SINCE2013 Singapore International NDT Conference & Exhibition 2013, 19-20 July 2013

The Application of Voxel Size Correction in X-ray Computed for Dimensional Metrology

Joseph J. LIFTON1,2, Andrew A. MALCOLM2, John W. MCBRIDE1,3, Kevin J. CROSS1

1The University of Southampton, United Kingdom; Email: [email protected]. 2Singapore Institute of Manufacturing Technology, Singapore; Email: [email protected]. 3The University of Southampton Malaysia Campus (USMC), Malaysia; Email: [email protected].

Abstract X-ray computed tomography (CT) is a non-destructive, radiographic scanning technique that enables the visualisation and dimensional evaluation of both internal and external features of a workpiece; it is therefore an attractive alternative for measurement tasks that prove problematic for conventional tactile and optical instruments. The data output of a CT measurement is a volume of grey value integers that describe the material distribution of the scanned workpiece; the relative spacing of volume-elements (voxels) is termed voxel size and influences all dimensional information evaluated from a CT data-set. Voxel size is defined by the position of a workpiece relative to the X-ray source and detector, and is therefore prone to axis position errors, errors in the geometric alignment of the CT system’s hardware, and the positional drift of the X-ray focal spot. In this work a method is presented for calculating a voxel scaling factor that corrects for voxel size errors, and this method is then applied to a general X-ray CT measurement task and demonstrated to reduce measurement errors.

Keywords: X-ray computed tomography, dimensional metrology, calibration, uncertainty, voxel size.

1. Introduction

X-ray computed tomography (CT) is a non-destructive radiographic scanning technique for imaging cross-sections of a workpiece. The X-ray CT measurement process can be summarised as follows: an X-ray source emits a cone-beam of X-rays that are attenuated as they propagate through a workpiece; X-rays that fully penetrate the workpiece fall incident on a flat-panel detector, the output of which is a transmission image, termed a projection. Projections are made for multiple angular positions of the workpiece; based on these projections cross-sectional images of the workpiece are calculated. The cross-sectional images, termed CT images, describe the material density distribution of the workpiece in the considered plane; the process of calculating CT images from projections is termed reconstruction. Individual CT images can be visualised as grey value images, alternatively multiple CT images can be ‘stacked’, thus forming a CT volume, in which three-dimensional are termed voxels. For a workpiece made from a single material, voxels will either belong to the workpiece, or to air; in order to evaluate dimensional information from a CT volume, the points at which voxels transition from air to workpiece must be identified, i.e. the position of the workpiece’s surface must be defined; this is termed surface determination. Surface determination is typically realised by a global grey value thresholding operation, whereby a single grey value is selected such that voxels with grey values higher than the threshold belong to the workpiece, and voxels with grey values lower than the threshold belong to air [1]. The data output from surface determination is a set of coordinates that describe the position of the workpiece’s surface; these surface coordinates are then used to evaluate the dimensions of the workpiece. The workflow for an X-ray CT measurement is summarised in figure 1.

Projection Surface Dimensional Reconstruction acquisition determination evaluation

Figure 1: Summary of the workflow for an X-ray CT measurement.

The relative positions of surface coordinates depends on the dimensions of the voxels, termed the voxel size; as a result of this dependency, voxel size directly influences all dimensional information evaluated from a CT volume. Voxel size, in the CT image plane, is a function of geometric magnification, , the number of detector pixels perpendicular to the axis about which the workpiece is rotated, , the detector size, , and the number of voxels into which the data is reconstructed, .

(1)

The magnification is in turn a function of the Source-to-Detector-Distance, , and the Source-to-Object distance, :

(2)

Figure 2 shows each term used in the voxel size calculation.

X-ray source D D O D S S Workpiece

Number of voxels (NoV) Centre of rotation y

Detector x Pixel size (d) Figure 2: Geometry of a CT system, and terms used in voxel size calculation. Looking to equation (1), there are two terms that are susceptible to error: (i) magnification and (ii) pixel size. Magnification is a function of , this being an axis position readout; Welkenhuyzen et al. [2] showed the magnification axis of an industrial CT system to have poor bi-directional repeatability; such axis position errors directly influence Additionally, the rotational stage is subject to radial run-out errors, i.e. radial deviation of the centre of rotation from its mean position; radial run-out errors cause to vary throughout projection acquisition, alongside a shifting of the position of the projection on the detector. Kumar et al. [3] showed detector tilt to cause the magnification to vary locally within projections, and suggested that this error causes distortions in the CT volume, thus leading to errors in length measurements. Hiller et al. [4] showed the position of the X-ray focal spot to move within the projection plane; this also has the effect of shifting the position of the projection on the detector. All of the aforementioned sources of error impact magnification and therefore voxel size when evaluated via equation (1). With regard to pixel size errors, Weiß et al. [5] showed scintillation type detectors to suffer from localised distortions; these distortions were attributed to the scintillator crystals not being perfectly perpendicular to the photodiode array. Weiß proposed the localised detector distortions were the reason for dimensional measurements varying depending on which portion of the detector was used.

It is clear that equation (1) is prone to errors originating from imperfect axes, geometric misalignment of the CT hardware and the positional drift of the X-ray focal spot. A more robust method for defining voxel size is to scan a reference workpiece under the same conditions as the actual workpiece; by evaluating the dimensions of the reference workpiece the voxel size of the actual workpiece is adjusted accordingly. This reference dimension must be independent of surface determination; an example is the centre-to-centre distance between two spheres, since it is assumed that the centre coordinates of a sphere remain unchanged regardless of the surface determination step; such dimensions are hereafter termed threshold independent dimensions [1]. Consider the reference dimension , where is evaluated via an accurate measurement instrument, and is the distance between the centres of two spheres; the centre coordinates of the two spheres are ( ) and ( ), thus is calculated as:

√( ) ( ) ( ) (3)

Consider the same dimension evaluated via X-ray CT, ; the ratio of and gives a voxel scaling factor, such that multiplying the original voxel size by this scaling factor, gives a new, corrected voxel size:

(4)

The coordinate systems of both the reference and CT measurements do not need to be the same, since it is the resultant lengths that are evaluated in equation (4), and not the individual components. Evaluating voxel size according to equation (4) considers all errors present in the CT measurement procedure, alongside all data processing steps (reconstruction & surface determination). Numerous authors [6, 7, 8] have discussed the use of equation (4) and proposed reference workpieces for the purpose of voxel size correction; however, few have shown the application of this method and how it might be used for the measurement of an actual workpiece; this is the purpose of the present work.

The remainder of this paper is organised as follows: in section 2.1 a reference workpiece featuring three threshold independent reference dimensions is introduced, and the methodology adopted in defining its reference dimensions is described. In section 2.2 a method for voxel size correction is introduced based on a linear regression model. In section 2.3 an actual workpiece is introduced, and the methodology adopted in defining its dimensions is described. In section 2.4 the X-ray CT measurement procedure is described. Finally in section 3 a comparison of measurement results is given; the dimensions of the actual workpiece are evaluated with and without voxel size correction.

2. Methodology

2.1 A reference workpiece for voxel size correction

Figure 3 shows the reference workpiece used for the purpose of voxel size correction, it features three ruby spheres, each with a nominal radius of 1.5 mm. The ruby spheres are mounted on pultruded carbon fibre tubes, which are glued into holes drilled in an aluminium base. Ruby spheres are chosen due to their low form errors and low roughness; ruby is dimensionally stable, and its X-ray attenuation properties are comparable to aluminium, meaning it is well suited for the scanning energies typically used for industrial X-ray CT. The top, middle and bottom sphere centres are denoted 1, 2 and 3 respectively, the centre-to- centre distances are therefore denoted , , and ; with each centre-to-centre distance presenting a threshold independent distance.

The reference workpiece is measured on a TaiCaan XYRIS 4000 optical profiler (TaiCaan Technologies, Southampton, UK); this non-contact instrument features an motion table with a resolution of 10 nm, coupled with a white-light (WL) confocal sensor with a resolution of 10 nm, a gauge range of 0.35 mm, an angular tolerance of 25°, and a focal spot size of 7 μm. The gauge range and angular tolerance of the WL sensor limit the maximum measurable area of each sphere; for this reason the highest of each sphere, termed the crown, is centred in a square measurement area, the side length of which is 1.27 mm; 201 by 201 data points are measured for each sphere and each measurement is repeated five times. The centre coordinates of each sphere are calculated by fitting spheres to each measurement result using a non-linear least-squares sphere-fitting algorithm (NLLS) [9], thus the centre-to-centre distances , , and are calculated using trigonometry.

Figure 3: Reference workpiece for voxel size correction.

2.2 A method for voxel size correction

The reference workpiece features three different reference dimensions, each of which has a different orientation in the resulting CT volume. Each of the three reference dimensions can be evaluated to derive a different voxel scaling factor as per equation (4); if however, all three reference dimensions are evaluated simultaneously a ‘best-fit’ voxel scaling factor is derived that considers a larger portion of the CT volume, and is therefore more applicable for general measurement tasks.

Consider the three reference dimensions of the ruby sphere workpiece, evaluated via the optical profiler: , and the same dimensions evaluated via X-ray CT: . A voxel scaling factor is derived which considers all three dimensions accordingly:

[ ] [ ] (5)

Equation (5) represents a linear regression model; the left hand term is the dependent variable, the right hand term is the independent variable, and is the unknown coefficient. The ordinary least-squares (OLS) method selects such that the sum of the squared difference between the dependent and independent variables is minimised; thus the ‘best-fit’ voxel scaling factor is derived. This method is similar to that briefly described by Muller et al. [8]; who similarly used a linear regression model to calculate a voxel scaling factor; however, Muller et al. defined their voxel scaling factor as ⁄( ).

2.3 The actual workpiece

The actual workpiece is a plastic brick, see figure 4; the workpiece features eight cylindrical knobs aligned in a row. The centre-to-centre distances of the cylindrical knobs present threshold independent dimensions; thus the performance of the voxel size correction method can be evaluated irrespective of the influence surface determination may have. Reference measurements of the actual workpiece are made on a Zeiss PRISMO Ultra tactile CMM (Zeiss, Germany). Circular profiles around six of the knobs are measured 1 mm above the top face of the brick, see figure 4, these measurements are repeated three times; for all measurements, a ruby stylus of 0.5 mm radius is used with a probing force of 100 mN. In total, the actual workpiece features 15 centre-to-centre distances, these are denoted: L12, L13, L14, L15, L16, L23, L24, L25, L26, L34, L35, L36, L45, L46, and L56.

Figure 4: The actual workpiece; a plastic brick.

2.4 X–ray CT measurement procedure

To demonstrate the application of voxel size correction for the measurement of the actual workpiece the following measurement procedure is adopted: (i) the scan settings are configured for the measurement of the actual workpiece; (ii) the actual workpiece is scanned; (iii) the actual workpiece is removed from the CT system and replaced by the reference workpiece; (iv) the reference workpiece is scanned, all scan settings remain unchanged. This procedure is repeated three times, and the scan settings and magnification remain unchanged throughout.

The CT scans are performed on an YXLON Y. FOX (XYLON International GmbH, Hamburg, Germany); this instrument features a 160 keV tungsten transmission target X-ray source and a Varian PaxScan detector (Varian Medical Systems, Inc. CA, USA) with dimensions 1480 by 1848 pixels, of size 0.127 mm; a summary of the scan settings is given in table 1, and the temperature in the CT scanner is recorded as 26 ± 1°C. All projections are reconstructed using VGStudio’s implementation of the FDK algorithm (Volume Graphics, Heidelberg, Germany) and surface determination is performed using a global thresholding operation. After surface determination, the reference dimensions of the actual workpiece are evaluated according to the measurement strategy previously described: i.e. circular profiles around six of the knobs are measured 1 mm above the top face of the brick; thus the distance between the centres is calculated using trigonometry. Similarly, the centre coordinates of the reference workpiece’s ruby spheres are estimated using the NLLS sphere-fitting algorithm; the three centre-to-centre distances are then calculated using trigonometry; all the measurements are corrected to 20°C.

Table 1: Scan settings for the CT measurements of the actual workpiece and the reference workpiece.

Scan setting Value Acceleration voltage (keV) 70 Filament current (μA) 24 Detector exposure time (s) 2 Number of voxels 5123 Number of projections 804 Initial voxel size & (μm) 128, 160

3. Results

Figure 5 shows the X-ray CT measurement results of the reference workpiece plotted against the optical profiler reference measurements; the gradient of the best-fit line through the origin is equal to the voxel scaling factor in equation (5). It is observed that is less than one, meaning the X-ray CT measurements are larger than the optical profiler measurements; by multiplying the voxel size by the voxel size is rescaled accordingly.

Figure 6 shows the measurement error for each of the 15 centre-to-centre distances of the actual workpiece, evaluated with and without voxel size correction. The measurement error is calculated as X-ray CT result minus CMM result, and the error bars consider the standard deviations of both the repeated reference measurements, and the repeated CT measurements. Figure 6 confirms the previous observation of the voxel size being too large, in that all the centre-to-centre distances of the actual workpiece are too large without voxel size correction. Figure 6 shows that when the voxel size is multiplied by , and the centre-to-centre distances re-evaluated, the measurement error is drastically reduced; with voxel scaling, the measurement error for several measurement results is reduced to within the bounds of the error bars. The horizontal line in figure 6 represents the voxel size in the plane; without voxel size correction only a few dimensions are evaluated with sub-voxel accuracy, however, with voxel size correction, all dimensions are evaluated with sub-voxel accuracy. One further observation, that is to be expected, is that measurement error scales with the length of the dimension evaluated; as such, the percentage error for the same 15 centre-to-centre dimensions are plotted in figure 7. Figure 7 shows the percentage error is approximately constant, and similar in magnitude to the voxel scaling factor derived from figure 5. A constant percentage error would be expected if the only source of error is the , i.e. a systematic scaling error, however, the slight variation in the percentage error suggests some amount of random error exists; this may be as a result of distortions introduced to the CT volume as a result of the sources of error previously described.

14 y = 0.9911x

12

10

8

6

4 via optical profiler profiler (mm) via optical

2 Reference workpiece dimensions evaluated workpiece dimensions Reference 0 0 2 4 6 8 10 12 14 Reference workpiece dimensions evaluated via X-ray CT (mm)

Figure 5: X-ray CT measurements of the reference workpiece plotted against the optical profiler measurements; the gradient of the least-squares line gives the voxel scaling factor δ in equation (5).

450 400 350 300

250

m) μ 200

Error ( Error 150 100 50

0

L15 L24 L46 L13 L14 L16 L23 L25 L26 L34 L35 L36 L45 L56 -50 L12 Corrected Uncorrected Voxel size

Figure 6: Measurement error for each of the 15 centre-to-centre distances of the actual workpiece evaluated with and without voxel size correction; the horizontal line represents the voxel size of the measurement, thus sub- voxel accuracy has been achieved.

1.2

1.0

0.8

0.6

0.4 Percentage error % errorPercentage

0.2

0.0

L15 L26 L46 L12 L13 L14 L16 L23 L24 L25 L34 L35 L36 L45 L56 Corrected Uncorrected δ

Figure 7: Percentage error for each of the 15 centre-to-centre distances of the actual workpiece evaluated with and without voxel size correction; the horizontal line is the voxel scaling factor derived from the reference workpiece.

4. Conclusions

A reference workpiece has been introduced, alongside the methodology adopted in defining its reference dimensions; based on the three threshold independent dimensions presented by the reference workpiece, a method for correcting voxel size has been described. The reference workpiece and voxel size correction method have then been used to rescale voxel size for an actual workpiece; the application of voxel size correction was shown to successfully reduced measurement error.

From this work it can be concluded that the measurement error was predominantly systematic in nature, however, some amount of random error was present. These initial results suggest the presented voxel size correction scheme is suitable for general X-ray CT measurement tasks, in that no prior knowledge of the actual workpiece is required. However, this method is limited to threshold independent dimensions, since surface determination is still a highly influential factor in X-ray CT for dimensional metrology.

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