Improving the dimensional accuracy of 3D x-ray microscopy data

In traditional industrial x-ray systems, designed for nondestructive evaluation and industrial metrology tasks, the main approach has been to use projection-based architectures in which two-dimensional images, or radiographs, are created by x-rays in a cone-beam that passes through an object and projects radiographs onto a flat panel detector (figure 1(a)). From several radiographic images collected at different angular positions, using localized x-ray absorption measurements, a CT reconstruction algorithm can create a volumetric representation of the object from which internal and external features can be extracted for dimensional measurements [1].

Zoom In Zoom Out Reset image size

With flat panel-based CT systems, the object must be positioned as close to the x-ray source as possible, while remaining within the cone beam, to obtain highly magnified radiographic images at the detector and produce 3D data reconstructions with the highest resolution possible. The geometrical magnification (M_g) of a projection-based system is a function that depends on the source-to-object distance (${d_{{\text{SO}}}}$) and the object-to-detector distance (${d_{{\text{OD}}}}$),

$$\begin{align}{M_{\text{g}}} = \frac{{{d_{{\text{SO}}}} + {d_{{\text{OD}}}}}}{{{d_{{\text{SO}}}}}}.\end{align} \tag{ 1 }$$

The maximum operating M_g value in industrial and laboratory-based x-ray projection instruments is generally determined by the minimum working distance ${d_{{\text{SO}}}}$, which is limited by the sample size—the sample needs to spin in the rotary stage without hitting the x-ray source (see figure 1). In general, the spatial resolution of a CT system depends on parameters such as the focal spot size of the x-ray source, the degree of collimation of the x-ray beam, the projection geometry, the size of the sample, the dimensions of the detector, the precision of mechanical motion controllers, and on details of the computer algorithms used for 3D reconstruction [13–16]. With an x-ray source focal spot in the order of 2–5 μm and a flat panel detector of 2048 × 2048 pixels with a pixel pitch in the range of 127–200 μm, the best spatial resolution achievable is typically limited to the range of 4–10 μm for cylindrical objects with diameters in the 2–25 mm range. Larger samples limit the geometrical magnification M_g, thus reducing the best achievable resolution of CT scans to several tens of micrometers.

There are two common approaches to further increasing the image resolution capabilities of flat panel CT instruments: a reduction in the x-ray focal spot and/or the use of a higher resolution flat panel detector. However, this would not eliminate the limitations on geometric magnification imposed by larger samples (>25 mm diameter). An alternative would be to incorporate optical lenses after x-ray detection to create a scintillator-lens-CCD ⁵ detector coupling, as shown in figure 1(b), to optically magnify the image before reaching the CCD detector. This strategy enables XRM images with spatial resolutions down to 500 nm—see table A1 and figure A1 (in appendix A). There are other means to improve XRM resolution, e.g. using x-ray focusing elements such as Fresnel zone plates or Kirkpatrick-Baez mirrors [17–20], but those techniques (commonly known as nano-XRM) are beyond the scope of this article.

Whereas typical pixel resolutions of commercial flat panel detectors are within the range of 75–200 μm, effective detector pixel sizes in XRM systems can be as low as 16 nm [4, 7]. Figure 2 illustrates scenarios of system resolutions versus M_g for four diferent detector resolutions: 200 μm and 75 μm for flat panel CT systems and 0.5 μm and 50 nm for XRM detectors. When comparing x-ray CT with XRM instruments, system resolutions behave differently for M_g values below 100$\times$. In x-ray CT, the resolution of the system worsens as M_g decreases (i.e. when the sample size increases). On the other hand, by leveraging optical magnification, XRMs allow to preserve (and improve) the system's spatial resolution as M_g decreases, without major limitations on sample size ⁶ —as long as the sample physically fits inside the instrument and does not limit x-ray transmission (the measuring range will be determined by the penetration depth of x-rays into the sample, which is dependent on the sample's material composition and the energy spectrum of the x-ray source [13]). The curves in the graph (figure 2) were derived from the approximated system's resolution formula given in [4],

$$\begin{align}{r_{\text{s}}} \approx \frac{1}{{{M_{\text{g}}}}}{ }\sqrt {{D_{\text{r}}}^2 + {{\left( {S\frac{{{d_{{\text{OD}}}}}}{{{d_{{\text{SO}}}}}}} \right)}^2}} ,\end{align} \tag{ 2 }$$

Zoom In Zoom Out Reset image size

where ${D_{\text{r}}}$ is the spatial resolution of the detector and $S$ is the x-ray source focal spot size.

After the reconstruction phase of x-ray CT or XRM data is completed, a representation of the object's internal and external surfaces can be produced with the use of thresholding algorithms for edge detection and material boundary/surface determination [21–25]. Thereafter, the geometric characteristics of those surfaces can be used as a reference for dimensional metrology. But to minimize errors and improve the accuracy of dimensional measurements, a suitable method is required to determine the voxel size ⁷ associated with 3D data reconstruction is required. In flat panel CT systems, voxel size dimensions are determined by M_g and the x-ray detector's pixel pitch [1, 13, 26]. In the last two decades, methods have already been proposed to determine and correct voxel size errors in industrial CT scanners using multisphere phantoms ⁸ [27–29]. In fact, since around 2005, several manufacturers have marketed flat panel x-ray CT systems for dimensional metrology. As a result of sub-voxel interpolation algorithms commonly used for surface determination, dimensional measurements with sub-voxel accuracies are now ordinarily reported from CT data. Previous studies have shown that when geometric features ranging from 0.5 mm to 60 mm are dimensioned, the expanded uncertainties (k = 2) associated with flat panel-based CT measurements are typically in the 5–50 µm range [10, 11, 30].

Still, depending on the sample size, flat panel-based CT systems may have limited spatial resolution capabilities (section 2.1). The spatial resolution of the data generated by such systems is, at best, on the order of micrometers or tens of micrometers. Spatial resolution limitations can hinder essential surface details that, in addition to improve the accuracy of CT-based dimensional data, could enable other nondestructive analyses, such as morphological characterization of internal walls and evaluation of material porosity, which require high spatial resolution features. As an alternative, this paper proposes the use of sub-micrometer resolution XRM instruments for dimensional metrology, an idea that has not been widely explored until some recent work introduced (by the authors) at conference settings [8, 9]. This article builds on that work and introduces new data that verify the dimensional accuracy of metrology workflows with XRM data. The repeatability and reproducibility of the XRM metrology workflow proposed in this paper, which employs a small version of a multisphere phantom (contained in a cylindrical volume of 4 mm diameter and about 1.8 mm height), are also verified by various data comparisons.

The test evaluation for CT (or XRM) systems generally includes comparisons between length measurements obtained from CT data and other more precise and accurate reference measurements, such as those acquired from a tactile CMM. In this work, the reference measurements for the different sphere distances in the XRM Check standard were calculated from the center positions of the spheres measured by the Federal Institute of Metrology METAS, Switzerland, with an ultra-precise tactile CMM dedicated for calibrating small size objects (METAS µCMM [41]). With three laser interferometers that measure the displacement of its table and equipped with a probe head that uses a spherical sapphire stylus (diameter 208 µm) with a weak probing force (<0.5 mN), this machine ensures accurate reference measurements. Each sphere was probed on its upper hemisphere along the equator and two half meridians. All measurement points where probed using scanning with a point density of 300 points/mm and using a Gaussian filter of 15 undulations per revolution cut-off frequency. The reported uncertainty for the reference measurements, expressed as a combined uncertainty multiplied by a coverage factor k = 2, is

$$\begin{align}{U_{{\text{ref}}}}\left( {k = 2} \right) = 0.14\,\,{{ \unicode{x03BC} {\text{m}}}}.\end{align} \tag{ 3 }$$

This is the associated expanded uncertainty for the reference measurements, for a confidence level of 95.45%, which was estimated following the guidelines of the Guide to the Expression of Uncertainty in Measurement (usually referred to as the GUM) [42].

To evaluate the metrological performance of an XRM instrument (for length measurements), after implementation of the MTX workflow, the deviations between the XRM data and the CMM references can be plotted on a graph (e.g. see figure 6). Table 1 lists the parameter settings used for x-ray imaging of the XRM Check multisphere phantom. Following the measurement workflow of figure 3, to perform coordinate-based dimensional measurements on the 3D XRM data, after surface determination with a local adaptive threshold method [21], spherical features were defined on each ruby ball of the XRM Check phantom using a Gaussian multipoint least-squares fit with Calypso software (Carl Zeiss Industrielle Messtechnik GmbH). Figure 5(b) illustrates the sampling strategy used to fit each spherical feature, which resembles a common CMM probing strategy [10]. To compensate for outliers, only data points in a range of ±2 σ (corresponding to approximately 95% of the 800 points), where σ is the standard deviation of all measured points, are used to calculate the fitted sphere elements. After determining the centers of the spheres, a choice of 35 different center-to-center length measurements, covering at least seven different spatial directions, as suggested by the VDI/VDE 2630-1.3 guideline, was evaluated and grouped into five sets based on their nominal values. The 35 measurands are listed in table 2.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

As seen from figure 6, the deviations of CT dimensional measurements (from reference CMM data) are confined to a range between ±0.7 µm. This range is well within the specification of MPE, for center-to-center sphere distance (SD), of a ZEISS Xradia 620 Versa XRM [8],

$$\begin{align}{\text{MP}}{{\text{E}}_{{\text{SD}}}} = \left( {1.9 + L/100} \right)\,\,{{\unicode{x03BC} {\text{m}}}},\end{align} \tag{ 4 }$$

with the measuring length $L$ given in mm. The results shown in figure 6 provide a verification of the accuracy of the ZEISS Xradia 620 Versa instrument's metrological capabilities (at FOV = 5 mm) with the added MTX workflow, enabling reliable dimensional measurement performance in small-scale volumes, on the order of (5 mm)³ or less. It is worth noting that, without MTX, the typical deviations of XRM data from calibrated/reference values can be anywhere in the range of 1–30 μm. Appendix C shows several sets of MTX-corrected and uncorrected measurements, for center-to-center sphere distances, extracted from 3D XRM data for a couple of multisphere phantoms obtained with different XRM systems. When looking at the least-squares lines that fit the uncorrected data from figures C1–C4 (in appendix C), the XRM dimensional measurements performed without MTX would suffer deviations that are largely dependent on the length being measured (i.e. with errors that are directly proportional to $L$).

To verify the repeatability and reproducibility of the MTX measurement workflow, 30 different scans of an XRM Check phantom were performed with three different instruments (i.e. 90 scans in total). Figure 7 shows the results of dimensional data for three different measurands: SD 11-4, SD 3-22, and SD 8-18 (naming convention of table 2). The measured value for SD 11-4 (∼3.605 mm) is one of the largest measurements of center-to-center length in the XRM Check multisphere standard. The repeatability standard deviations for measurements of SD 11-4, evaluated by the measurement system are ${S_1} =$ 0.08 μm, ${S_2} =$ 0.06 μm, and ${S_3} =$ 0.11 μm; and the reproducibility standard deviation across the three systems is ${S_R} =$ 0.36 μm. From these measurements for SD 11-4, the deviations between measurements made on the same length feature, repeated through 90 different scans, are in the range of ±0.48 µm. The repeatability and reproducibility standard deviations for the measurement results associated with SD 3-22, and SD 8-18 are listed in figure 7.

Zoom In Zoom Out Reset image size

Although not all measurement results can be explicitly listed in this document due to space limitations, all measurands listed in table 2 were measured and analyzed. Of a total of 35 characteristics measured per scan, in 30 scans obtained per measurement system, using three different systems, i.e. 3150 measurements in total, the repeatability standard deviations of the measurements were in the range of 0.04–0.15 μm and the reproducibility standard deviations ranged from 0.05 to 0.45 µm. These results serve as a verification of the repeatability and reproducibility of the dimensional data produced by the ZEISS Xradia 620 Versa XRM when coupled with MTX.

Lastly, in figure 7, the deviations between coordinate-based 3D XRM measurements and CMM reference data are confined to the range ±0.95 µm. This range is also within the MPE_SD specification of equation (4).

In the assembled state of the smartphone camera lens modules, the evaluation of the geometric properties of the lenses, such as the thickness of the annular wedges, the centering interlock diameters, the spaces between the wedges, the lens-to-lens tilt, the vertex heights and centration, etc, require a non-contact and nondestructive measurement method. These measurements are important for the functional inspection of camera lens modules and the improvement of designs and manufacturing processes to enable the production of versatile cameras that improve the image quality of mobile phones. Figure 8 shows the 3D rendering of a smartphone camera lens assembly reconstructed from XRM data, including a view of cross-sectional images (2D slices). Since the largest dimension of interest in the assembly is around 4.4 mm, a cylindrical region of about 5 mm diameter was scanned from the full camera lens module for dimensional measurements. Here it is worth noting that the 5 mm FOV diameter limitation of the MTX workflow strictly refers to the measurement volume. Although an object should ideally have a diameter of the order of 5 mm or less, so that a full field-of-view scan can be performed with MTX, interior tomographies can be performed on somewhat larger samples, as shown in figure 8, where a semi-transparent view shows the entire camera lens assembly with an outer diameter greater than 5 mm. Before the acquisition of the XRM data in the region of interest (on a 5 mm diameter cylinder, half shown in blue in figure 8), the MTX workflow was applied to verify the measurement accuracy of the XRM system, so that the dimensional measurements extracted from the virtual reconstruction of the camera lens module are accurate. Figure 8 shows examples of typical dimensions of interest in the camera lens in its assembled state.

Zoom In Zoom Out Reset image size

To facilitate efficient fuel spraying in internal combustion engines (and to keep emissions requirements lower), fuel injectors use high injection pressures with small nozzle orifices. Since the variability of the geometric characteristics of the injector nozzle, such as the sharpness of the inlet corners and the diameters of the holes, can influence the internal flow and fuel spray in combustion engines, manufacturing deviations of the nominal nozzle design are typically measured for dimensional quality control. In this case, to measure the smallest nozzle holes in the fuel injector (features that are difficult to access with measurement systems using tactile or optical probes), XRM is especially well suited for dimensional measurement. Figure 9 shows cross-sectional image views and cropped/semi-transparent sections of the reconstructed XRM data for a fuel injector nozzle tip, revealing the roughness of the wall surface in the internal cavities and the presence of material porosity. The size of the region of interest, the tip portion of the fuel injector, is revealed by the dimensions shown in the image (∼2.4 mm for the largest diameter). Since the region of interest fits in a volume of less than (5 mm)³, the MTX workflow was ran, prior to scanning the fuel injector tip, to verify the accuracy of the XRM system measurement. Then, dimensional inspection of the small holes in the nozzle was carried out.

Zoom In Zoom Out Reset image size

It is worth noting, in figure 9, that the blue spots in the image reveal the presence of voids within the walls of the injector nozzle. This is an added benefit of measurement inspections based on XRM volumetric data, so in addition to reconstructing all internal and external surfaces of a mechanical workpiece, material tests can be performed on the same data set. In this case, the resolution capabilities of the ZEISS Xradia Versa system used for XRM scanning allowed for the visualization of the porosity present within the injector nozzle.

For the inspection of internal and external structures in small plastic injection molded parts, x-ray CT or XRM may be more suitable for geometric measurement compared to contact or vision inspection techniques. Non-contact measurement is important to avoid distortion of flexible or easy-to-deform components. In injection molded workpieces, dimensional measurements help determine deviations in manufactured parts compared to the nominal geometry specified in the original computer-aided design (CAD) models of the part. This is important for product development, quality assurance of functional characteristics of parts, and evaluation of structural integrity of industrially manufactured components and assembled devices. Figure 10 shows the example of a small plastic connector, reconstructed from XRM data, with dimensional measurements and CAD-to-part comparison data. Again, since the largest length measurement of the connector is approximately 4 mm, this sample fits well within a volume of less than (5 mm)³. Therefore, the MTX workflow was used prior to x-ray scanning the sample to verify the accuracy of the XRM system. Then, the XRM data shown in figure 10 was acquired for dimensional measurements.

Zoom In Zoom Out Reset image size