Investigating the inﬂuence of workpiece placement on the uncertainty of measurements in industrial computed tomography

Industrial X-ray computed tomography (CT) is a powerful technique for producing a three-dimensional model of an object and for performing dimensional measurements of an object’s inner and outer features. The diversity of the measuring objects causes a variety of CT setup parameters. The chosen CT setup parameters inﬂuence the uncertainty of dimensional measurements. A CT user determines the workpiece orientation and the distance between the workpiece and the X-ray source before a CT scan is commenced, and all other CT setup parameters are based on the workpiece placement inside the CT system. Hence, the work-piece placement is the trend-setting parameter of a CT scan. This paper presents a method for optimising a workpiece placement for dimensional measurements of multi-material workpieces. The method works with the stereolithography (STL) model of the workpiece and analytically tests the whole placement range to ﬁnd the placement that minimises the attenuation power of a workpiece. The method is validated by analysing the standard deviation of CT measurements on a multi-material workpiece. An analysis of variance on the uncertainty contributions demonstrated that the predicted workpiece placement was optimal.


Introduction
X-ray CT was developed for medical purposes in the 1970s and has extended its application boundaries.Computed tomography enables three-dimensional (3D) reconstruction of both the inner and outer features of a workpiece.In recent years, CT has gained a leading role in the field of non-destructive testing (NDT) and dimensional metrology [1][2][3].The user choice of CT setup parameters causes the most subjective and unstable input into the measurement uncertainty of a CT acquisition.The first parameters chosen before starting a CT acquisition is the workpiece placement that consists of orienting the workpiece inside the CT system and determining its position along the x-axis (Figure 1).The other CT parameters are affected by the workpiece placement.
There are different studies that focusing on the determination of appropriate CT setup parameters for NDT and dimensional Figure 1: The main variables for workpiece orientation metrology.A common approach to predict the optimal set of CT setup parameters is to simulate CT scans for various parameter combination and to evaluate the quality of CT acquisitions for each setup [4].Armikhanov et al. presented a simulation program for optimising the orientation of the workpiece [5].The program works with the 3D model of an object and analyses the stability and imaging quality for each tested orientation.However, this approach does not consider the influence of the distance between the X-ray source and the object (SOD) and the acquisition of the multi-material workpieces.Schmitt et al. have proposed an analytical method for optimising the imaging parameters of the CT system for mono-material and multi-material objects [6,7].However, this method is not optimising the workpiece placement.Currently, there is no established model that describes 9th Conference on Industrial Computed Tomography, Padova, Italy (iCT 2019) the minimisation of measurement uncertainty of the placement of multi-material workpieces.This paper presents a method to optimise of the SOD and the workpiece orientation in space.These aims are achieved with the simulation of the CT measurements using a ray-tracing algorithm, the minimisation of the required photon energy used for the measurement, and the minimisation of the enclosing sphere around the object (section 2).The validation of the method demonstrates a concurrence between the theoretical results and the experimental investigations (section 3).

Method
The SOD is equal to the distance between the X-ray source and the centre of the smallest sphere that contains the whole workpiece.The position of the rotary table in the CT system corresponds to the SOD.Lowering the SOD decreases the voxel size of the CT scan and increases its resolution.The optimal SOD is the shortest distance that still guarantees the complete reconstruction of the workpiece.The position of the sphere is minimised when it is tangent to the surface of the X-ray cone beam geometry.ϕ, θ , and γ characterise the workpiece orientation along the x-axis, y-axis, and z-axis (see Figure 1).γ is the projection angle that performing a full rotation of 360 • during the CT acquisition.ϕ and θ define the orientation of the workpiece.The determination of the most favourable orientation is achieved with a ray-tracing algorithm, takes the STL file of the workpiece, which is generated from the workpiece computer-aided design (CAD) model or produced by a preprocessing CT measurement.Ray-tracing guides each X-ray along a straight line originating from the X-ray source to a detector pixel.The X-ray source is modeled as a monochromatic point source.The penetration length of the j-th ray is a sum of the path lengths, d i , that the ray travels inside the m materials.The attenuation of the X-rays by the workpiece is considered to be the projection integrals along the penetration lengths of the rays.The algorithm calculates the projection integrals for the penetration length of each ray for each projection angle γ and each orientation of ϕ and θ .The projection integral of the j-th ray is expressed as a function of the photon energy and considers the linear attenuation coefficients, µ i , and the path length, d i , for each i-th material The algorithm calculates the minimum required photon energy for each orientation (ϕ,θ ) by maximising the contrast-to-noise ratio (CNR).The maximum of CNR corresponds to the mean projection integral that minimises the absolute error of that integral for all rays and all projection angles γ of the k-th orientation, (ϕ, θ ) k , [6, 7] The optimal orientation is the one with the smallest minimum photon energy: Furthermore, the optimal orientation must avoid Feldkamp reconstruction artefacts.Therefore, the algorithm evaluates the cosine of the angles between the normal vectors of the mesh surface triangles of the workpiece and the z-axis.

Experimental investigation
The aim of the experimental investigation is to demonstrate that the predicted workpiece placement is optimal.Here, a polyoxymethylene (POM) cube (100 mm) with two aluminium cylinders as inserts (length 60 mm, radius 10 mm) is used as a multi-material reference artefact (see Figure 2).Cylindrical holes break the axial symmetry of the cube and provide more complexity than the simple mono-material object.
Three measured orientations are depicted in the Table 1.These where adjusted using polystyrene clumping.Orientation 1 is the result of the prediction algorithm.Orientation 2 is the middle orientation, according to the prediction algorithm.Orientation 3 is affected by Feldkamp artefacts and represents the worst orientation, according to the algorithm.Twenty measurements were performed for each orientation.All other setup parameters (u = 195 kV, i = 230 µA, t = 250 ms, 1 mm aluminium as a prefilter, 1400 projections) were chosen in accordance with the minimum recommended photon energy for the first orientation and were constant for all measurements.The CT measurements were performed with the Werth Tomoscope HV Compact (Werth Messtechnik GmbH, Germany).Tomoscope HV Compact has a transmission target X-ray tube with a maximum tube voltage of 225 KV and a maximum tube power of 50 W. Twenty-four of the investigated features belonged to the aluminium cylinders and 24 belonged to the cube (see Tables 2, 3).The metrology measurements were performed with the software application WinWerth (Werth Messtechnik GmbH, Germany) based on the STL models calculated with the volume section algorithm for each of the two materials.The volume section algorithm enables the choice of an area for the stable calculation of a threshold value for multi-material workpieces.POM surface is determined between POM and glue, used to fix the cube, aluminium surface -between aluminium and POM to reduce the artefacts on material intersection areas.The measured features also specify high-attenuating aluminium as low-attenuating POM areas.The target variable of the experiment was the standard uncertainty of the measurement process by the workpiece orientation as an input.An analysis of the equality of the variance (ANOVA) demonstrates if the difference between the standard uncertainty values is significant.For significant differences, the probability (p-value) of the hypothesis that the orientations are equal should be less than 5%.Standard deviations and confidence intervals of three orientations for flatness of POM face 3 and roundness of aluminium cylinder 3 at the height 10 mm are demonstrated in Figure 3.The standard deviations and ANOVA results for all features are demonstrated in the Tables 4, 5.The minimum standard deviation is highlighted in bold and the features with significant differences between all three orientations are marked blue.The analysis proves that the predicted orientation is optimal for the performed measurements.Tests with another (not recommended by the algorithm) CT setup parameters indicate poor results for measurement uncertainty and neglect the difference between orientations.

Conclusions
The paper proposed a method for the determination for the most suitable placement of multi-material workpieces inside a CT system.The method is based on the calculation of the SOD with the enclosing sphere and the ray-tracing algorithm with an estimation of the minimum photon energy.The method was validated by examining a multi-material workpiece and analysing the equality of the variances of the standard uncertainties of the measurements of the workpiece features.All CT setup parameters except of orientation were chosen in accordance with the minimum recommended photon energy for the first orientation and were constant for all measurements.The results demonstrate the agreement between theoretical and experimental data.Tests with another (not recommended by the algorithm) CT setup parameters and another surface determination indicate poor results for measurement uncertainty and neglect the difference between orientations.At this moment, further research to improve the generality of the method are ongoing.

Figure 2 :
Figure 2: a) POM cube CAD model and b) STL produced by a CT measurement with investigated features

Table 1 :
Workpiece orientations and optimal photon energy

Table 2 :
POM features of interest

Table 3 :
Aluminium features of interest

Table 4 :
Standard deviations of POM features

Table 5 :
Standard deviations of aluminium features