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Abstract
In this paper, super-resolution imaging is described and evaluated for x-ray tomography and is compared with standard tomography and upscaling during reconstruction. Blurring is minimized due to the negligible point spread of photon counting detectors and an electromagnetically movable micro-focus x-ray spot. Scans are acquired in high and low magnification geometry, where the latter is used to minimize penumbral blurring from the x-ray source. Sharpness and level of detail can be significantly increased in reconstructed slices to the point where the source size becomes the limiting factor. The achieved resolution of the different methods is quantified and compared using biological samples via the edge spread function, modulation transfer function, and Fourier ring correlation.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
The achievable spatial resolution of a laboratory x-ray imaging setup depends on several parameters. Considering the detector, a small pixel size while maintaining a small point spread function (PSF) is desirable. The conventional scintillator-based charge coupled device (CCD) and complementary metal oxide semiconductor (CMOS) cameras typically have a Gaussian-like PSF covering several pixels. Further, blurring from the finite scintillator thickness increases the size of the PSF.
As an alternative, hybrid photon counting detectors provide a much narrower PSF, which can typically be assumed to be the size of a single pixel. However, such detectors are bound to have larger, and much more expensive, pixels due to the construction of individual counting electronics behind each pixel. Large field of view (FOV) can be achieved by tiling detector elements [1]. For many applications of x-ray imaging, however, high spatial resolution is the sought parameter. With scintillator-based CCD x-ray cameras, the achievable spatial resolution is typically limited by the PSF, which is often the size of several pixels. Moreover, photon counting detectors provide additional features, such as spectral imaging using multiple thresholds [1,2], direct measurement of photon energy [3,4], time-resolved measurements [3,4], and integrated correction for charge sharing [2].
Photon counting detectors are a recently developed type of direct x-ray detector. The first photon counting detector was the Medipix1 chip in 1998 [5]. Such detectors count individual photons, provide a noise-free readout, and a typically box-like PSF corresponding to a single pixel. Moreover, photon counting detectors show only limited charge sharing between pixels compared to CCD and CMOS cameras. These properties make photon counting detectors interesting for applications in biomedical imaging [6].
Photon counting detectors directly convert x-rays into a charge cloud using a semiconductor material such as silicon (Si), gallium arsenide (GaAs), cadmium telluride (CdTe), or cadmium zinc telluride (CdZnTe) to which a high voltage is applied. The semiconductor sensor is bonded to an underlying readout application specific integrated circuit (ASIC), e.g., via bump-bonding.
The detectable photon energy range is determined by the sensor material. While Si is easy to manufacture in high quality at low price, its efficiency decreases significantly for energies higher than 10 keV. GaAs, CdTe, and CdZnTe can detect much higher energy photons but are much more difficult to manufacture, more expensive, and less stable in operation [7]. CdTe offers a high detection efficiency above 30 keV making it particularly useful in medical imaging and material studies [8]. However, fluorescence from high density sensor material can occur, i.e., a photon is emitted by the detector material, which then might be registered by a different pixel [9].
The main disadvantage of photon counting detectors is their relatively large pixel size of 55 µm (Medipix2-3, Timepix1-3) [2–4,10,11], 75 µm (Eiger) [1], and 172 µm (Pilatus) [12]. The newly developed charge-integrating detector Mönch [13] with a pixel size of 25 µm provides comparable capabilities.
Super-resolution describes a technique estimating a high-resolution image from several slightly shifted low-resolution images. To obtain super-resolution images, the shifts between the images are precisely registered using a cross correlation algorithm [14], the images are interpolated on a high resolution grid, and then they are finally combined into a single image [15]. Depending on the sample and the setup parameters, an advanced super-resolution algorithm can be utilized including filters and regularization steps. However, here we utilize the basic approach for a more direct comparison reducing the amount of image processing steps.
Many common approaches to super-resolution take detector blurring into account by incorporating a deconvolution filter to deblur the estimated super-resolution image [16]. A simple approach uses, e.g., a Wiener filter and a Gaussian kernel with the size of the detector PSF [16]. Blind deconvolution provides a method where the PSF is estimated from the provided images [17]. An alternative approach is image regularization [18]. During an iterative process, multiple image parameters can be optimized to achieve an optimal result. A commonly used method is iterative back-projection [19], which has been utilized in x-ray-based super-resolution applications before [20]. In clinical applications of super-resolution using positron emission tomography - computed tomography (PET-CT) and magnetic resonance imaging (MRI) machines, a projection onto convex sets (POCS) algorithm [21] has been demonstrated to provide usable results. Also, neural networks can be used to create high-resolution images from a single image [22].
It has been demonstrated that limiting geometric magnification in favor of super-resolution also reduces blurring in a Talbot–Lau interferometer setup, where the sample placement is limited by the required grating placement in a setup that is not limited by source blurring [20].
In this study, we demonstrate that super-resolution with photon counting detectors is a viable approach to improve spatial resolution in x-ray tomography, achieving significantly improved resolution. Using a micro-focus x-ray source, in a geometry with low magnification, assures that the re-scaled x-ray spot in the detector plane is comparable to the super-resolution pixel size. By electromagnetic stepping of the x-ray spot, precise movements can be executed without the need for mechanical movements, avoiding jitter and vibrations in the setup [23,24]. We quantify the achieved resolution of super-resolution, upscaled, and regular reconstructions using real-world biological samples. Further image processing is avoided for a more generalized comparison.
2. METHOD
A. Detectors
Here we utilize two different photon counting detectors: an Eiger 2R 500 K with 75 µm pixel size and a 450 µm silicon sensor from Dectris Ltd. (Baden, Switzerland) and a Lambda 350 K (Medipix3 [25]) with 55 µm pixel size and a 500 µm gallium arsenide sensor from X-spectrum GmbH (Hamburg, Germany).
B. Minimizing Blurring from X Ray Detector and Source
The PSF of photon counting detectors can be assumed to be a single pixel. However, charge sharing might occur depending on the energy and impact location of photons on the detector. Higher energy photons and impacts in the corner of a pixel or along the border between two pixels are more likely to cause charge sharing [26]. In imaging, these charge sharing effects are often neglected for simplicity considering sufficient photon statistics.
Neglecting detector blurring, the x-ray spot size becomes the limiting factor. Depending on the geometry, the re-scaled x-ray spot in the detector plane can be larger than a pixel, resulting in penumbral blurring from the source. The sample position can be moved closer to the detector to reduce penumbral blurring, while the size of the re-scaled x-ray spot in the detector plane will be reduced. However, this reduces the magnification of the sample, increasing the effective pixel size ${P_{{\rm eff}}}$, and, thus, requires more oversampling, i.e., more spot positions, to increase the spatial resolution using super-resolution.
C. Super-Resolution Tomography
To create super-resolution images, the shifts between individual images are registered with cross correlation [14]. All images are interpolated on a new high-resolution grid considering their shifts using a spline [27]. Finally, the resulting high-resolution images are averaged [15]. No further image processing is applied. Alternatively, the ASTRA toolbox [28,29] allows reconstruction onto a finer grid.
The achievable increase in resolution $R$ is calculated in Eq. (1) based on simulations [17,18] and previous experiments [27] adjusted for regular grids of images based on the experiments performed in [23,27],
where ${n_{{\rm img}}}$ is the number of low-resolution images. Regular shifts between images assure sufficient oversampling. Results presented in this paper have been obtained with total shifts from 1 to 3 pixels in $(x,y)$ directions of the sample image on the detector. While a total shift of 1 pixel is theoretically sufficient, good results have been obtained using a slightly larger sample translation [27]. However, larger shifts can cause astigmatism of the spot; thus, it is desirable to limit the movement distance.Raw data obtained from the Eiger detector contain masked pixels that have to be removed from the images and the corresponding flat-fields. The Lambda detector does not utilize a pixel mask, i.e., noisy, dead, and under- or over-responsive pixels have to be identified first. In both cases a mask is created to select the pixels that should be removed. Masked pixels are replaced with an average of their surrounding pixels. The values are obtained by applying a median filter to the image, extracting the pixel values at masked positions, and replacing the corresponding pixels in the original image. For tomography, rotation is used as the fast axis, i.e., the spot position is only changed after every sample rotation, thus allowing for the shifts between images to be obtained from a single projection. Depending on the sample, a region containing high contrast features or edges can be defined to be used for shift registration by cross correlation. The raw images are then combined into projections. Tomographic reconstruction is performed using the Compute Unified Device Architecture (CUDA) implementation of the Feldkamp–Davis–Kress (FDK) [30] algorithm provided by the ASTRA toolbox [28,29] in a MATLAB (Mathworks) environment, with corrections for both center-of-rotation displacement and detector tilt. Reconstructions are performed on a computer with an Intel Core i9-10900X (10 cores at 3.7 GHz), 256 GB of RAM, and a Nvidia GeForce RTX 2080 Ti. See Supplement 1 for details on the volume size and reconstruction times. Further, a wavelet-based ring filter [31] is added to the reconstructions of the data obtained with the Lambda detector [32].
D. Electromagnetic Source Stepping
To achieve super-resolution in a laboratory setup, the sample image has to move on the obtained projection radiographies. This can be implemented in several ways, such as motorizing the sample stage, the detector, or the source itself [33]. However, considering the weight of the equipment that has to be moved, mechanical stepping can be slow, imprecise, and expensive [34]. Highly precise mechanical alignments are also associated with increased sensitivity to thermal effects [35]. Replacing mechanical movements with an electromagnetically movable x ray spot abrogates many issues of motorized setups [24], such as vibration and instabilities. The movement distance on the x-ray source target is dependent on the sample magnification. Further, electromagnetic source stepping offers precise and repeatable movements without sensitivity to vibrations and with no need to add further motorization into the setup.
A Talbot interferometer setup with 1D electromagnetic source stepping has been previously described [24]. Two-dimensional (2D) electromagnetic source stepping has been demonstrated to work reliably in conventional x-ray radiography [27] and single-grating dark-field radiography [23]. Further, phase-contrast imaging with sub-pixel resolution has also been demonstrated by utilizing a mask with a data-driven Timepix3 detector [36]. An alternative approach combining shifted sinograms has been presented as well [37]. Here we demonstrate this technique in x-ray tomography, extending the concept into three-dimensional imaging.
E. Resolution Estimation
A standard method to measure the resolution of an image is the edge spread function (ESF), which is obtained by measuring the characteristic width $\sigma$ of a sharp edge. Typically, standard objects are used, such as Siemens stars or JIMA charts, containing sharp metal features [38]. Thus, measuring the ESF using a biomedical sample might cause varying results due to the available features and lower contrast. Further, using the ESF data, a resolution limit can be estimated via the modulation transfer function (MTF). Following the slanted edge method [39,40], first the line spread function (LSF) is obtained by derivation of the ESF, which is then Fourier transformed to obtain the MTF. The resolution limit is then estimated via the spatial frequency at 10% modulation. However, the obtained values are affected by the selected feature.
To obtain a generalized resolution estimation of a real-world biomedical sample, Fourier ring correlation (FRC) is used [41–44]. FRC works on identical slices from two independent reconstructions of the same scan, where the projections have been split into two datasets of even and odd projections. Translation, rotation, and scaling of the used slices are matched with sub-pixel precision. Features are matched using an oriented FAST and rotated BRIEF (ORB) classifier [45], and outliers are removed using an iterative random sample consensus (RANSAC) algorithm [46]. On the aligned slices or volumes, a correlation function is calculated utilizing equivalent rings in Fourier space [41]. From an FRC curve, the resolution limit can be obtained by intersection with a threshold criterion [47]. In imaging, this criterion is typically based on a signal-to-noise ratio threshold [43,44].
F. Experiment Parameters
Our experimental setup (Fig. 1) consists of a prototype tungsten anode micro-focus source from Excillum AB allowing free movement of the x-ray spot in the $(x-y)$ direction. The x-ray spot is focused to 10 µm, which is internally verified, at an acceleration voltage of 70 kV. In the estimation of the ESF, we assume that the source intensity distribution is Gaussian. Downstream from the source, a rotation stage is mounted on two linear stages for vertical and horizontal positioning of the sample. The detector is an Eiger2 500 K R from Dectris Ltd. with a 450 µm thick Silicon sensor and $1030 \times 514$ square pixels of size $P = 75\,\,\unicode{x00B5}{\rm m}$ providing a sensitive area of $77.25 \;{\rm mm} \times 38.6\;{\rm mm} $. An additional scan was acquired with a Lambda 350 K from X-spectrum GmbH with a 500 µm thick gallium arsenide sensor and $768 \times 512$ square pixels of size $P = 55\,\,\unicode{x00B5}{\rm m}$ with a gap of 6 pixels between the individual $256 \times 256$ pixel chips resulting in an active area of $42.3 \;{\rm mm} \times 28.2\;{\rm mm} $.
Fig. 1. Laboratory setup consisting of an Excillum micro-focus source with a movable x-ray spot and a Dectris Eiger 2 R 500 K detector with 75 µm pixel size and a 450 µm Si sensor or a X-spectrum Lambda 350 K detector with 55 µm pixel size and a 500 µm GaAs sensor. (a) Schematic drawing of the geometry, the source-detector distance SD is 0.55 m. Two sample position are used: Position A with a source-object distance ${{\rm SO}_{\rm A}} = 0.15\;{\rm m} $ (for higher magnification) and Position B with ${{\rm SO}_{\rm B}} = 0.346\;{\rm m} $ for the Eiger and 0.37 m for the Lambda have been selected to limit the penumbral blurring of the x-ray spot to half a pixel. (b) Computer-aided design (CAD) model of the setup, with the sample holder and detector mounted on rails for manual adjustment along the beam direction.
In this study, the detector is always placed at 0.55 m from the source, assuring the full FOV is illuminated by x-rayS. Two general sample positions are evaluated. Position A is selected to be 0.15 m from the source. Position B is chosen to minimize penumbral blurring to half a pixel. Table 1 summarizes the different geometries and their effective pixel sizes. The sample location ${{\rm SO}_{\rm B}}$ is selected so the projected spot size is limited to half a pixel of the used detector. At Position A, geometric magnification ${M_A} = {\rm SD}/{{\rm SO}_{\rm A}}$ of 3.667 is achieved. However, penumbral blurring will limit the resolution in this geometry. At Position B, the Eiger achieves a magnification of ${M_B} = {\rm SD}/{{\rm SO}_{\rm B}} = 1.59$, with ${{\rm SO}_{{\rm B,Eiger}}} = 0.346\;{\rm m} $ resulting in an effective pixel size of ${P_{{\rm eff}}} = P/M = 47.18\,\,\unicode{x00B5}{\rm m}$ and a super-resolution pixel size of ${P_{{\rm super}}} = {P_{{\rm eff}}}/S = 7.86\,\,\unicode{x00B5}{\rm m}$ with super-resolution factor $S = 6$ or 11.8 µm with $S = 4$. The Lambda achieves an effective pixel size of 37.5 µm, where ${{\rm SO}_{{\rm B,Lambda}}} = 0.375\;{\rm m} $ and, thus, ${M_B} = 1.467$. Using a super-resolution factor $S = 4$, a super-resolution pixel size of ${P_{{\rm super}}} = 9.37\,\,\unicode{x00B5}{\rm m}$ is achieved. In this geometry, penumbral blurring is no longer the main limiting factor for the resolution; potential detector blurring and blurring from image processing and reconstruction will define the resolution limit instead.
Table 1. Setup Geometries
Acquiring projections on a regular grid and creating super-resolution projections as described in Section 2.C and using Eq. (1) allows us to increase the spatial resolution, thus reducing the effective pixel size. Most scans have been obtained using a $4 \times 4$ grid of images, i.e., a resolution increase by a factor of 4. This amount of images has been selected as a compromise of acquisition time and resolution increase. The achieved effective pixel sizes with super-resolution ${P_{{\rm super}}}$ at Position B are around or slightly smaller than the x-ray spot size and are listed in Table 1. The spot spacing ${X_{{\rm spacing}}}$ on the x-ray target is calculated via ${X_{{\rm spacing}}} = P \times {s_{{\rm step}}} \times {M_S}$ with $P$ as the physical pixel size, ${s_{{\rm step}}}$ as the pixel shift per step, and the source magnification ${M_S} = ({\rm SD - SO})/{\rm SO}$. At Position B, 63.6 µm is denoted for a 3 pixel shift, 21.2 µm for a 1 pixel shift with the Eiger detector, and 58.9 µm for a 2 pixel shift with the Lambda detector.
Fig. 2. Reconstructed slices from the rose bud sample at Position A. The blue square marks the zoomed region shown in the panels below, and the red line marks the edge used to estimate the resolution via ESF. (a) Regular CT with zoomed area in (d), (b) $4 \times$ upscaled CT with zoomed area in (e), and (c) super-resolution CT with zoomed area in (f).
At Position A with ${\rm SO} = 0.15\;{\rm m} $ and $4 \times 4$ images, a super-resolution pixel size of ${P_{{\rm super,A}}} = 5.11\,\,\unicode{x00B5}{\rm m}$ is achieved for the Eiger detector. The resulting super-resolution pixel size is significantly smaller than the x-ray spot limiting the resolution. The Lambda detector was only used at Position B. The spot spacing ${X_{{\rm spacing}}}$ on the x-ray target was 21.1 µm for a 3 pixel shift and 7 µm for a 1 pixel shift.
Since the accumulated exposure time is significantly increased, reducing the exposure time per individual image would be a viable approach to reduce the total scan time. However, this is not evaluated in this paper.
When acquiring a tomographic scan, the rotation axis is used as the fast axis, i.e., a full CT is acquired before the x-ray spot position is moved. This protocol simplifies data processing, i.e., only data from a single projection need to be registered by cross correlation since it can be assumed that the x-ray spot did not shift. Although the x-ray spot movements are very precise, slight variations in position repeatability may require to register all projections if a protocol with more frequent source movement was chosen.
Two different samples have been used: a dried flower bud (rose bud) and a cricket (see Figure S1 in Supplement 1. All samples have been scanned with the Eiger detector. For comparison, the rose bud has also been scanned at Position B with the Lambda detector. The source was calibrated to 10 µm spot size, and 1080 projections have been acquired with varying source emission powers and exposure times. The imaging settings can be found in Table S1, and details on the reconstructed volumes can be found in Data File 1, Ref. [48]. Selected levels in the volumes are illustrated Visualization 1 for the rose bud sample and Visualization 2 for the cricket sample.
3. RESULTS AND DISCUSSION
For the rose bud sample at all positions, similar levels close to the optical axis have been selected. Figures 2(a)–2(c) show equivalent reconstructed slices of a regular CT using a single x-ray spot position, an upscaled CT using the same projections, and a super-resolution CT created using $4 \times 4$ x-ray spot positions per projection. Further, the lower panels [Figs. 2(d)–2(f)] show a zoomed in feature, which demonstrates that the different CTs are not limited by the acquisition time; thus, the image noise is not a limiting factor. Due to the large amount of projections (1080), upscaling [Figs. 2(b) and 2(e)] significantly improves the reconstruction. However, the most significant improvement can be seen in the super-resolution reconstruction [Figs. 2(c) and 2(f)], where the features are significantly sharper.
Measuring the ESF of the same edge in the three different reconstructions [Fig. 3(a)] gives a FWHM of 1 pixel for the regular CT, i.e., the resolution of this particular feature is limited by the pixel size of detector. However, utilizing FRC [Fig. 3(b)] as a more generalized resolution estimation considering the full slice results in a resolution limit of 32.14 µm, larger than the effective pixel size ${P_{{\rm eff,A}}} = 20.45\,\,\unicode{x00B5}{\rm m}$, which is caused by penumbral blurring. The upscaled CT gives a FWHM of 25.25 µm via the ESF, a resolution limit of 17.05 µm at 10% MTF, and 11.68 µm via FRC [Fig. 3(b)] with an effective pixel size of ${P_{{\rm super,A}}} = 5.11\,\,\unicode{x00B5}{\rm m}$. FRC shows that the resolution has been improved almost up to the size of the x-ray spot (10 µm). The super-resolution CT achieves slightly better values with a FWHM of 16.17 µm, 11.36 µm at 10% MTF, and 7.44 µm resolution limit according to FRC [Fig. 3(d)]. Measured MTF curves are shown in Figure S2a in Supplement 1. The wobble of the FRC for the super-resolution CT is most probably caused by the shift between images acquired for each projection, where a total shift of 3 pixels (0.75 pixels between individual images) was selected, while for the cricket sample a total shift of 1 pixel was used, which leads to a smoother FRC (see FRC plot in Supplement 1). Further, a resolution limit smaller than the configured spot size was obtained, which might be caused by the aforementioned wobble of the FRC curve, but also is due to the Gaussian shape of the of the x-ray spot. All measured resolutions can be found in Table 2.
Fig. 3. Edge profiles and Fourier ring correlations of the rose bud sample imaged with the Eiger at Position A. (a) Edge profiles for the three different reconstructions with an estimated resolution via ESF (FWHM) of 1 pixel, 16.42 µm, and 10.53 µm for the regular CT, $4 \times$-upscaled CT, and super-resolution CT, respectively. (b) FRC of the regular CT showing that the scan is limited by penumbral blurring with a resolution limit of 32.15 µm. (c) FRC of the $4 \times$-upscaled CT with a resolution limit of 11.68 µm. (d) FRC of super-resolution CT with a resolution limit of 7.44 µm.
Moving the sample to Position B and comparing equivalent reconstructed slices (Fig. 4) at a similar level as at Position A yields comparable results. Again, the upscaled CT manages to significantly increase the resolution while the super-resolution CT performs better. Comparing the zoomed areas between the upscaled and super-resolution CT [Figs. 4(e) and 4(f)] shows significantly more artefacts in the upscaled slice. Here the effective pixel size is ${P_{{\rm eff,B}}} = 47.18\,\,\unicode{x00B5}{\rm m}$, and $6 \times$ upscaling or super-resolution is used giving a super-resolution pixel size of ${P_{{\rm super,B}}} = 7.86\,\,\unicode{x00B5}{\rm m}$, slightly below the x-ray spot size.
Fig. 4. Reconstructed slices from the rose bud sample at Position B (see Visualization 1). The blue square marks the zoomed region shown in the panels below, and the red line marks the edge used to estimate the resolution via ESF. (a) Regular CT with zoomed area in (d), (b) $6 \times$ upscaled CT with zoomed area in (e), and (c) super-resolution CT with zoomed area in (f).
For the regular CT, the FWHM obtained via the ESF is again limited by the pixel size as expected as shown in Fig. 5(a), which is confirmed by the FRC in Fig. 5(b). At this position, penumbral blurring is no longer limiting the resolution as it does in the high magnification geometry. The ESF for the upscaled and super-resolution CT yields FWHM of 34.02 µm and 23.47 µm, respectively. Thus, the resolution limits at 10% MTF are 35.73 µm and 23.82 µm, and the corresponding curves can be found in Figure S2b in Supplement 1. Further, the resolution limits obtained via FRC are 17.0 µm and 13.81 µm for the upscaled and super-resolution CT, respectively [Figs. 5(c) and 5(d)], comparable to the values achieved at Position A [Figs. 3(c) and 3(d)]. All measured resolutions can be found in 2.
Fig. 5. Edge profiles and Fourier ring correlations of the rose bud sample imaged with the Eiger at Position B. (a) Edge profiles for the three different reconstructions with an estimated resolution via ESF (FWHM) of 1 pixel, 34.02 µm, and 23.47 µm for the regular CT, $6 \times$-upscaled CT, and super-resolution CT respectively. (b) FRC of the regular CT, which confirms the resolution limit to be the pixel size by not falling below the 1-bit threshold. (c) FRC of the $6 \times$-upscaled CT with a resolution limit of 17.0 µm. (d) FRC of super-resolution CT with a resolution limit of 13.81 µm.
While the scans with the Eiger detector in Figs. 2 and 4 were not limited by the noise in the images, a comparable acquisition with the Lambda detector, shown in Fig. 6, can be observed to be limited by image noise. In this case, the super-resolution scan [Fig. 5(c)] shows significantly improved contrast compared to the regular and $4 \times$ upscaled CT, seen in Figs. 6(a) and 6(b). The ESF of the regular CT results in a FWHM of 1 pixel, as the scans before, while the upscaled CT yields a FWHM of 39.81 µm and 34.09 µm at 10% MTF, which is about the effective pixel size of the regular CT with ${P_{{\rm eff,B}}} = 37.5\,\,\unicode{x00B5}{\rm m}$. However, considering the FRC, a resolution limit of 24.59 µm is found. The super-resolution CT yields a FWHM of 17.11 µm, 17.52 µm at 10% MTF, and 22.16 µm using FRC. ESF and FRC curves are shown in Figure S3 in Supplement 1, and the MTF curves are shown in Figure S2c in Supplement 1. The resolution was most probably also affected by the applied ring filter. All measured resolutions can be found in 2.
Fig. 6. Reconstructed slices from the rose bud sample at Position B imaged with the Lambda detector. The red line marks the edge used to estimate the resolution via ESF. (a) Regular CT, (b) $4 \times$ upscaled CT, and (c) super-resolution CT. Here the (a) regular CT and (b) upscaled CT are limited by the exposure time; thus, the (c) super-resolution CT looks significantly clearer due to the longer accumulated exposure time.
When interpolating projections, here by a factor of 4 using spline interpolation matching the method during super-resolution, before reconstruction or interpolating the reconstructed slices [Figs. 7(a) and 7(b)], the ESF [Fig. 7(c)] gives values close to those of the super-resolution reconstruction (10.53 µm) with 10.54 µm and 13.72 µm. However, FRC [Figs. 7(d) and 7(e)] yields 18.97 µm and 32.01 µm with interpolation before and after reconstruction, respectively, compared to 7.44 µm with super-resolution and 11.68 µm with upscaling. This shows that interpolation may perform well on sharp features, but when considering the resolution of a full slice, it will perform worse than super-resolution and upscaling during reconstruction. With regards to FRC resolution, interpolation before reconstruction is in this case better than interpolation after reconstruction, which shows no improvement compared to using no interpolation.
Fig. 7. Rose bud sample at position A with interpolation. The red line marks the edge used to estimate the resolution via ESF. (a) Interpolated projections before reconstruction. (b) Interpolated slices after reconstruction. (c) ESF yielding 10.54 µm and 13.72 µm for interpolation before and after reconstruction, respectively. (d) FRC of the reconstruction with interpolated projections. (e) FRC of the interpolated reconstruction.
Additionally, a cricket has been scanned at Positions A and B as well with $4 \times$ upscaling/super-resolution. Figures 8(a)–8(c) show reconstructed slices for the regular, upscaled, and super-resolution CT at Position A 0.15 m from the source, and Figs. 8(d)–8(f) show reconstructed slices at a similar level at Position B. In Figs. 8(a)–8(c), some features are highlighted (arrows) where the improvement in resolution can be observed in particular.
Fig. 8. Reconstructed slices of the cricket sample (see also Visualization 2) at Positions A (upper row) and B (lower row). (a) Regular CT, (b) $4 \times$ upscaled CT, and (c) super-resolution CT at Position A. (d) Regular CT, (e) $4 \times$ upscaled CT, and (f) super-resolution CT at Position B. The red line mark edges used for resolution estimation via the ESF. In (a)–(c) features are marked with arrows where a difference between the three methods can be observed.
Fig. 9. Reconstructed slices orthogonal to the beam of the cricket sample. (a) Regular CT, (b) $4 \times$ upscaled CT, (c) super-resolution CT, and (d) super-resolution CT using $2 \times 2$ images and 270 projections matching the total exposure time of the regular and upscaled CT. (e)–(l) Zoomed areas [blue squares in (a)–(d)] showing that upscaling (f), (j) significantly improves the image, (g), (k) super-resolution performs best, and (h), (l) super-resolution with fewer images per projection and fewer projections increases the resolution significantly but suffers from artefacts due to the low amount of projections. The arrows in (a)–(d) mark features that improve visually or show reduced artefacts in the (c) super-resolution and (d) equivalent super-resolution slices.
At Position A [Figs. 8(a)–8(c)], the FWHMs of 1 pixel, 14.18 µm, and 8.75 µm have been measured for the regular, upscaled, and super-resolution CT, respectively. At 10% MTF, the resolution of the upscaled CT was measured to be 13.64 µm and 9.47 µm for the super-resolution CT. At Position B [Figs. 8(d)–8(f)], the FWHMs have been measure to be 1 pixel, 34.73 µm, and 21.81 µm. Again, the resolution at 10% MTF was measured as 39.32 µm for the upscaled CT and 23.13 µm for the super-resolution CT. These values show a very similar behavior as observed for the rose bud sample in Figs. 2 and 4. Comparing the resolution limit obtained via FRC confirms again that the regular CT at Position A is limited by penumbral blurring yielding a resolution limit of 43.3 µm, while at Position B the effective pixel size is the limiting factor. The upscaled CT results in a resolution limit of 46.2 µm and the super-resolution CT at 26.61 µm. At Position B, the upscaled CT reaches a resolution limit of 28.25 µm, and the super-resolution CT a resolution limit of 26.85 µm. The corresponding ESF, MTF, and FRC plots can be found in Figures S4 and S5 in Supplement 1. Comparing these values to the ones obtained for the rose bud sample in Figs. 3 and 5 shows slightly higher values for the cricket scans. This might be caused by the precision of alignment and tilt correction during the reconstruction, or the sample might have moved slightly during the acquisition. All measured resolutions can be found in 2.
Due to the cone beam geometry, upscaling is expected to perform better close to the optical axis. Thus, farther away from the optical axis, more artefacts are expected. Comparing slices from four different reconstructions that have been resliced orthogonal to the beam direction are used to illustrate this in Fig. 9. The optical axis is just below the cricket’s head. Four different reconstructions are shown using data from the cricket at Position B. Figures 9(a)–9(d) show a regular CT, upscaled CT, super-resolution CT, and super-resolution CT created using $2 \times 2$ images per projections and 270 projections in total matching the total exposure time of the regular and upscaled CT, respectively. While the limited super-resolution reconstruction [Fig. 9(d)] shows significant artefacts due to the limited number of projections, features are still comparable to the upscaled or even the super-resolution scan. Further, particularly features in the lower part of the cricket appear visually clearer than in the upscaled scan [Figs. 9(b)]. Zooming in on particular features confirms this assessment [Figs. 9(e)–9(i)].
Table 2. Measured Resolutions of the Rose Bud and Cricket Samples at Both Positions
4. CONCLUSION
We have demonstrated and compared two methods to improve the resolution in x-ray micro-tomography. While upscaling can significantly improve the resolution of a reconstructed volume given sufficient projections, super-resolution will typically perform better at the cost of longer accumulated exposure time. Further, both methods outperform interpolation before or after reconstruction. Interpolation before reconstruction is expected to always perform better than interpolation of reconstructed data due to having more projections than pixel columns in the data. Individual acquisitions are not limited by noise, and the total exposure time can be neglected as a factor. Thus, increasing the exposure time would only yield a very minor improvement of the image quality. Increasing the number of angles is expected to also increase the resolution when fewer projections than optimal are acquired, according to the sampling theorem. Both methods can resolve features that are not visible in a regular CT, but upscaling will also amplify artefacts in the images. Further, the sample position was optimized to minimize penumbral blurring from the source by reducing geometrical magnification, which is then compensated via upscaling or super-resolution. Analyzing reconstructed slices at low magnification showed that the resolution was not limited by the source as observed for scans at high magnification. Finally, we have shown that, at both sample positions, super-resolution can improve the resolution up to the theoretical limit given by the size of the x-ray spot.
To facilitate a meaningful estimation of resolution of a full reconstructed slice, FRC was used. Estimations via ESF and MTF are limited since no standard sample has been used, and they are generally more common in 2D imaging. Further, obtained values depend on the selected feature and the position of the feature in the volume. Using photon counting detectors greatly limits the amount of data points as well, which can be seen in the regular CTs at both positions where the selected feature usually contains only a single data point on the edge. Thus, the resulting FWHM will be a single pixel.
As demonstrated by the rose bud scans with the Eiger detector in Figs. 2 and 4, the total accumulated exposure time of a scan has only a very minor effect on the reconstructions given the individual images have sufficient photon statistics. In comparison, the scan using the Lambda detector in Fig. 6 shows this limitation. The individual projections are limited by noise through their exposure time and, thus, significantly improve the quality of the reconstruction using super-resolution. Even though the image quality is affected, the resolution could be improved in both cases. Consequently, a possible improvement of super-resolution scan time could be to reduce the exposure time of individual images compensating for the much larger amount of images being acquired.
A disadvantage of the super-resolution approach is that the resolution can be limited by the method used to combine the different images. In the approach presented in this paper, there are two contributing factors: precision of image cross correlation and stability of the x-ray spot. Particularly lower contrast images can be more difficult to register precisely. In the presented approach, only a single projection is registered and, thus, drifts or shape and size changes of the x-ray spot can affect the images. Alternatively, every projection can be registered individually at the cost of increased computing time. During the experiments presented in this paper, the x-ray spot has been found to be sufficiently stable to not affect the images. However, other environmental effects can still occur, such as thermal drifts and vibrations, which can cause the sample to move during acquisition.
The resolution of a reconstructed volume largely depends on the alignment and tilt correction of the rotation axis. Thus, precise alignment and corrections are major aspects during the reconstruction step. Further, the selected filter for the FBP algorithm can have a significant effect on the reconstruction. In this paper, a Ram–Lak filter has been used for all samples. Other processing steps performed at this stage, such as ring filtering, will affect image quality and resolution.
Remaining artefacts in the reconstructed slices can be Gibb’s phenomenon, rings around objects, caused by FBP. These artefacts can be reduced by changing the reconstruction filter, which might then also affect the resolution. Similar, phase fringes can be seen in the reconstructed slices presented in this paper, particularly in the super-resolution slices. This effect shows up predominantly as edge enhancement and can be exploited to improve contrast at the cost of resolution by applying phase-retrieval. However, the focus of this paper is to compare resolution; thus, phase contrast is not discussed in detail.
Funding
Carl Tryggers Stiftelse för Vetenskaplig Forskning (CTS17:45); Stiftelsen för Strategisk Forskning (ID17-0097); Vetenskapsrådet (2014-06054).
Acknowledgment
The authors acknowledge X-spectrum GmbH for providing a Lambda 350K GaAs detector and for their support setting it up.
Disclosures
Till Dreier, Excillum AB (E); Ulf Lundström, Excillum AB (E). The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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