Utilization of single point uncertainties for geometry element regression analysis in dimensional X-ray computed tomography

The work outlined in this paper is examining ways to incorporate previously determined single point uncertainties of computed tomography (CT) measurements into a suitable regression analysis in order to evaluate standard geometry elements. The goal is to determine and reduce the measurement uncertainty for the specific parameters of a geometry element by using selective point rejections based on their single point uncertainty. Examinations show that the systematic error as well as the associated uncertainties can be reduced in certain scenarios compared to the statistical interpretations of repeated (simulated) measurements without considering the single point uncertainty. Nonetheless, these results still need to be verified for different data series and measurement setups.


Introduction
Geometrical work piece deviations are unavoidable and directly affect the function and quality of technological products.Therefore, the control of those deviations during the product engineering process is crucial for the creation of efficient products, which satisfy the customers' functional needs and performance specifications in order to be successful on the market.Consequently, tolerance management is regarded as an elementary subtask of the development of technological products, because it ensures the function as well as a sufficient product quality while maintaining reasonable production costs.That means, that geometric tolerances as an essential part of the product description greatly affect the functional capability, manufacturability, mountability, verifiability and the costs of the final product.The geometrical product specifications (GPS) standard [1] describes the framework used to verify specifications of a work piece.The properties and relations of geometry elements of the imperfect work piece surface are compared against the specification of the nominal geometry.Different operations are used to extract the properties and their relationships of ideal (as defined during the design) and non-ideal (result from real measurements) geometry elements; they are "partition", "extraction", "filtration", "association", "collection" and "construction".In this contribution, the regression analysis required for execution of the operator "association" is demonstrated for the cylinders geometry element.The distinction between different types of reference cylinders defines how the associated geometry element is constructed mathematically into the measurement point cloud representing that element.There are four reference cylinders (similar reference elements also exist for circles and other geometry elements): Least Squares Reference Cylinder (LSCY), Minimum Zone Reference Cylinders (MZCY), Maximum Inscribed Cylinder (MICY) and Minimum Circumscribed Cylinder (MCCY) [2].Least squares sizes used for the evaluation of geometrical parameters mathematically weigh all measurement coordinates associated to the geometry element equally, thus leaving the possibility open for faulty measurement vertices to negatively influence the measurement result.This contribution aims to increase the accuracy and precision of the resulting measurements by the introduction of suitable weighting factors for each processed measurement vertex.

Measurement setup and the acquisition of single point uncertainty values
A measurement result in dimensional metrology is obtained after the complete measurement chain processed a nonzero amount of physical quantities.The last step of the measurement chain is defined by the data evaluation, which in case of CT systems includes the processing of the X-ray projections until a surface representing the measurement object is obtained.After that, geometric evaluations can take place using the known GPS operators.Because the quality of the association operator using weighted least square sizes should be evaluated in the context of this contribution, the reference geometry of the evaluated geometry element needs to be known.In case a manufactured work piece is used, the necessary calibration using a coordinate measurement machine (CMM) can only be performed with a limited amount of single measurement points.Additionally, the measurement result of the calibration of a standard geometry element also includes the association GPS operator, which is the original target of the following investigations.Because of that, a virtual X-ray CT system, which is based on aRTist 2.8.3 [3,4] and was successfully adapted [5] to the real CT system Werth TomoCheck 200 3D, was used.The usage of a virtual measurement chain makes sure that the reference geometry of the (virtual) measurement object is equal to its nominal geometry (CAD), which is known exactly by definition.This principle was exploited in [6] for the correction of local systematic measurement errors, which were determined from repeated virtual CT measurements.The least squares measurement result is realized by a regression analysis, which takes all measurement points into account and subsequently minimizes a certain error function.Because the reference geometry is known, the weighted regression results can be evaluated by comparison against the (unweighted) results of the measurement results using VGStudio Max 3.2. 1 [7].Figure 1 shows the nominal geometry of the test specimen used for demonstration purposes.The perforated plate has a thickness of 10 mm and consists of eight holes of three different diameters.The material of the test specimen was set to aluminium in aRTist.The aim of this contribution is to apply a weighted regression analysis for the determination of the cylinder parameters defining the different holes in the perforated plate.At first, virtual measurements of the test specimen were repeatedly performed 20 times.The (virtual) CT settings were chosen as follows: 130 kV tube voltage, 275 µA tube current, X-ray pre-filtration with 0.5 mm Fe, 44 µm voxel size, 800 projections.The process of surface determination out of the reconstructed volume data was performed using the software VGStudio Max 3.2.1.The second step is to determine the uncertainty parameters needed to perform a weighted regression analysis, which was done using the single point uncertainty framework, already applied in [6,8,9].The concept of the single point uncertainty describes a methodology to statistically evaluate the local measurement results of a measurement series (n repeated single measurements) with respect to the associated reference geometry.Originating from each sampling point on the reference geometry, the distances to each single measurement are calculated in the direction of the reference surface normal vector.The resulting sets of distances ( , ,…, ) are assigned to their respective sampling points which effectively maps the computed distances onto the reference geometry.In the context of this contribution, the term "single point uncertainty" describes the local precision of a measurement series, which is represented by the standard deviation of the intersection distances .These local uncertainty parameters can then be used as weighting factors.Mathematically, this problem is described by a ray-triangle (the measurements are represented as triangulated surfaces) intersection test, which is a well-known problem in the field of computer graphics.Equation ( 1), describes a commonly used minimum storage ray-triangle-intersection test, which was formulated by Möller and Trumbore in 1997 [10].(1), [10] The sampling points were obtained using VGStudio Max.The CAD file of the perforated plate was imported and the cylinder parameters were determined for each drilling.For this procedure, the standard settings were used with the number of sampling points set to (maximum value possible) and .This results, dependent on the different hole sizes, in a fit point density of approx.274 to 1092 fit points per mm 2 (using ).After that, the fit points and the associated normal vectors of each cylinder were saved and exported as a CSV file.These fit points then represent the sampling points , which together with the associated normal vectors define the sampling vectors for the single point uncertainty framework introduced above.After that, the measurement results of the cylinder shell surfaces (holes) are defined by the measurement points (2), with denoting the number of measurement repetitions (20).The scatter (single point uncertainty) ∈ of each measurement vertex is defined by the standard deviation of the distances (3).The uncertainty is expected to be normally distributed. • (3) Figure 2: Distribution of uncertainty parameters for all cylinders (drillings) of the perforated plate (Figure 1) Figure 2 shows the resulting distribution of uncertainty parameters for all geometry elements.The observed uncertainties of the simulated data set are unrealistically small, because unfortunately we neglected to correctly consider various scattering effects within the simulation framework as used in this paper.Nonetheless, this measurement series is used for demonstration purposes in this contribution to support the intended proof of concept.

Cylinder regression error function
A key component of the weighted regression analysis conducted in the context of the GPS operator "association" is the choice of a suitable error function.There are two categories of least-squares surface fitting error functions referenced in various literature resources in this context, i.e. algebraic fitting and geometric fitting.These approaches differ by their respective definition of the error measure to be minimized.Although the terminology originated with the fitting problems of implicit surfaces, it reflects the research efforts made in past decades to develop fitting algorithms for general curves/surfaces.Geometric fitting designates the computation of the shortest distance (also known as geometric distance, orthogonal distance, or Euclidean distance in literature) between the model feature and the given point from a measurement [11] (4).The error function (4) minimizes the distance between each given (measurement / fit) point and its associated nearest point on the (continuous) model feature , with model parameters .The variable denotes the number of fit points.min , ∈ ‖ ‖ (4), [11] Unfortunately, contrary to its mathematically clear definition, the geometric distance generally depends nonlinearly on the model parameters.Taking this fact alone into consideration, the tasks involved in computing and minimizing the square sum of the geometric distances for general features are highly complex, with the exception of a few model features (line/plane and circle/sphere) [11].To bypass the analytical and computational difficulties associated with geometric fitting, many sources state that most data processing software tools use approximating measures of the geometric error [12][13][14][15].Many approximation measures (algebraic fits) try to avoid the nonlinearities to make a closed form solution possible, which results in significantly lower implementation and computation cost.In order to nonetheless verify certain standards for regression algorithms implemented in various commercial software tools, ISO 10360-6 was established as an objective method for testing software tools without requiring CMM manufacturers to disclose their software techniques (black-box approach) [11,16].Because of the issues reported above, the error function used in this contribution does not use true geometric fitting, but the model is nonetheless based on physical cylinder parameters.The mathematical background for the error function was formulated by Eberly [17].An infinite cylinder is specified by an axis containing a point and having unit-length direction , the radius of the cylinder is ∈ .Any point in space may be written uniquely as (5), where , , are unit-length vectors defining a right-handed orthogonal set.
(5) For to be on the cylinder shell surface, (6) has to hold, with denoting the identity matrix.
The formula (7) can then be rearranged as a function of .Because of the nonlinear nature of (7), the solution procedure needs a first good enough estimate of , which is set to the nominal cylinder axis vector, defined by the CAD model of the perforated plate (Figure 1).The algorithm was implemented using MATLAB and the final solution was obtained using the nonlinear programming solver "fminunc" [18].Note that the error function is based on the quadratic definition of a circle (6).The algebraic distance can be interpreted as proportional the annulus area between the fitted cylinder and the concentric cylinder passing through the given point .Thus, the minimized error is not equal to the orthogonal distance, which makes this function an algebraic fit function.An alternative formulation of an algebraic cylinder fit routine was published in [19].The mathematical formulation of an equivalent unweighted geometric error function, which is substantially more complex, can be found in [11].

Hypothesis testing by application of the Monte Carlo method
In order to evaluate the quality of the weighted regression analysis, a test framework based on the Monte Carlo method was set up.The goal of this setup is to evaluate the stability of the fit routines in case small scatter is introduced into the data sets.Each measurement (fit) point is associated with an uncertainty parameter (see section 2).Now a large amount (10 4 ) of normally distributed random numbers are generated for each point with (8).The associated distributional widths of two different evaluation modes and are then defined as 99 and , respectively.This procedure applies a defined scatter onto the measured surfaces to test the performance of the fit routine.
• with ∈ , , … , (8) min max min ( 9 ) After that the regression analyses for all indexes are performed.The recalculation routine for the uncertainty parameters into weighting factors for the error function ( 7) is defined in (9).This routine normalizes the uncertainties to the interval [0 1] and associates high uncertainties with low weights and vice versa.The results in form of the geometrical parameters of the cylinders (position vector, cylinder axis vector and radius) are compared against the unweighted evaluation of all 20 single measurements using VGStudio Max.The statistical analysis of the differences of the geometrical parameters against the nominal values from the CAD definition can then be visualized using box plots.The positional deviations are defined as the perpendicular distance of the fit cylinder position vector to the nominal cylinder axis.The angular deviations are defined as the angle between the fit cylinder axis and the nominal cylinder axis.All evaluations were performed using and fit points per cylinder (see section 2), before the application of the subsequently listed selection criteria.The following nomenclature is valid for Figures 3-8:  Color "black" denotes the geometric parameters from the VGStudio Max fitting routine for the 20 measurement repetitions.The Monte Carlo simulation was not performed for this type of evaluations. Color "red" denotes the results of the Monte Carlo fits using all fit points . Color "green" denotes the results of the Monte Carlo fits using all fit points satisfying .This criterion performs a pre-selection of the best fit points based on the local uncertainty value. Cyl_g_01 designates the cylinder with diameter 12 mm (Figure 1). Cyl_m_01, Cyl_m_02 and Cyl_m_03 denote cylinders with diameter 6 mm (Figure 1). Cyl_k_01, Cyl_k_02, Cyl_k_03 and Cyl_k_04 denote cylinders with diameter 3 mm (Figure 1).Consequently, the results of are not directly comparable to and and are only presented to show a reference.The deviations of the cylinder radius decrease for all cylinders if selection is applied in contrast to .This could indicate a positive correlation between low single point uncertainty and improved measurement result.Additionally, the results for show a small improvement of the radius deviation of some cylinders compared to .Nonetheless, the differences between the results are very small (mean values) to non-existent (scatter).The trend observed in Figure 3 cannot be extended to Figure 4 showing the positional deviations, because the results are very close together without a clearly observable trend.Figure 5 shows that the angular orientation of the fitted cylinder tends to be more unstable for compared to the other methods.The reduced effective number of fit points probably causes this.Again, there is no improvement observable for the weighted regression analysis compared to the VGS routine.Figures 6-8 show the fit results for fit points.The scatter of the geometric cylinder parameters is clearly increased compared to the examinations using fit points.The deviations of the radii are again improved for compared to .Overall, no clear trend of improvement can be determined by the usage of the weighted regression analysis with respect to the fit results using VGStudio Max, even the opposite is observable in case of the angular deviations of the cylinder axis.It is assumed that lower deviations with respect to the nominal geometry element parameters reflect a better regression result.Nevertheless, this premise only holds if the measurement (here: simulation) itself shows no systematic error (which should be the case for aRTist), because the regression analysis applied is only an interpretation of the measurement result.

Summary and outlook
In this contribution the weighted regression analysis for cylinders, which implemented single point uncertainty values as weighting factors, was compared against the solution of industry standard software solution VGStudio Max (unweighted).Comparing the statistical results of the evaluated geometrical cylinder parameters, no clear trend of improvement could be observed.There are several possible reasons for this:  The simulated measurement series exhibits only very small uncertainty values, which is caused by the omitted consideration of various scattering effects within the simulation framework. The implemented regression error function is rated among the algebraic error functions because it does not minimize the true geometric distance between fit points and the fitted geometry element.This introduces an additional uncertainty regarding the results obtained compared to a geometric error function. The Monte Carlo simulation of scattered fit points does not include the local lateral geometrical / physical correlation between neighbouring points.The random numbers are generated independently of each other such that the position of adjacent points could potentially be shifted in opposite directions of each other, which is not a realistic behaviour. Potentially better results can be achieved by adjusting the recalculation function of the weights (9) in such way that lower uncertainties are considered much stronger than higher uncertainties.In some scenarios, a positive correlation was observable between a per-selection of fit points associated with a lower single point uncertainty and a better measurement result.However, the scatter of the evaluated cylinder parameters is also increased for compared to .Future research will focus on improving data processing pipeline, especially the measurement data, the fit error functions and the setup of the Monte Carlo simulation need to be revisited.The ultimate goal is it to perform simulated measurements for a real measurement task and then improve the geometrical inspections by application of single point uncertainty parameters.Additionally, a qualitative evaluation of the implemented improved regression should be performed using the established testing criteria defined in ISO 10360-6.Finally, it should be possible to observe an improved measurement result by consideration of local uncertainty parameters if the data processing pipeline is capable enough.

Figure 1 :
Figure 1: Drawing of test specimen with thickness 10 mm and eight holes of three different sizes triangle plane, , : triangle barycentric coordinates of hit location , .: triangle edge points, , : ray start point / normalized direction vector

Figures 3 -
Figures 3-5 show the comparison of the deviations of the determined cylinder parameters radius, position and axis alignment with respect to the nominal values for the number of fit points set to .

Figure 3 : 5 Figure 4 : 5 Figure 5 :
Figure 3: Deviation of cylinder radius from nominal value, number of fit points is

Figure 6 :Figure 8 :
Figure 6: Deviation of cylinder radius from nominal value, number of fit points is