research papers
Evaluation of the improved threedimensional resolution of a synchrotron radiation computed tomograph using a microfabricated test pattern
^{a}Department of Applied Biochemistry, School of Engineering, Tokai University, Kitakaname 1117, Hiratsuka, Kanagawa 2591292, Japan, and ^{b}Research and Utilization Division, JASRI/SPring8, Kouto 111, Sayo, Hyogo 6795198, Japan
^{*}Correspondence email: ryuta@tokaiu.jp
A micro test pattern prepared by focused ion beam milling was used to evaluate the threedimensional resolution of a microtomograph at the BL20B2 beamline of SPring8. The resolutions along the direction within the tomographic slice plane and perpendicular to it were determined from the modulation transfer functions. The throughplane resolution perpendicular to the tomographic slice was evaluated to be 8 µm, which corresponds to the spatial resolution of twodimensional radiographs. In contrast, the inplane resolution within the slice was evaluated to be 12 µm. Realspace interpolation was performed prior to the tomographic reconstruction, giving an improved inplane resolution of 8.5 µm. However, the 8 µm pitch pattern was resolved in the interpolated slice image. To reflect this result, another resolution measure from the peaktovalley difference plot was introduced. This resolution measure gave resolution limits of 7.4 µm for the inplane direction and 6.1 µm for the throughplane direction. The threedimensional test pattern along with the interpolated reconstruction enables the quantitative evaluation of the spatial resolution of microtomographs.
Keywords: computed tomography; microCT; spatial resolution; FIB; MTF.
1. Introduction
Threedimensional structural analysis by computed tomography is performed by recording twodimensional images of a sample by rotating either the sample itself or the imaging system. The obtained images are subjected to reconstruction calculation giving the threedimensional density distribution. In clinical and laboratory computed tomographs (CTs), Xray radiation is obtained from a focused Xray tube and used for recording radiographs. The Xray tube gives divergent radiation called a fan beam or cone beam. In contrast, synchrotron radiation CT uses an almost parallel beam for obtaining the radiographs. The application of synchrotron radiation to tomographic analysis allowed highresolution threedimensional analysis at micrometer to submicrometer resolution (Bonse et al., 1994; Salome et al., 1999; Uesugi et al., 2001; Takeuchi et al., 2002). The imaging geometry and achieved spatial resolution of such highresolution CT, hereafter called microCT, are very different from those of Xray tube CT. Therefore, the resolution of the synchrotron radiation microCT should be evaluated from test objects appropriate for micrometer to submicrometer analyses.
The spatial resolutions of microCTs have been deduced from the spatial resolutions of twodimensional radiographs. Although threedimensional images have been used for estimating the approximate resolution of a microCT (e.g. Tang et al., 2006), quantitative evaluation of the threedimensional resolution of microCTs has not been performed. This is because the fabrication of threedimensional test patterns appropriate for evaluating micrometer to submicrometer resolution is rather difficult. This paper describes the microfabrication of a threedimensional test pattern by focused ion beam (FIB) milling. The spatial resolution of the microtomograph at the BL20B2 beamline of SPring8 was examined using this test pattern.
According to sampling theorem, the presampling resolution can be retrieved from data sampled at a rate corresponding to twice the resolution spacing. Therefore, images for the tomographic analysis are acquired with a pixel width of less than half the spatial resolution of the twodimensional image. The threedimensional structure is then determined by the convolution backprojection method (Huesman et al., 1977), which primarily cuts off the Fourier transform at half the frequency bandwidth. This reconstruction calculation is repeated for each tomographic slice, giving the threedimensional structure. Therefore, the resolution in the tomographic slice, hereafter called the inplane resolution, should be affected by the reconstruction calculation, while the resolution along the sample rotation axis, hereafter called the throughplane resolution, is largely independent of the tomographic reconstruction. The influence of the reconstruction calculation on the inplane resolution should be determined in comparison with the throughplane resolution.
In the study reported here, a threedimensional micropattern was used for evaluating resolution from the modulation transfer function. The inplane resolution was improved by calculating a tomogram after realspace interpolation. The improved resolution was comparable with the throughplane resolution. Another resolution measure given by the peaktovalley difference plot is introduced, which gives a critical limit for the spatial resolution in each direction.
2. Materials and methods
2.1. Threedimensional test pattern
A threedimensional test pattern for the resolution evaluation was microfabricated by using a FIB apparatus (FB2000, Hitachi HighTechnologies, Japan) operated at 30 kV. An aluminium wire with a diameter of 250 µm was attached to the sample holder and subjected to gallium ion beam milling. A series of square wells was carved along the wire axis and along the direction perpendicular to the axis. The pitches of the resultant squarewave patterns were 16, 12, 10, 8 and 6 µm. Along the axis direction, an additional 20 µm pattern was prepared. Each pattern had a 50% duty cycle, i.e. 5 µm well and 5 µm interval for a 10 µm pitch. The beam aperture was selected to regulate the ion beam current. A gallium beam of 4.5 nA was used for carving the 20, 16, 12 and 8 µm patterns and a 1.7 nA beam for the 6 µm pattern, giving rectangular wells with a typical depth of 10 µm. Secondary electron images of the obtained test pattern are shown in Fig. 1. Minor defects in well shape originated from the strong beam current for carving the larger wells. The test pattern was then recovered and mounted on a stainless steel pin by using epoxy glue. Air exposure had no effect on the pattern structure.
2.2. MicroCT
The tomographic analysis of the test pattern was performed at the BL20B2 beamline (Goto et al., 2001) of SPring8. The steel pin was mounted on the goniometer head of the microtomograph, using a brass fitting designed for the pinhold sample. Transmission radiographs were recorded using a CCDbased Xray imaging detector (AA40P and C488041S, Hamamatsu Photonics, Japan) and 12.000 keV Xrays. The density was determined to be 3.86 × 10^{9} photons s^{−1} mm^{−2} at the sample position. The number of detector pixels was 2000 in the horizontal direction perpendicular to the sample rotation axis and 1312 along the vertical axis. The field of view and effective pixel size of the detector were 5.50 mm × 3.61 mm and 2.75 µm × 2.75 µm, respectively. The acquisition area, which included the test pattern image, was set to 600 × 812 pixels corresponding to an area of 1.65 mm × 2.23 mm. The images of this area were acquired with a rotation step of 0.10° and an exposure time of 600 ms per image. The data acquisition conditions are summarized in Table 1.

2.3. Tomographic reconstruction
The convolution backprojection method using a Hannwindow filter (Chesler & Riederer, 1975; Heusman et al., 1977) was used for the tomographic reconstruction. The noninterpolated reconstruction was performed with a frequency cutoff at half the total bandwidth. The filter function profile is shown in Fig. 2. For the interpolated reconstruction, the horizontal pixel strip extracted from the twodimensional was subjected to double or quadruple linear interpolation. The double interpolation was performed by placing one additional pixel between observed pixels, and the quadruple interpolation by placing three pixels. Each interpolated strip was subjected to a Fourier transformation. The filter functions applied to the Fourier transform of the interpolated strips are superposed in Fig. 2. Although the higher half of the frequency space was cut off, the same as in the noninterpolated reconstruction, this higher half spectra corresponds to the Fourier transform of the interpolated points and does not in principle convey the frequency information of observed data. Then the inverse Fourier transformation and backprojection calculations were performed to obtain the sample tomogram.
The periodical profile of the squarewave pattern was analyzed by taking a spatial trajectory along the test pattern. The inplane resolution was examined from the non, double and quadrupleinterpolated images of the pattern perpendicular to the sample rotation axis. The throughplane resolution was examined from the noninterpolated images of the pattern along the axis.
The reconstruction and examination protocols were implemented in the program RecView (available from https://pubweb.cc.utokai.ac.jp/ryuta/ ). Volumerendered figures of the obtained threedimensional structures were produced using the program VG Studio MAX (Volume Graphics, Germany). CT densities were rendered using the scatter HQ algorithm.
3. Results
3.1. Tomographic reconstruction
Tomographic images and spatial trajectories of the 10 µm inplane pattern are shown in Fig. 3. Although the pattern cannot be resolved in the noninterpolated image, the double and quadrupleinterpolated images gave the periodical structure of the test pattern. The spatial trajectories of the indicate that the doubleinterpolated reconstruction is sufficient for improving the spatial resolution, while the coefficients obtained from the quadrupleinterpolated reconstruction gave slightly larger peaktovalley differences than those from the double interpolation.
Tomograms of inplane patterns obtained by the noninterpolated and quadrupleinterpolated reconstruction are shown in Fig. 4. Threedimensional renderings of the pattern structures are shown in Fig. 5. The double or quadrupleinterpolated reconstruction resolved the inplane pattern up to a pitch of 8 µm, whereas the noninterpolated reconstruction, which has been used for microCT analyses until the study reported here, resolved only 12 and 16 µm patterns.
The tomographic data acquisition should be performed by taking a sufficient step of RΔθ in the circumferential direction, where R denotes the distance from the rotation axis and Δθ the rotation angle per frame. The center of the inplane pattern was located at 400 µm from the rotation axis, corresponding to the position where the circumferential step is 0.70 µm. The throughplane pattern was located at 515 µm, corresponding to a 0.90 µm step. Therefore, the circumferential step was fine enough for reconstructing tomograms of the test pattern from the observed images.
The reconstruction program used in this study was partially written in native machine language and accelerated by multithread processing. The convolution backprojection calculation took only 1.08 s per noninterpolated tomogram of 600 × 600 pixels on a Windows PC equipped with a 2.1 GHz Core 2 Duo processor. Although the reconstruction calculation of the doubleinterpolated image took three times longer than the noninterpolated image, the faster reconstruction routine implemented in this program executed the doubleinterpolated calculation taking only 1.5 times longer than the former noninterpolated reconstruction (Mizutani et al., 2007, 2008).
3.2. Resolution evaluation
The spatial resolution can be evaluated from the peaktovalley contrast, which is calculated from the spatial trajectory of the test pattern, as shown in Fig. 3. However, the observed contrast in linear absorption coefficients depends on the depth position of the trajectory in the carved well. A schematic drawing of observed and ideal slices of the test pattern is shown in Fig. 6(a). If the carved well is deep enough compared with the spatial resolution, outlines along the depth direction of the observed image coincide with those of the ideal sample structure. In such cases the peaktovalley contrast should be virtually independent of the position of the trajectory line. The depth dependence of the peaktovalley contrast of the inplane 10 µm pattern is shown in Fig. 6(b). This depth dependence gave a plateau where the contrast was almost independent of the depth position. Therefore, the contrast can be evaluated from the spatial trajectory along the middepth line of each carved pattern.
The ratio of the observed contrast to the intrinsic contrast between aluminium and air (41.3 cm^{−1}) was calculated separately from each pattern: the throughplane pattern and the inplane pattern. As the spatial frequency approaches the resolution limit, this contrast transfer ratio becomes identical to the modulation transfer function (MTF). It has been reported that the contrast in the Xray image of a squarewave test chart showed good agreement with the predicted MTF (Takeuchi et al., 2001). Therefore, the contrast ratio can be treated as the MTF. Plots of this squarewave MTF in each direction are shown in Fig. 7. All the MTFs obtained from the throughplane and inplane patterns gave positive values. The inplane MTF plot of the noninterpolated image (Fig. 7a) showed a steep descent. The spatial resolution at 5% MTF was approximately evaluated to be 12 µm. This inplane resolution is consistent with the result estimated from the tomogram of abrasives (Uesugi et al., 1999). In contrast, the double and quadrupleinterpolated MTFs gave an improved resolution of 8.5 µm. This inplane resolution is comparable with the throughplane resolution, which was evaluated to be 8 µm at 5% MTF (Fig. 7b). These results indicate that the double interpolation is sufficient for improving the inplane resolution.
Although the inplane resolution of the interpolated image was evaluated to be 8.5 µm at 5% MTF, Fig. 4(f) shows that the inplane 8 µm pattern was resolved by the interpolated reconstruction. This indicates that the finer structure can be resolved in practice beyond the spatial resolution at 5% MTF. Plots of average peaktovalley differences of the linear absorption coefficients versus pattern pitch are shown in Fig. 8. They indicate that the peaktovalley difference is linearly correlated with the pattern pitch. As discussed in Appendix A, the slope of the fitted line reflects the width of the point spread function. From this plot, the limit of the spatial resolution can be defined as the x intercept of the fitted line, giving resolution limits of 7.4 µm for the inplane direction and 6.1 µm for the throughplane direction. These resolutions are finer than those obtained from the MTF. However, the appearance of tomograms (Figs. 3 and 4) showed agreement with the spatial resolution evaluated from the peaktovalley difference plots rather than the resolutions from the MTF.
The spatial resolution of twodimensional radiographs taken by the Xray optics used in this study has been estimated to be 8 µm (Uesugi et al., 1999; Mizutani et al., 2008). This twodimensional resolution corresponds to the inplane resolution of the interpolated tomogram, but not to that of the noninterpolated tomogram. This indicates that the reconstruction calculation without interpolation did not retrieve the spatial resolution achieved by the Xray optics, even when the radiographs were taken with the sampling theorem being taken into consideration.
4. Discussion
Quantitative evaluation of the threedimensional resolution has not been made for the major synchrotron radiation microCTs. Our microfabricated threedimensional test pattern along with realspace interpolation can be used to determine the spatial resolution of any type of microCT. This evaluation method will give the exact resolution of the threedimensional structure, which has not previously been determined from twodimensional test charts.
The zoomed reconstruction has been used to obtain the point spread function of clinical CT (Mori et al., 2004). In the present study the observed densities were zoomed by linear interpolation prior to the reconstruction calculation. However, the linear interpolation does not reproduce the ideal structure of the observed sample, especially around density peaks and troughs. Other interpolation methods such as quadratic models would be appropriate for precise reproduction of the sample structure.
The convolution backprojection method was performed via the Fourier transformation. In frequency space the realspace interpolation extends the frequency bandwidth. The filter function applied in the frequency space strengthens highfrequency structures. This would result in the linear absorption coefficients being distorted around the density edge. The spatial resolution and the density resolution countervail each other, suggesting that interpolated reconstruction should be performed only when the spatial resolution is rather important.
The spatial resolution is the fundamental parameter in the structural analysis. The threedimensional test pattern prepared by FIB milling allowed the quantitative evaluation of the spatial resolution of the microCT. The machining accuracy of FIB is 100 nm or finer, so test patterns on the nanometer scale can be fabricated by FIB. Such a nanometer test pattern should allow the resolution evaluation of the nanoCTs that are planned in the major synchrotron radiation facilities.
APPENDIX A
Peaktovalley difference plot
Since the tomographic image is obtained as a threedimensional structure, the spatial resolution can be defined along arbitrary vectors in threedimensional space. Therefore, the resolution is evaluated in onedimensional space along the predefined measure vector, such as the inplane or throughplane vector. Assuming that a point density in the sample is spread in the observed image by the point spread function (PSF) f_{PS}(u), the integral of the PSF should be
Hereafter, the PSF is considered to be symmetric about its center. The observed image F_{obs}(x) can be obtained by the convolution of the PSF and the ideal sample structure F_{S}(x) as
If the sample function has square waves composed of rectangular peaks with width 2w and density amplitude a_{0} (Fig. 9), the observed density at the center of the peak is given by
where the first term corresponds to the central peak under consideration and the other terms to the distal peaks.
If the half spread of f_{PS}(u) is narrower than 3w, the contributions of the distal peaks can be ignored. This assumption gives
where F_{PS}(u) = . The integral of the symmetric PSF is 1/2 at the origin. Therefore, the center density is determined solely by F_{PS}(w).
Similarly, the observed density at the center of the rectangular well with width 2w should be
Then the peaktovalley difference of the observed density can be obtained by
In Fig. 8 we have fitted a line on the peaktovalley difference plot against the pattern pitch 4w. Since the peaktovalley difference is proportional to the PSF integral F_{PS}(w), the slope of the fitted line corresponds to an average PSF height around the PSF peak. Assuming that the PSF can be approximated by a trapezoidal profile (Fig. 9) with a full width at halfmaximum of 2d, the average height of the PSF peak is estimated to be 1/(2d). From this approximation, the PSF integral is given by
around the peak. When this integral is applied, equation (6) becomes
This equation explains the linear correlation between the peaktovalley difference and the pitch 4w.
The width 2d corresponds to an extrapolated point where the fitted line becomes zero, 4w = 2d. At this point the peak spreads to adjacent wells resulting in an unresolved image of the squarewave pattern. Therefore, a critical limit of the spatial resolution can be defined by this intercept 2d of the peaktovalley difference plot.
Since the MTF is a Fourier transform of the PSF, the peaktovalley difference plot is a realspace equivalent of the MTF. While the resolution evaluated from the MTF is defined by a particular threshold such as 5% MTF, the resolution from the peaktovalley difference plot is evaluated without any threshold parameters, giving the limit of the spatial resolution.
Acknowledgements
We thank Yasuo Miyamoto, Technical Service Coordination Office, Tokai University, for helpful assistance with FIB milling. The synchrotron radiation experiments were performed at SPring8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal Nos. 2007A2072 and 2007B1894).
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