Empirical approaches to uncertainty analysis of X-ray computed tomography measurements: A review with examples

The substitution method-an empirical approach for uncertainty assessment (adapted from the ISO 15530-3 guidelines) that is based on a comparison between repeated measurements of a calibrated standard workpiece and measurements of a test (uncalibrated) sample-has been the approach most adopted over the past decade for estimation of measurement uncertainties in dimensional metrology with X-ray computed tomography (CT). However, questions about how to apply the substitution (or use calibrated workpieces) for X-ray CT metrology persist because the substitution method does not always encompass all the most relevant CT measurement influencing factors. This paper discusses some issues with the direct application of the ISO 15530 series for the estimation of CT measurement uncertainties and reviews other empirical methods that can be applied in uncertainty analyses in CT metrology. Special attention is placed to the treatment of uncertainties in the case of 'uncorrected' measurement results (i.e., not compensated for bias), which for X-ray CT has traditionally been limited to the use of the root-sum-of-squares of standard uncertainties (RSSu) approach. This article investigates other possibilities for 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 U-epsilon method that sums the expanded uncertainty with the absolute value of the bias scaled by a factor epsilon assigned for a 95% distribution coverage. In addition, the alternative of using a maximum permissible error (MPE) statement-typically specified by the manufacturer of the CT instrument-to generate a rough estimate of the expanded uncertainties of CT measurements is considered. Through two examples using dimensional X-ray CT data, these possibilities are analyzed. From all the possibilities for estimation of uncertainties associated with CT dimensional measurements that are not compensated for bias, the RSSu method produced the largest uncertainty estimates and thus seems to be the most conservative approach. For dimensioning geometric features mostly ranging between 10 mm and 60 mm, the expanded uncertainties (k = 2) computed with the RSSu method ranged from 0.6 mu m up to 72.7 mu m. It was with the asymmetrical SUMU approach that the smaller uncertainty intervals were generated. On the other hand, uncertainty bounds estimated with the MPE based approach changed little from a constant value (around +/- 9.5 mu m), and, therefore, risk creating significant underor over-estimation of the uncertainty intervals.