that overestimate the CT uncertainties for ranges of low measurement bias, but underestimate them when the expected biases of the CT measurements are larger.