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

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The Application of Voxel Size Correction in X-Ray Computed Tomography for Dimensional Metrology SINCE2013 Singapore International NDT Conference & Exhibition 2013, 19-20 July 2013 The Application of Voxel Size Correction in X-ray Computed Tomography 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 pixels 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 pixel 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
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