Effect of the number of projections on dimensional measurements with X-ray computed tomography

Highlights

• X-ray computed tomography (CT) is studied under changes in number of projections $N_p$

• Reducing $N_p$ degrades image quality & dimensional accuracy of CT data reconstruction.

• Decreasing $N_p$ mainly affects the CT dimensional accuracy of form measurements.

• Accuracy degradation (as $N_p$ decreases) is less severe for CT measurements of size.

• A loss of image quality can be traded off for time reduction in CT data acquisition.

Abstract

In X-ray computed tomography (CT) reducing the number of projections (N_p) acquired for data reconstruction will reduce the measurement acquisition time and, thereby, the cost of the measuring process. However, reducing N_p can also reduce reconstruction quality and the accuracy of dimensional information provided by the CT measurement. This paper assesses changes in dimensional accuracy of X-ray CT data as a function of N_p. The variance of CT dimensional measurements, with respect to reference data obtained from tactile coordinate measurement machines (CMMs), is studied. Two cheese-like hole-cubes made of aluminum material and nylon (a polyamide thermoplastic) are used as measuring workpieces. It is determined that using N_p between 600 and 2000 does not produce major changes in accuracy for size measurements (lengths and diameters), for which absolute deviations between CT and reference data were mainly within the range of 5–10 μm. This range reflects sub-voxel accuracies of CT measurement (the voxel size for CT data reconstruction was 73 μm) when determining component size dimensions. However, for N_p < 600 the accuracy of CT measurements rapidly deteriorates, with deviations that can be much larger than 100 μm. This is worse for measurements of form (flatness and cylindricity). Reducing N_p strongly affects form measurements. To assess form properties of CT reconstructed surfaces, using a large N_p ($\ge$ 2000) is preferable. Although the accuracy of measurements of size also deteriorates with reducing N_p the accuracy loss is less severe, particularly if averaging over several data points. When measuring size, a loss of image quality can be tolerated as a trade-off for time optimization in CT data collection. Assessments of image quality further complement the conclusions presented in this paper.

Introduction

X-ray computed tomography (CT) is a technique widely used for nondestructive evaluation and quality control of industrial manufactured components. More recently, it has been expanded in its field of application to dimensional metrology and is being used for geometric dimensioning and tolerancing, of both the internal and external features, of mechanical parts and other device components [[1], [2], [3]]. Applications of industrial X-ray inspection are abundant in industries such as electronics, medical devices, aerospace, defense, automotive, and several modalities of manufacturing (e.g., casting, injection molding, and additive manufacturing or 3D printing). However, one of the main drawbacks of CT is the time required for data acquisition; the time required to measure complex parts can take anywhere from several tens of minutes to several hours, which can lead to significant bottlenecks for inspecting multiple parts in manufacturing companies or assembly factories. Therefore, reducing data acquisition times for CT measurement is desirable to improve the measurement throughput. But, for the sake of fair comparisons, it is worth noting here that even if the time required for CT data acquisition might be a drawback, in several cases, this time is relatively short if compared with other techniques, e.g., tactile CMMs, especially when a high density of points must be acquired for an accurate representation of a complex component. Moreover, the long acquisition time is often tolerated if the characteristic to be evaluated cannot be evaluated with other instruments, e.g., for nondestructive evaluation of internal features.

Several approaches can be adopted to increase throughput of X-ray CT measurement, the most direct being to reduce detector exposure or acquire less radiographic data, but this may compromise the quality of the CT reconstruction. For several industrial inspection tasks, mainly feature identification such as defects and voids, these compromises can be tolerated for optimizing the CT scan time. In dimensional metrology applications, a loss in image quality can be accepted as a trade-off for measurement time reduction, if the CT data provides adequate dimensional accuracy for the purposes of the measurement. This paper evaluates variations in the accuracy of industrial CT based dimensional data by reducing the number of radiographic images used for CT reconstruction.

To place this work in a context of the extensive research carried out over the last forty or so years [1], it is necessary in this introduction to provide outlines of the specific technology evaluated in this work and previous optimization studies. From the theoretical fundamentals of X-ray CT, a filtered-backprojection (FBP) image reconstruction resembling the full spatial distribution of an object (from measurements of sets of projections across it) can only be realized with an impractically large number of acquired projections N_p [[4], [5], [6]], and unlimited camera pixel resolution and sensitivity. In the case of analytical reconstruction methods typically used in commercial CT scanners, e.g., Feldkamp-Davis-Kress or FDK [7,8], to produce an accurate volumetric reconstruction of a sample it is necessary to acquire hundreds of projection views (or radiographs), equiangularly imaged over a full 360° rotation of the part, or at least through a rotation that covers 180° plus the fan-angle/cone-angle of the X-ray beam. Using only a few projections will cause reconstruction artifacts due to under-sampling [9,10], but acquiring a large number (several hundreds or even thousands) of projections can result in scanning times in the order of hours.

It is already known that reducing N_p degrades the quality of imagery data produced after reconstruction [[11], [12], [13], [14], [15], [16], [17]]. Less well-known are the effects on CT dimensional data. Artifacts in the form of shadow bands and dark streaks—known as view aliasing—and noise-like distortions in the CT reconstructed images are the main effects typically associated with an insufficient projection data (see Section 4). Whereas for medical applications there are several studies (and developments) that deal with the acquisition of limited projection views, e.g., see Refs. [[18], [19], [20], [21], [22], [23]], little research exists in the field of industrial metrology. In a recent article [24], it was shown that N_p can significantly be reduced for CT dimensional measurements if iterative algorithms with prior information are used. In another investigation [25], which explored a task-specific selection of certain projection angles for data reconstruction, i.e., sparse-view CT scan, an improvement on dimensional accuracy was reported through the use of iterative reconstruction methods in comparison to data reconstructed with the FDK algorithm. Other alternatives for CT dimensional metrology, when a low N_p number is required, include the use of task-specific scan trajectories that do not necessarily follow a circular-orbit [26,27], and a frequency-based method to optimize N_p depending on the workpiece geometrical feature that has to be measured [28]. However, there is still a need for systematic studies of the effect of N_p on dimensional measurements with traditional industrial CT machines. To address this gap in the field, the present study sheds light on the behavior of the dimensional accuracy provided by CT data, with specific regard to view sampling (i.e., the projection data as input for CT reconstruction), when standard industrial setups are used. This study, which was based on research conducted at the University of North Carolina at Charlotte [29,30], is split between the assessment of reconstructed image quality and the evaluation of dimensional accuracy.

Image quality of the CT data is qualified as a function of the N_p number used for CT reconstruction by using measures such as the root-mean-square error (RMSE), peak signal-to-noise ratio (PSNR), and the structural similarity (SSIM) index. A brief introduction to these quantitative performance metrics is presented in Section 2. These metrics are used for the assessment of image quality produced by both simulations (with Matlab software) and experimental measurements from an industrial cone-beam CT setup with a flat panel detector and a circular-orbit scanning trajectory. Details on the configurations employed for data collection are presented in Section 3. The RMSE, PSNR and SSIM metrics are used, in Section 4, to evaluate several simulated and experimental CT cross-sectional images. Here, it is worth noting that the traditional assessments of image quality for CT analysis are the mean-square error-based metrics, i.e., evaluating quantities such as RMSE and the PSNR, but up until this point, in the field of industrial CT for metrology applications, no other research has evaluated image quality with SSIM-based metrics. Similarly, there have been no investigations that have compared RMSE, PSNR, and SSIM evaluations for the assessments of dimensional measurement performance with X-ray CT data. Additionally, this article expands the use of the SSIM measures for the assessment of image quality in the field of X-ray CT metrology.

To assess changes in dimensional accuracy of CT data as a function of N_p, in Section 5, deviations between CT dimensional data and reference measurements obtained from tactile coordinate measurement machines (CMMs)¹ are evaluated. (Workpieces made of aluminum and a polyamide thermoplastic were used as measuring specimens for such comparisons.) Lastly, Section 6 provides some concluding remarks that summarize the contributions of this paper to the field of knowledge and its limitations. Overall, the main findings presented in this paper suggest that, for optimization of the measurement process, a loss of image quality can be tolerated as a trade-off for time reduction in acquisition and handling of CT dimensional data.