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Abstract
Scanning X-ray fluorescence tomography was once considered impractical due to prohibitive measurement time requirements but is now common for investigating metal distributions within small systems. A recent look-ahead to the possibilities of 4th-generation synchrotron light sources [J. Synchrotron. Radiat. 21, 1031 (2014)] raised the possibility of a spiral-scanning measurement scheme where motion overheads are almost completely eliminated. Here we demonstrate the spiral scanning measurement and use Fourier ring correlation analysis to interrogate sources of resolution degradation. We develop an extension to the Fourier ring correlation formalism that enables direct determination of resolution from the measured sinogram data, greatly enhancing its power as a diagnostic tool for computed tomography.
© 2017 Optical Society of America
Corrections
18 September 2017: Typographical corrections were made to the author affiliations.
1. Introduction
Scanning x-ray fluorescence computed tomography promises access to three-dimensional elemental distributions with extremely high sensitivity and resolution [1]. However, realisation has been hindered by measurement effects such as self absorption [2] and implementation issues [3]. Recent (2005–2015) improvements to x-ray energy-dispersive detectors [4–6] have addressed critical implementation bottlenecks resulting from inadequate detector electronics architecture, relocating limitations to motion control and specimen scanning implementation.
Scanning is integral to scanning x-ray fluorescence microscopy and can be achieved by moving the x-ray source, the optical components, or the specimen. While it may be possible to scan an electron beam within a synchrotron storage ring using electrostatics [3], that modality is not presently within operational scope at any terrestrial facility. The fact that photons do not carry charge means that there is no further possibility for employing the electromagnetic interaction directly, and so one must resort to scanning via physical movements of massive objects, viz the specimen or the optical components. Scanning often requires millisecond-scale movements of micrometer-scale distances of kilogram-scale masses, and presents a significant motion controls challenge. In view of this challenge, reconfiguration of the measurement into a spiral scanning trajectory, where the specimen is rotated continuously while simultaneously undergoing a slow traverse, presents some advantages and simplifications of the motion that greatly affect the realisability of scanning tomography.
Computed tomographic datasets are usually visualized as measurements of some projected quantity acquired at a sequence of angular orientations. Full sampling of the information content in a projection series is described by the Crowther criterion, the angular-space variant of Nyquist sampling, requiring that the number of evenly-spaced angular projections nθ is proportional to the width of the region of interest in pixels nx according to: [7]. The time T required for a single-slice scanning tomographic measurement acquired as a series of projections is:
where D is the pixel dwell time, and tx and tθ are the overheads associated with incremental motion in the x and θ axes, respectively. The locus for this motion is represented in Fig. 1(a). Fortunately, the dominant pixel overhead tx can be reduced to zero with the use of the fly scan [8, 9], and event-mode data-streams [10]. Accordingly, Eq. (1) reduces to: where D is now to be interpreted as the pixel transit time.
Fig. 1 Loci of specimen motion in single-slice scanning tomography. The grids represent the encoder or pixel boundaries and the red lines indicate the specimen loci. (a) ‘Slice’ tomography, employing traditional ‘rotation-series’ addressing via traverse-and-rotate. Overheads are incurred at the beginning and end of each distinct motion due to the need to accelerate inertial bodies. (b) ‘Spiral’ tomography. The specimen rotates and traverses continuously, with the traverse achieving one resolution element per rotation. Acceleration occurs only at the beginning and end of the scan. The discontinuities seen at the left and right of the spiral scan locus are the 0° to 360° step discontinuity in angle, and are managed by employing modulo arithmetic in the pixel counter.
The overhead at the end of each line tθ is associated with the rotational motion, but includes also the acceleration of the linear stages and any detector and stage overheads that occur on a once-per-line interval. For a test case of D = 3 ms, nx = 100, and tθ = 0.3 s [parameters taken from the XFM beamline of the Australian Synchrotron [11]], 50% of the total measurement time is consumed by motion overheads. This overhead prevents proper measurement fractionation [12] or reduction of the overall measurement time by reducing the dwell time.
The spiral scan presented in Fig. 1(b) presents an alternative measurement sequence. In this case a specimen is continuously rotated and translated at a rate of one resolution element of translation for each entire rotation. The acceleration phases occur at the beginning and end of the scan; these are the only motion overheads.
In this article we demonstrate spiral scanning x-ray fluorescence tomography, and evaluate its performance by comparison against the slice scanning approach. As the two approaches sample the sinogram very differently, direct comparison is difficult; nevertheless, Fourier Ring Correlation (FRC) methods indicate that the present implementation of spiral scanning produces inferior results. In order to diagnose the likely cause of this resolution degradation, we extend the existing FRC analysis so as to act directly on the sinogram, and thereby assess the data independently of data preprocessing, alignment, and reconstruction. We anticipate that this extension will be of great interest to the tomographic community, as it greatly facilitates identification of resolution-degrading stages within the reconstruction analysis pipeline. We thereby determine the likely cause of the resolution degradation as originating from random components in the rotation stage eccentricity. While this eccentricity is present for both measurement modalities, correction by data alignment can be achieved in the case of the slice measurement, but is not possible for the spiral scanning approach; in that case interferometric encoding of the specimen position (including eccentric motion components) could be used as input for data alignment or a sufficiently precise rotation stage could be used to avoid the problem altogether.
2. Measurement
We have a history of continuing research into metals fate in biological systems such as Caenorhabditis elegans and Drosophila melanogaster, with investigations studying two and three-dimensional distributions [11, 13], XANES imaging [14] and XANES tomography [15]. The present work directly informs our ability to perform tomographic studies on statistically significant populations of animals. Figure 2 presents an overview of Mn, K, and Cu in a specimen of C. elegans that has been freeze-dried and mounted freestanding on the end of a narror capillary. The worm was brought to within 10 µm of the centre of an Aerotech rotation stage (ANT-95-360-R) using an orthogonally-mounted pair of small, manual adjustable stages and a visible light microscope.
The ANT-95-360-R rotation stage encoder has 11840 lines, and we have used an FX50 interpolator with a factor of 50 interpolation, producing 4 counts per digital quadrature cycle, giving 11840 × 50 × 4 = 2,368,000 counts/rev, but we have down-scaled these by a factor of 512 to provide 4,625 counts/rev at an angular increment of 0.07784°. The continuous spiral trajectory is effected by using a virtual or software ‘gearing’ between the linear and angular motions; the software resides in the motion controller [16]. The entire two-dimensional scan is thus effected via a single coordinated motion from (x, θ) = (0 µm, 0°) to (x, θ) = (250 µm, 90,000°). For this work we have used a spiral gear ratio of 1 µm/rev with an angular velocity of 1 rev/s, so the 250 µm-wide single slice tomography measurement required 250 s of measurement time. Figure 3 presents the measured sinogram for Ca, K, and Cu as an RGB overlay. Mn, clearly visualised in Fig. 2, was not present in the plane of this measurement.
Fig. 3 Spiral sinogram of the C. elegans specimen shown in Fig. 2, indicating the distributions of Ca, K, and Cu (RGB). The measurement was recorded with a gearing of 1 µm/rev, an angular velocity of 1 rev/s, and at an angular resolution of 4,625 steps/rev. The image presented here comprises 250 pixels across the specimen (vertical as displayed) and 4,625 in the rotation direction (horizontal as displayed). At this angular velocity and degree of fractionation, the pixel transit time is 1/4,625 s ≈ 216 µs. The sequence of the data acquisition was as per Fig. 1(b), starting from the lower left of this image, heading right (fast rotational axis), across the image until the right-edge is reached, whereupon the angular coordinate wraps to the left-hand side of the image. This angular wrapping was introduced into the data by effecting a modulo counter in the Maia detector system [10], so that the angular pixel modulo 4,625 was reported in the data-stream. The spatial traversal pixel increment (vertical direction in this image) occurs when the relevant encoder pulse is received by the Maia detector system.
In order to evaluate the performance of the spiral tomography measurement scheme, we have also acquired data using the traditional slice addressing scheme, as per Fig. 1(a). This measurement, presented in Fig. 4, was acquired over 221 unique angular projections with a transit time of 14.3 ms per 1 µm pixel. As the x-ray intensity was essentially constant for the spiral and slice measurements, the relative imaging statistics for the two measurements are proportional to the traverse times, which were 3.15 s per linear µm over the slice rotation series, and 1 s per linear µm over the spiral measurement.
Fig. 4 Left-to-right: distributions of K, Cu, Zn in a single plane of C. elegans, as recorded using the ‘slice’ addressing modality. The linear scan covered 71 µm, and was measured at 221 unique angular orientations. The frequent, sudden positioning errors, of order 2–4 µm in size, are easily seen as discontinuities in the sinogram, but significant low-frequency distortions are also present. Both effects are the result of eccentricity in the rotation stage.
2.1. Reconstruction
Pre-processing of the elemental projection data involved: i) binning of 5 adjacent projection angles to reduce the dataset from 4,625 to 925 unique angles; ii) identification and replacement of absent values resulting from incomplete pixel addressing (see §3.3); iii) geometric alignment, and iv) motion correction. The angular binning to 925 projection angles still provides sufficient angular sampling, but improves signal-to-noise per row of the sinogram, which aids the application of subsequent iterative alignment and motion correction algorithms. The smoothness of the spiral sinogram indicates the validity of this angular averaging.
A global horizontal offset of the rotation axis in the sinogram, (i.e., geometric misalignment), was determined using an ‘autofocus’ technique [17]. This technique implements a multiscale search to determine the offset that maximises the ‘sharpness’ of the reconstructed image. Since all elemental sinograms share the same alignment, this was performed on the average sinogram of all emission data.
Imperfections in the trajectory through motion inaccuracies (in particular, rotation stage eccentricity, with systematic and random components) and sample motion (due to e.g., thermal expansion during the experiment) were identified as a positional shift per sinogram row. These corrections were achieved through the reprojection alignment method proposed for electron tomography [18]. The process is summarised as follows: i) reconstruct tomogram; ii) reproject tomogram; iii) align measured projection data to reprojection data; iv) repeat until converged. The locus of the determined correction of the specimen rotation centre as a function of projection angle is presented in Fig. 5. As was the case for geometric alignment, this operation was performed on the average sinogram of all emission data.
Fig. 5 Horizontal specimen alignment correction for the spiral and slice measurement modalities, estimated using the reprojection alignment method [18].
Since experimental noise is predominantly Gaussian, the Bayesian maximum likelihood solution to the reconstruction problem may be found via L2 norm minimisation. Classic simultaneous iterative reconstruction technique (SIRT) type reconstruction schemes are therefore ideal. The reconstructed elemental distributions presented in Fig. 6 were generated by up to 64 iterations of SIRT with a windowed, ramp-filtered back-projection forming the initial guess. Our use of a GPU-enabled reconstruction software suite [19] enabled this reconstruction to be performed rapidly, and has scope for further optimisation by employing an ordered-subsets implementation of SIRT.
Fig. 6 Left-to-right, top-to-bottom: distributions of K, Ca, Mn, Fe, Cu, and Zn, the mass-density determined from the Compton scatter signal, and the absorption-contrast signal in C. elegans, as reconstructed from measurements made using the spiral addressing modality.
2.2. Estimation of data quality
We use Fourier ring correlation (FRC) analysis [20, 21] to estimate reconstructed resolution. In essence, FRC analysis provides an estimate of resolution by interrogating correlations of two independent measurements of the same object, as a function of spatial frequency. At a given spatial frequency a loss of correlation indicates that signal has been overwhelmed with noise, and establishes a limit on the resolution of the measurements. Ideally, the two data sets should be produced from repeated, independent measurement, but this can be impractical when the measurement is not intrinsically repeatable. Here we estimate resolution by performing the FRC analysis on reconstructions obtained from sub-sets of the data and also on reconstructed distributions that appear to be similar (if not identical). In particular, we have used the potassium (K) emission and Compton scatter, as these show a high degree of correlation, as can be observed in Fig. 7.
Fig. 7 Tomograms generated by standard filtered back-projection from the potassium and Compton scatter signals from the slice (leftmost panels, respectively), and spiral (rightmost panels, respectively) measurement modalities. Slice data was acquired over a different region of the C. elegans and has a smaller field-of-view (FOV); the spiral tomogram has been cropped to present the two at comparable FOV (64µm) and pixel size (1µm). The structural similarity of the potassium and Compton scatter signals indicates that they might be used as independent inputs for the Fourier Ring Correlation resolution measure.
Figure 8 presents the FRC analysis of reconstructions of sinograms obtained using the spiral and slice acquisition modalities. For the spiral data we present FRC analysis of reconstructions from: i) the potassium (K) elemental and the Compton scatter distribution; ii) two subsets of the K elemental distribution, K1 and K2, obtained from the odd and even rows of the sinogram; and iii) as for (ii), but for the Compton scatter reconstructions, labeled C1 and C2. Subject to the assumption that the K and Compton scatter distributions are sufficiently correlated (see Fig. 7), the K-C FRC analysis should yield a more accurate estimate of resolution as it uses two complete datasets. On the other hand, the K1-K2 and C1-C2 FRC analyses are each measurements of exactly the same object, and so should in principle be correlated, but suffer resolution degradation due to each using only half of the total available data.
Fig. 8 Fourier Ring Correlation (FRC) analysis for reconstructions derived from (a) the potassium (K) and Compton data, and from reconstructions derived from the odd and even angular increments from the potassium (K1-K2) and Compton scatter (C1-C2) data. In order to evaluate the spiral addressing modality, we present also the FRC analysis of data acquired in the slice measurement modality, which shows considerably improved resolution over the spiral scheme. Three commonly-used resolution estimates are presented, being the 1-bit information threshold, the 1/2-bit information threshold, and the 2σ information limit. Numerical estimates presented in this article use the 1/2-bit threshold.
We observe in Fig. 8 that the FRC resolution estimated from the K-C reconstructions is slightly better than that from the K1-K2 and C1-C2 reconstructions, which validates the use of this ‘cross channel’ approach. Figure 8 also presents a FRC analysis of the K-C reconstructions obtained from the slice data, which clearly exhibits a significantly higher resolution than the spiral measurement.
The focussed beam has a cross-section of around 1 µm (1-σ), and the measurement interval is 1 µm, so we expect an overall spatial resolution that is no better than around 2 µm. Any significant degradation from this value might suggest loss of resolution through the addressing and/or the preprocessing and tomographic reconstruction process. The FRC analysis of the slice tomogram yields a resolution of around 3 µm, in broad agreement with this expectation. However, the FRC analysis of the spiral tomogram indicates a spatial resolution of around 6 µm, suggesting an issue in the measurement or reconstruction process.
The FRC analysis presented in Fig. 8 estimates the resolution of the reconstructed tomograms, and as such does not distinguish between contributions from the measurement and the preprocessing and reconstruction algorithms. Here we develop a method for disentangling these contributions by applying FRC theory directly to the measured sinogram data. Let us first define the 1D Fourier transform (FT) of a 1D function g as:
and the 2D FT of an image f likewise. The FRC of two images f1 and f2 at spatial frequency ω > 0 is then defined as: where the summations are evaluated over all points R with radius ω, i.e., |R| = ω describes the set of Cartesian Fourier coordinates {(kx, ky): kx2 + ky2 = ω2, ω > 0} F1 and F2 are the 2D Fourier transforms of f1 and f2, and ∗ denotes complex conjugation.To obtain F1 and F2 directly from the sinogram prior to tomographic reconstruction, we use the Fourier slice theorem (FST, see e.g., [22]). The FST describes the relationship between the projection of f (or sinogram) and the FT of f (i.e., F). Let us define the projection of f along the line ρ = x cos θ + y sin θ as follows:
where δ(η) = 1 when η = 0 and is zero otherwise. The 1D FT of gθ, denoted Gθ, is therefore: The 1D Fourier transform of the sinogram rows, i.e., constant angle, yields the values of the 2D Fourier transform of the tomogram in a polar coordinate system. Combining Eqs. (4) and (6), and re-parameterising the set |R | = ω as {(kρ, θ): |kρ| = ω}, we see that the FRC can be calculated directly from the sinogram: The set |R | = ω that describes the ‘rings’ used for correlation in the tomogram, in polar coordinates describes a pair (corresponding to ±ω) of columns in the sinogram. Comparing the FRC calculated from the sinogram and tomogram, we are able to separate the resolution loss derived from i) measurement, ii) preprocessing, and iii) the reconstruction procedure.Figure 9 presents the result of FRC analysis applied directly to the K-C sinogram data as per Eq. (7). Observe that the thresholds in Fig. 9 are now constant across resolution: the curved thresholds in Fig. 8 arise due to the increasing number of samples in each ‘ring’ with radius in the Cartesian representation. The FRC applied to the sinogram yields a polar representation of F and so the number of samples is constant with radius. Note that by analysing un-processed data, we directly probe the resolution of the measured data, free of the influence of preprocessing and reconstruction algorithms.
Fig. 9 Fourier Ring Correlation (FRC) analysis as per Figure 8, but here evaluated directly from the un-processed sinogram data using Eq. (7). The number of sinogram rows (i.e., projection angles) was binned down to approximately π/2 times the number of measurements per row.
The FRC resolution estimates determined from Fig. 9 are very similar to those observed in Fig. 8, indicating that the loss of resolution is unrelated to the alignment preprocessing operations or the tomographic reconstruction, and that it is intrinsic to the imaging process. Conversely; had this FRC analysis suggested a superior resolution in the raw sinogram data, we would then apply sinogram FRC analysis [Eq. (7)] to the pre-processed sinograms, to determine exactly which step resulted in the degraded resolution.
Interestingly, the FRC resolutions of both the slice and spiral tomograms are slightly improved relative to those estimated from the sinograms. However, we expect alignment pre-processing operations and reconstruction by filtered back-projection can only degrade intrinsic resolution. We therefore propose that the apparent improvement in resolution is due to the introduction of correlated high-frequency artifacts in the reconstruction process.
This analysis clearly indicates that the resolution of the data collected using the slice trajectory is much better than that using the spiral trajectory. Figure 4, presenting the as-measured slice sinogram data, provides some clues as to the potential cause of the resolution loss: eccentricity in the rotation stage. The 6 µm eccentricity that is clearly apparent as positional discontinuities in the sinogram (Fig. 4) is accurately characterised in the alignment phase of the preprocessing (Fig. 5). However, this eccentricity does not affect the ultimate resolution of the slice data because there the eccentricity is constant for each traverse and is fully addressed by the alignment correction. The discontinuities in the alignment correction for the slice data (Fig. 5), are not visible in the alignment correction for the spiral data, due to two effects: (a) mechanical smoothing, due to continuous rotational motion of the stage and; (b) averaging of the angular offset over the 250 rotations of the stage. That is, we believe that some aspects of the poor rotational motion indicated in the slice sinogram also affect the spiral data on a per-rotation basis, but that these cannot be corrected by the present method. We attribute the degraded resolution of the spiral data to these errors, and note that they could be overcome by use of more sophisticated approaches to stage metrology and encoding (see §3.1).
3. Discussion
3.1. Post-processing: alignment
One significant but often under-reported aspect of the computed tomography is the requirement for and benefit derived from post-processing and post-acquisition alignment. Unconstrained specimen eccentricity or drift results in misaligned projections, and these in turn degrade the tomographic resolution if not corrected.
Full-field tomographic data differs from scanning-probe data in that the data is acquired from pixels in a camera in the form of a frame; the pixels within the frame are usually highly orthogonal and well aligned, and are always completely repeatable. The underlying internal integrity of the frame means that there are immutables of the measurement process which can be relied upon. Alignment of full-field tomographic data can occur at the level of the projection and at the level of the slice. For example, alignment markers provide readily-identified features that can be automatically located and then used to bring a series of projections to a common rotation centre. Additionally, a sinogram extracted from a single slice of such a rotation series can be further aligned to optimise reconstruction resolution.
In the scanning microscopies, the pixels that make up an image are discrete samples made at a series of spatial locations at various different times [23]. There is no definite, fixed relationship between the image pixels in one projection and the next. It is therefore reasonable to question the validity of employing image alignment methods that assume fixed relationships to scanning tomography modalities. However, practically speaking, reported alignment methods are not notably different to those used for full-field tomography. Scanned projections are often shifted using rigid-body translations, even though the ‘pixels’ do not inherently carry a fixed relationship to one-another. Non rigid-body operations are sometimes called for [24], but even these assume some level of ‘rigidity’.
In scanning tomography, the hierarchy of repeatability extends from the fast axis (the most ‘rigid’) to the slow axis [23]; in 3-D tomography, the slow axis is used to align projections in both the vertical and horizontal, and from this moment the vertical alignment is often not refined [25]. Once a sinogram is extracted from a particular slice of the tomogram, its alignment is further refined according to some criteria, usually relating to internal consistency.
Spiral scanning tomography presents new challenges for alignment. What are the rigid-body entities, and how can they be shifted? Perhaps one of the biggest influences on sinogram alignment in the spiral scanning geometry is the influence of eccentricity, inducing an angle-dependent translation to the specimen position. This eccentricity will have systematic (repeatable) and random (non-repeatable) components.
What can we do in the spiral scanning approach? The repeatable component of eccentricity will be compensated in the usual manner, by aligning e.g., the centre of mass, but the non-repeatable component (which presents no particular problem to traditional slice addressing method) simply cannot be corrected. In our case the sinogram was recorded with some 250 rotations of the stage, with the non-repeatable component of the rotation deeply embedded in the data set. We believe that the inability to correct this non-repeatable component of the eccentricity is the cause of the reduced resolution.
Is this the end of spiral tomography? This discussion relates most directly to scanning implementations where there is uncertainty about the registration of the specimen with the beam. However, modern implementations of high-precision interferometric encoding [9] will mitigate several of the issues present here, with direct implications for the feasibility of the spiral scanning modality.
3.2. Oversampling
The spiral tomography data is oversampled in several dimensions, and these inherent redundancies in the data can provide opportunities for further studies. The angle, here almost a continuous variable, is oversampled by a factor of 2nθ/πnx relative to Crowther; additionally, the position is oversampled by a factor of two: a full sinogram can be acquired over the full width of the specimen spanning 180° of rotation, or half the width of the specimen spanning 360° of rotation [26]. This oversampling can be used to determine an independent measurement, in a similar manner to our use of even and odd angle partitioning of the sinogram, or to check (e.g.) for radiation damage during a measurement.
3.3. Pixel addressing
Figure 10 presents a sinogram of another C. elegans specimen, this time displaying Ca, K and the Compton scatter image. The Compton scatter can be an excellent indicator of the distribution of mass-density within a specimen, and is so sensitive as to pick up signal from the measurement environment (air). The series of black streaks parallel to the rotation axis motion (horizontal as displayed here) each indicate instances where one or several pixels have not been addressed by the scanning stages.
Fig. 10 Spiral sinogram of the C. elegans specimen indicating the distributions of Ca, K, and Compton scatter (RGB). Measurement parameters were similar to Fig. 3. This figure has been thresholded so as to emphasise the pixels that were not addressed by the scanning stages, particularly noticeable in the Compton scatter image. The image is 550 µm high and 360° wide.
Initially we thought that the motion controller was not functioning correctly, but in fact the cause of these black streaks is our unreasonable expectation of the encoders and the motion control system in this spiral scanning mode. In most typical scanning implementations, such as that indicated in Fig. 1(a), pixel boundaries are traversed at right angles, and are thereby generally well-defined. The locus presented in Fig. 1(b) shows a perfectly ordered trajectory where the x-pixel crossing occurs at exactly the same angular location on each rotation of the specimen. However, it is also clear from this figure that the x-pixel traverse occurs at a very shallow angle of approach. Accordingly, when the x-pixel traverse occurs at a different angular location to that on the previous rotation there will be some region of x-θ space that is either multiply visited or not visited by the motion system – leading to the black pixels shown in Fig. 10.
Multiple visits are just as likely as null visits, indeed: if the x-θ gearing ratio is correct then the two should ultimately be a zero-sum. However, the accumulative analysis of the event-mode data-stream means that the double visits are not visible in the elemental images, as they correspond only to regions of higher statistical precision, but with the mean elemental content unchanged.
Null visit pixel addressing generally involves less than 5% of a revolution. Accordingly, for cases where the effect is much less severe than that illustrated in Fig. 10 we have replaced the missing data with the average of the neighbouring traverse values. We note that the null-visit effect can be entirely removed by modifying the x-θ gearing ratio so that slightly more than one rotation is completed for every x-pixel traverse, deliberately aiming to multiply-visit some fraction of pixels on each turn, with a commensurate increase in the acquisition time.
4. Conclusion
Spiral scanning tomography reduces to virtually nil the overheads intrinsic to measurements where sample scanning is unavoidable, and provides direct access to fully-fractionated tomography and higher dimensional methods such as volumetric tomography and XANES tomography. Here we have demonstrated the method and explore some of the issues raised by this novel measurement modality. Future implementation will require better control over rotational motions, probably through improved hardware such as use of an air-bearing rotation stage or by use of improved metrology to determine the specimen locus.
Funding
CSIRO Science and Industry Endowment Fund.
Acknowledgments
The measurements presented in this work were performed at the X-Ray Fluorescence Microscopy beamline of the Australian Synchrotron. Deconvolution of x-ray fluorescence spectra were performed using GeoPIXE running on the MASSIVE environment (www.massive.org.au). Reconstructions were performed on the GPU cluster at the CTLab facility (www.ctlab.anu.edu.au) at the Australian National University. This research is supported by the CSIRO Science and Industry Endowment Fund. Gary Ruben was supported by a CSIRO OCE Postdoctoral Fellowship.
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