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Abstract
Hollow glass microcapillaries or x-ray waveguides very efficiently confine x-rays to submicron or nanospots, which can be used for point projection imaging. However, x-ray beams exiting from such devices have ultranarrow cones that are limited by the critical angle for the total external reflection to a few milliradians. Narrow cone beams result in small fields of view, and the application of multiple-reflection optics to cone beam tomography is challenging. In this work, we describe a new nonconventional tomographic geometry realized with multiple confocal ultranarrow cone beams. The geometry enables an increase in the effective radiation cone to over 10° without resolution reduction. The proposed tomographic scans can be performed without truncations of the field of view or limitations of the angular range and do not require sample translations, which are inherent to other multibeam x-ray techniques. Volumetric imaging is possible with a simultaneous iterative reconstruction technique or with a fast approximate noniterative two-step approach. A proof-of-principle experiment was performed in the multipoint projection geometry with polycapillary optics and a multi-pinhole mask inserted upstream of the optics. The geometry is suited for phase-contrast tomography with polychromatic laboratory and synchrotron sources.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Cone beam x-ray tomography is an important 3D imaging geometry implemented in medicine and industry [1]. This technique is also routinely used in x-ray laboratory micro-[2] and nanotomography [3,4] scanners. The spatial resolution of a cone beam tomographic system depends mainly on the size of the point-like source that generates the x-ray projections. To achieve a high geometrical magnification, the source-to-sample distance is kept very small [Fig. 1(a)] and to obtain a large field-of-field (FOV) wide cone beams are used.
Fig. 1. Principle of x-ray tomography with multiple ultranarrow cone beams. (a) A phantom and its calculated conventional cone beam projection. (b) Ultranarrow x-ray cone beam projection calculated for a single capillary fiber. The projection is formed relative to the tip of the capillary ($s$). The FOV is truncated. (c) The cone beam projection calculated for the polycapillary optics. The projection is formed relative to the optics focal spot ($S$). The FOV is larger, but the resolution is low. (d) Multipoint projection with ultranarrow beams calculated for the polycapillary optics and a multi-pinhole inserted upstream of the optics. Projection splits into a set of multiple ultranarrow cone beam projections generated by a set of sources $\{s_{i}\}$ and sharpens. The FOV is large, and the spatial resolution is high. The images on the left show the geometry and the images on the right show the calculated x-ray projections.
The generation of wide cone x-ray beams with small primary sources produced by electrons in x-ray tubes is straightforward. However, there are fundamental limitations in the size of the x-ray sources. The thermal damage limits the power load to approximately 1 W/$\mu$m. Therefore, the finest spots of x-ray nanotubes can reach at best sizes of ${\sim }0.3$ $\mu$m at a power of ${\sim }0.2$ Watts [4].
Secondary submicron or nano x-ray sources for point projection can be generated using multi-bounce x-ray optical elements such as hollow glass capillaries [5] or x-ray waveguides [6]. Single capillaries were shown to produce spots smaller than 100 nm [7], and x-ray waveguides are capable of providing sub-10 nanometer confinement of the x-rays [8]. However, applications of multi-bounce x-ray optics to tomography are rare. While holo-tomographic synchrotron setups with waveguides exist [9], tomographic cone beam setups based on capillary optics have not been reported. The main reason is the fundamental limit set by the critical angle $\theta _{c}$ for the total external reflection of the x-rays [10]. Cone beams generated by single capillaries or waveguides have openings that are on the order of $2\theta _c$. For example, for glass or silicon, at a photon energy of 10 keV, the x-ray cone has a divergence of a few milliradians or a few tenths of a degree. Therefore the FOV is truncated, as shown in Fig. 1(b).
The cone angle can be slightly increased using highly tapered optics [11] or high-Z materials in Kirkpatrick-Baez mirrors (KB) [12,13] or multilayer KB optics, which can be used for submicron or nanoscale zoom tomography [14,15] or full-field x-ray microscopy [16]. The FOV in magnifying cone-beam tomographic systems based on reflective optics can be also increased by placing the sample further from the source. However, this requires the positioning of the detectors at a very large distance $D$ of several meters, which can significantly lower the geometrical magnification. In addition, such a geometry is incompatible with hybrid pixel detectors, which have large pixels but can facilitate multi-energy experiments for quantitative phase retrieval [17] or hyperspectral imaging (color-CT) for material discrimination [18].
A dramatic increase in the radiation cone can be achieved using concentrating polycapillary optics [19,20]. A polycapillary optics is a structured device that is composed of hundreds of thousands of bent microcapillaries that are “confocal” or, in other words, point toward a common focal point. Individual capillaries can have submicron inner channels and produce beams with a divergence of ${{\sim }} 2\theta _c$. Hence, a focal spot of a polycapillary optics has lateral dimensions of ${\sim } 2f\theta _c$, where $f$ is the optics focal distance. Most importantly, the radiation cone that is generated by polycapillary optics can be greater than 10$^{\circ }$ and is over one order of magnitude wider than the cones generated by single capillaries. However, commercially manufactured polycapillary devices are fabricated to maximize the intensity at the focal spot and are composed of densely packed periodic arrays of capillaries. Therefore, these devices are not suited for high-resolution x-ray imaging. Due to the finite divergence of each beam, signals from individual capillaries are just smeared out [21]. The dense packing of capillaries causes the image formed at the detector to be a cone beam projection relative to the focal spot [22,23], and the spatial resolution is limited (typically to over 10 $\mu$m) [24], as shown in Fig. 1(c).
Recently, in our laboratory, it was shown that individual capillaries in polycapillary optics can be used for plenoptic x-ray imaging [25] in the multipoint projection geometry [26]. Over one hundred multiview projections of small volumes located in the focal spot region were recorded in a single exposure, and slices at various depths near the focal plane were reconstructed in a way similar to tomosynthesis [27,28]. Plenoptic x-ray microscopy uses the inside-focal spot geometry that was initially introduced for coded-aperture imaging [22,29]. Therefore, this method has an FOV comparable to that of a single capillary. In addition, the limited angular range of x-ray plenoptic microscopy makes the depth resolution approximately 10 times lower than the lateral resolution as in laminography or tomosynthesis [27,28].
In this paper, we demonstrate that multiple confocal ultranarrow x-ray beams can be used for computed tomography for 3D imaging with isotropic spatial resolution. As demonstrated in Fig. 1(d), this novel tomographic geometry combines the high resolution provided by single capillary fibers with a large FOV characteristic of polycapillary optics. Second-generation roto-translational computed tomography scanners used “narrow” beams with fan beam angles of approx. $3^{\circ }-10^{\circ }$ [1]. In this paper, the cones of beamlets used for 3D imaging are over one order of magnitude narrower (${\sim } 0.3^{\circ }$). Hence, throughout this paper we use the term “ultranarrow”. The use of narrow cone beams in tomography is not rare but has always been accompanied by undersampling. For example, various kinds of multisource or multibeam systems [26,30–32] for single-shot experiments have either very limited angular ranges or are suited only for imaging small subvolumes of samples. Other multibeam methods based on beam shaping with gratings [33,34] or arrays of pinholes [35] result in a strong truncation in the spatial domain and require time-consuming roto-translational scans. The proposed concept of tomography with multiple ultranarrow cone beams eliminates all these problems by using a “confocal" geometry from Fig. 1(d), in which all beams are focused upstream of the sample without the necessity to undersample the data.
2. Geometry and reconstruction
2.1 Definition of the tomographic geometry
Proposed tomographic geometry is non-conventional. Nevertheless, it is tractable with a highly flexible ASTRA Toolbox for tomography [36,37], which is extensively used in this paper. The geometry of tomography with ultranarrow cone beams is defined in Fig. 2, which was prepared with the astra_plot_geometry function from the ASTRA Toolbox.
Fig. 2. Definition of the tomographic geometry with multiple ultranarrow cone beams. For clarity, the sample-to-detector distance $D$ and the detector pixel size were downscaled by factors of 20 and 2, respectively. $\{ s_{i} \}$ is a set of sources that generate ultranarrow beams, $\{\delta _i\}$ is a set of subdetectors that detect ultranarrow beams, $\omega _{j}$ is the $j$-th orientation of the sample and $\mathbf {p}_{ij}$ represents ultranarrow projections.
In Fig. 2, a set of sources $\{ s_i\}$, generates ultranarrow cone beams that are confocal or that illuminate a common focal point at $f$. In simulations, for clarity, we used a small number of sources with $N_{s}=87$, whereas, in the experiment, the number of sources is much higher, i.e., $N_{s}=630$. Each beam has a cone with an angle of $\gamma {\sim } 2\theta _c$. The sources have lateral coordinates $\mathbf {r}^s_i$ and a pitch of $a_s$, which is approximately equal to the lateral dimension of the focal spot $w\approx \gamma f$. The sample is located out-of-focal plane. The origin is placed at the sample position, and the sample-to-source distance is $d=f+\Delta z$. The beams are sampled by a set $\{\delta _i\}$ of small subdetectors (or just detectors in ASTRA terminology). In the experiment, all subdetectors are in fact small areas with $N_{p}\times N_{p}$ pixels (here, $N_p=19$) of a single large position sensitive detector. To mimic the experiment, a circular mask is applied to each sub-detector, so only pixels inside the circle inscribed in the detector are taken into account. The sample-to-detector distance is $D$. The position of each detector in the lateral plane is $\mathbf {r}^{D}_i=-M\mathbf {r}^{s}_i$, where $M=(D+\Delta z)/f$. The magnification of ultranarrow projections is $m=1+D/d$. The global cone of the illumination $\alpha$ is a superposition of all ultranarrow beams. As shown in Fig. 2, the narrow beams are fully multiplexed at the focal spot, but there is only small beam overlap at the position of the sample. This small overlap of the beams at the sample position is a prerequisite to avoid undersampling. At the detector plane, as shown in Fig. 1(d), the gaps between individual beams are present. In fact, any overlap of the beams at the detector plane is an unwanted effect, which should be minimized, to avoid artefacts in the reconstructed volume.
Tomography requires that during a scan each sample voxel is illuminated from all directions. The set of sources $\{ s_{i}\}$ simultaneously illuminates the sample within a relatively wide cone $\alpha$. However, almost every voxel is illuminated from a different angle. Therefore, for collection of the full data set, the sample is rotated around the $\omega$ axis. Individual sample orientations are denoted as $\omega _j$. Note that to avoid undersampling one has to use a rotation step of $\gamma >\Delta \omega \ll \alpha$. The required rotation step can be estimated by calculating the so-called effective aperture for the $\Delta z$ defocusing distance [25].
Hence, in the tomographic scan, one records ultranarrow projections $\mathbf {p}_{ij}$. Each $\mathbf {p}_{ij}$ is a column vector of length $N_p^2$ that represents projections from the $i$-th subdetector and for the $j$-th sample orientation. In ASTRA Toolbox terminology, the tomographic data set consists of $N=N_{s}\times N_{\omega }$ “angles”. The number $N$ can be very high (for simulations $N\approx 3\times 10^{4}$, for experiment $N=1.1 \times 10^{5}$). The tomographic forward projection is defined as [36]:
whereThe compound image from Fig. 1(d) built-up from multiple ultranarrow cone beam projections was calculated using geometry from Fig. 2 via Eq. (1) and for parameters that reflect the experimental geometry reported in next sections, namely, for $f=2.5$ mm, $\Delta z=1.9$ mm, $D=240$ mm, source pitch $\Delta a_s=17$ $\mu$m and the detector pixel size of $75$ $\mu$m.
2.2 Tomographic reconstruction
For the reconstruction of our nonconventional ultranarrow cone beam projection data, a general technique, i.e, the simultaneous iterative reconstruction technique (SIRT), was tested. SIRT alternates forward and back projections, and the sample volume is restored iteratively, according to [36]:
In an experiment, SIRT reconstruction may be dominated by noise. In addition, the experimental data need to be corrected for misalignment, frequently in a trial-and-error approach. Since the SIRT is slowly convergent, this may result in considerably long processing times. Moreover, the experimental data may require a phase retrieval procedure [38–40], which is not straightforward to perform directly on ultranarrow data. Post-reconstruction phase retrieval [41] or going beyond the projection approximation [42] could be necessary in the future for quantitative imaging. However, for the purpose of the present proof-of-principle experiment, we propose a fast approximate two-step (FATS) reconstruction.
In the first step, $N_s$ ultranarrow cone beam projections $\mathbf {p}_j=[\mathbf {p}_{1j};\ldots ;\mathbf {p}_{N_{s}j}]$ for a given sample orientation $\omega _j$ are simply back-projected according to $\mathbf {W}_{j}^{T}\mathbf {p_{j}}$, where ${W}_{j}$ is a projection matrix that is limited to projections for a single $\omega _j$, in the geometry from Fig. 2. Next, the geometry is changed; i.e., the reconstructed volume is forward projected using a standard cone beam geometry, with a virtual source at the focal point to obtain a single standard cone beam projection $\widetilde {\mathbf {p}}_j$:
In the second step, $N_\omega$ cone beam projections $\widetilde {\mathbf {p}}=[\widetilde {\mathbf {p}}_{1};\ldots ;\widetilde {\mathbf {p}}_{N_{\omega }}]$ can be filtered and weighted, and phase retrieval [43] can be optionally performed. This operation is symbolically denoted as $\mathbf {F}$. Finally, filtered wide cone-beam projections are again back-projected to yield the reconstructed sample volume: Importantly, Eq. (5) can be implemented using a standard Feldkamp, Davis and Kress (FDK) algorithm [44].Figure 3 compares various reconstructions of the phantom from Fig. 1. The phantom ($256\times 256\times 128$ voxels) was calculated with TomoPhantom software [45]. Tomographic computations were performed using CUDA procedures from ASTRA Toolbox 1.9.0 in MATLAB R2019a using a 4GB GPU. The SIRT reconstruction used data from $N=87\times 360$ ultranarrow cone beam projections that had a size of $19\times 19$ pixels. In the two-step reconstruction, 87 ultranarrow beam projections for each sample orientation are first transformed into standard 256$\times$128 pixel cone beam projections (relative to the focal spot of the optics) and subsequently were reconstructed using a standard FDK cone beam algorithm.
Fig. 3. Tomographic reconstruction of noiseless multiple ultranarrow cone beam projections of the phantom from Fig. 1. (a) Axial slice of the phantom. (b) Back-projection (BP). (c) Simultaneous iterative reconstruction technique (SIRT) from $N_s\times N_{\omega }=31320$ ultranarrow (19$\times$19 pixels, $\gamma \approx 0.3^{\circ }$) cone beam projections. (d) Fast approximate two-stage (FATS) reconstruction. Top row: comparison of the axial slices. Bottom: zoomed areas. (e) Comparison of the one-dimensional profiles. BP is scaled and offset.
Figure 3(b) shows the results of a single back-projection (the first SIRT iteration). As shown in Fig. 3(c), after 2500 iterations the phantom is properly reconstructed. However, a slight “overfitting” is present, which in the presence of noise can hinder proper convergence. The result of FATS reconstruction is shown in Fig. 3(d), and the comparison with the SIRT reconstruction is shown in Fig. 3(e). The main effect of the approximation used is a slight resolution deterioration. However, at least in the present work, this resolution deterioration is less significant than the expected experimental distortions that are due to geometry misalignment and thermal drifts. Generally, the applied approximation works better for $D\gg d$, a sample displaced from the focal plane by $\Delta z>w/\alpha$ and not too large opening angles $\alpha$. For reconstruction of noisy data see Appendix A.
3. Experiment
3.1 Experimental setup
For a proof-of-principle experiment, we adopted the experimental system that is described in Refs. [25,26]. Radiation from a Cu anode x-ray tube (50kV, 1mA, focal spot of ${\sim } 40$ $\mu$m) was collimated by polycapillary optics to a beam with a ${\sim } 8$ mm diameter and a total photon flux of ${\sim } 4\times 10^8$. The crucial part of the experimental system that is located $0.5$ m from the first polycapillary optics is shown in Fig. 4(a).
Fig. 4. Experimental details of x-ray tomography with multiple ultranarrow cone beams. (a) Fragment of the experimental setup. The asterisk marks the position of the sample during the tomographic scan at $\Delta z=1.9$ mm. The short dashed line marks the focal plane at $f=2.5$ mm. During experiments, the multi-pinhole mask is moved to an almost in-contact position with the input surface of the optics. (b) Test sample: borosilicate glass capillary filled with SiO$_{2}$ spheres. Scale bar: 100 $\mu$m. (c) Microscope image of the exit surface of the optics. The red-filled area marks the approximate distribution of sources $\{s_{i}\}$ that generate ultranarrow x-ray cone beams, which are captured by the detector. (d) Experimentally determined distribution of sources $\{s_{i}\}$. The number of sources that generate ultranarrow cone beams is $N_{s}=630$.
The central element in Fig. 4(a) is a second polycapillary optics with a taper ratio of ${\sim }$5 that concentrates x-rays to a focal spot with a FWHM of $w\approx 12$ $\mu$m at the focal distance of $f=2.5$ mm. The optics was composed of $~{\sim } 3\times 10^5$ capillaries, with inner channel diameters of ${\sim } 3.5$ $\mu$m and ${\sim } 0.7$ $\mu$m at the input and exit surfaces, respectively. To generate ultranarrow cone beams a multi-pinhole mask (tungsten foil with thickness of $50$ $\mu$m) was inserted upstream of the optics. The mask was placed as close as possible to the optics; it consist of a large set of a nearly hexagonal array of $7.5$ $\mu$m pinholes with spacing of $75$ $\mu$m. All optical elements and the sample can be translated and aligned using multiaxis piezo or stepper-motor stages. Originally, the setup was designed for single-exposure plenoptic imaging in the multipoint projection geometry and was not dedicated for sample rotation. To facilitate sample rotation, a miniature 6 mm diameter 18${^\circ }$ stepper motor with a 256:1 gearhead (Faulhaber DM0620) was placed on a 3D printed sample holder. A highly absorbing object (E symbol ion-beam milled in a golden foil) was also placed on the holder to facilitate the determination of ultranarrow beam distribution. In the experiment, the rotated object was placed at a distance $\Delta z=1.9$ mm from the focal plane. A hybrid pixel detector (Dectris Eiger2 R 500k with pixel pitch of 75 $\mu$m) was used for photon detection. The total flux measured by the detector was $1.4\times 10^{7}$ photons/s without the multi-pinhole mask and $9\times 10^{4}$ photons/s with the mask. The x-ray source used in our setup has a power that is too low to demonstrate all capabilities of ultranarrow tomography simultaneously. The present setup is capable of providing a spatial lateral resolution of $0.4$ $\mu$m (half-picth) [25], which is limited only by the inner diameter of the capillaries. In this experiment, to maximize the count rates and to capture a wide cone beam from the polycapillary, we put the detector relatively close to the sample ($D=240$ mm). This short distance limits the spatial resolution to approximately 1.5 $\mu$m, as determined with a JIMA RT RC-02B resolution chart (see Appendix B).
As a phantom, we used a thin-walled borosilicate glass capillary (Capillary Tube Supplies Ltd, density ${\sim } 2.2$ g/cm$^3$) filled with 25 $\mu$m diameter SiO$_2$ spheres (microParticles GmbH, density ${\sim } 1.8$ g/cm$^3$). An optical microscope image of the filled capillary after drying the spheres in solution is shown in Fig. 4(b). X-ray projections were taken at the tapered end of the capillary, where the outer diameter of the capillary is ${\sim } 210$ $\mu$m and the thickness of the wall is ${\sim } 19$ $\mu$m.
The positions of capillary tips that shine ultranarrow beams were determined using the method described in Ref. [25]. The determined distribution of sources $\{s_i\}$ is shown in Figs. 4(c) and (d). The relative positions of individual sources were determined with sub-pixel precision; however, the absolute position in the microscope image from Fig. 4(c) is only approximate.
3.2 Experimental results
Figure 5 shows x-ray images of the borosilicate glass capillary filled with SiO$_2$ spheres, recorded for $\omega =0$. The x-ray projection from Fig. 5(a) was recorded without the multi-pinhole mask. In this case, as shown in Fig. 1(c), the image is formed relative to the focal spot of the optics. Projections of spheres and capillary walls are smeared due to poor resolution that is limited by the focal spot size to approximately ${\sim } 7$ $\mu$m (half-pitch), as checked with the JIMA chart (see Appendix B).
Fig. 5. Experimental data for a phantom: borosilicate glass capillary filled with 25 $\mu$m diameter SiO$_2$ spheres ($\omega =0$). (a) Cone beam projection relative to the focal spot of the polycapillary optics. (b) Image recorded with polycapillary optics with a multi-pinhole mask inserted upstream of the optics. This image splits into $\mathbf {p}_{ij}$ ($N_{s}=630$) ultranarrow cone beam projections. (c) Conventional cone beam projection $\widetilde {\mathbf {p}}_{j}$ reconstructed from image (b). (d) Sample thickness $\mathbf {F}\widetilde {\mathbf {p}}_{j}$ retrieved with a Paganin filter. Insets show zoomed and contrasted fragments marked with rectangles. The second zoom in (b) shows a numerical mask that was used to eliminate the partial overlap of neighboring ultranarrow cone beams to construct a set of $\mathbf {p}_{ij}$ ultranarrow cone beam projections. Scale bars: 25 $\mu$m.
After insertion of the multi-pinhole upstream of the polycapillary optics, the x-ray image splits into a set of $N_{s}=630$ ultranarrow high-resolution projections, as shown in Fig. 5(b). Each beamlet has an opening angle of $\gamma {\sim } 0.3^{\circ }$ and the angular cone of the image is $\alpha {\sim } 10^{\circ }$. The FOV at the position of the sample is greater than 300 $\mu$m. Note that FOV of a single ultranarrow cone beam projection has an FOV as small as ${\sim }30$ $\mu$m.
To reduce the partial overlap of the beamlets, a numerical mask was applied [second inset in Fig. 5(b)]. However, no undersampling of the sample volume is present since projections of the same sample voxels are multiplexed among neighboring beamlets.
The set of ultranarrow projections was transformed via Eq. (4) to a conventional cone beam projection from the viewpoint of the focal spot [Fig. 5(c)]. The projections of capillary walls and spheres are clearly resolved. Moreover, the phase contrast is manifested as edge enhancement. The remnant effect of partial beam overlap at the detector is visible in Fig. 5(c). A kind of ghost images of the capillary walls are marked with arrows. At the expense of signal-to-noise ratio degradation, this effect could be eliminated by restricting the numerical mask to smaller regions or using a multi-pinhole mask with slightly larger pinhole spacing. Note that to avoid undersampling, a larger pinhole spacing requires the sample to be positioned at a smaller $\Delta z$, which decreases FOV.
For thickness retrieval, we used a single-material approximation (Paganin filter [43]) with $\delta =7.2\times 10^{-6}$, $\beta =6.9\times 10^{-8}$ and an energy of $E=8.05$ keV [Fig. 5(d)].
The explanatory x-ray image from Fig. 5(b) was acquired in a very long 3600 s exposure. In tomographic scans, we recorded images with an acquisition time of 180 s and with a rotation step of $\Delta \omega = 1^{\circ }$. The data acquisition in the $180^{\circ }$ angular range lasted 9 hours. The scan itself was longer because we recorded multiple images without the sample for normalization and checked the source distribution during the scan. The full data set is shown in Visualization 1.
The tomographic scan was reconstructed using the FATS reconstruction from Eqns. (4) and (5), and it was corrected for the geometry misalignment. X-ray projections calculated in the first FATS step are shown in Visualization 2. The second reconstruction step from Eq. (5) was performed with an FDK algorithm.
The full 3D ultranarrow multiple cone beam reconstruction of the sample is shown in Fig. 6 and it is compared with a conventional cone beam reconstruction. Conventional data was acquired before the sample was fully dried. Therefore the spatial arrangements of spheres are not identical. During the whole ultranarrow multiple cone beam scan, only $\sim 3\times 10^{9}$ photons irradiated the sample (${\sim }0.15$ photons/s/pixel). Hence, the quality of the reconstruction is mainly limited by noise and misalignment of the sample, which was difficult to correct for our noisy data. Despite the small number of recorded photons, the 3D arrangement of the spheres in the capillary is imaged with a high precision, which was not possible with polycapillary optics. For comparison, in the conventional tomographic scan $\sim 3\times 10^{10}$ photons irradiated the sample and the poor spatial resolution does not enable to resolve adjacent 25 $\mu$m-diameter spheres.
Fig. 6. Reconstructions of tomographic scans recorded for a phantom: a borosilicate glass capillary filled with 25 $\mu$m diameter SiO$_2$ spheres. (a) Slices from conventional cone beam tomography with polycapillary optics. (b) Slices reconstructed with FATS from an ultranarrow multiple cone beam dataset. Scale bar : 50 $\mu$m. (c) 3D view of ultranarrow multiple cone beam reconstruction. (d) Zooms of the regions marked with dashed orange rectangles. (e) Line profiles along white lines marked in (d). Spatial arrangements of spheres slightly differ in (a) and (b).
Quantitative resolution determination is presented in Figs. 6(d) and (e). In Fig. 6(e), the in-slice resolution was estimated by fitting a scaled, translated and offset error function $\textrm {erf}(x/\sigma /\sqrt {2})$ to the line profile of a single sphere [34]. A resolution value, which can be directly compared to the half-pitch resolution discussed in Appendix B was calculated as $2.35\sigma /2\approx 2.3$ $\mu$m. Therefore, the in-slice resolution is slightly lower than the lateral resolution of 1.5 $\mu$m, which is determined in Appendix B. The main reasons are errors in the sample alignment, approximate character of FATS and the application of the Paganin filter.
4. Summary and outlook
In this work, we demonstrated a novel cone beam x-ray tomographic geometry that is suitable for applications with total-reflection optics, in which the intrinsic beam divergence is limited to a few milliradians. It was shown that by means of tomography with multiple ultranarrow beams, it is possible to combine the high spatial resolution provided by single capillaries with a large field of view provided by polycapillary devices. For example, the FOV of an ultranarrow cone beam of a single capillary from Fig. 5(b) is only slightly larger than the diameter of a single glass sphere (${\sim }25$ $\mu$m), whereas the full FOV is larger than 300 $\mu$m. Simultaneously, the spatial resolution of the x-ray image from Fig. 5(b) is almost five times higher than the resolution of the conventional cone beam projection generated by polycapillary optics from Fig. 5(a). Hence, by using ultranarrow cone beam tomography, it is possible to obtain a large FOV at a small source-to-sample distance, at a high geometrical magnification and at a small effective voxel size. The geometry could be extended to arrays of x-ray waveguides.
Another novelty of our approach is that a large FOV is achieved without any undersampling and without the necessity of an additional translation [35] and performing time-consuming roto-translational scans [34]. Contrary to other multisource techniques, the “confocal” geometry of the ultranarrow beams eliminates truncation effects [30]. The present geometry is also different from the plenoptic geometry from our recent paper [25]. In the plenoptic geometry, for depth-resolved imaging, the sample is placed in the focal spot of the optics, and all beams illuminate nearly the same small region of the sample from different viewpoints. In multiple ultranarrow cone beam tomography, 3D information is obtained for a large sample that is placed out of focus. Multiple beamlets also illuminate the sample from different angles, but each beamlet illuminates a different sample region. There is only small angular overlap of a few neighboring beams. This overlap improves the signal-to-noise ratio and, most importantly, guarantees the absence of undersampling in the rotational scans. Contrary to the plenoptic approach, the present geometry enables 3D imaging with an isotropic spatial resolution.
The proposed geometry is nonconventional, but we have shown that an open source software, i.e., the ASTRA Toolbox [36,37] can be very efficiently used for reconstruction via the iterative SIRT technique. We have also proposed a fast approximate two-step reconstruction method and benchmarked it against the iterative SIRT reconstruction method.
The present experiment was performed with a low power x-ray source, near the detection limit; i.e., we observed that the extension or elongation of the tomographic scan did not improve the data quality, which for longer scans were obstructed by thermal drifts of the system. Therefore, the full capabilities of tomography with multiple ultranarrow cone beams, e.g., submicron resolution, has not yet been explored and demonstrated. Routine application will definitely require much more powerful laboratory sources [46,47]. For example, taking into account a high power, short focus-to-optics distance and a small spot of a liquid metal jet source [47], a tenfold increase of photon flux can be expected, compared to our setup. At synchrotrons, single glass capillaries withstand the flux of a focused undulator beam [48]. Therefore, tomography with multiple ultranarrow cone beams could also provide a means for imaging large samples at high spatial resolution at insertion device beamlines, where the limited size of a synchrotron beam is a problem [49].
We plan to adopt multiple ultranarrow cone beam tomography for routine experiments at a constructed bending magnet beamline PolyX of the SOLARIS National Synchrotron Radiation Centre [50]. Although SOLARIS is a low-energy (soft x-ray) source, we estimate that the total hard x-ray flux at the input surface of the concentrating polycapillary optics can be increased by a factor >100 with a 2$\%$ bandwidth multilayer monochromator at $E=8$ keV. The use of a filtered or mirror-reflected pink beam will increase the flux by another order of magnitude. This increase will shorten the total scan acquisition times to minutes or seconds. The high flux will also enable the use of polycapillaries with smaller channel diameters. Current technology enables to fabricate polycapillary devices with 300 nm channel diameter at the exit surface, which could provide single radiographs with a half-pitch spatial resolution at the level of 150 nm. Such a resolution is lower than the resolution of synchrotron systems based on KB mirrors or Fresnel zone plates [12–16,51]. However, resolution of tomography with multiple ultranarrow cone beams could be higher than the resolution of typical parallel-beam tomographic experiments at synchrotrons and would enable zooming [14]. Importantly, the multi-pinhole mask can be very quickly inserted and removed from the beam, enabling multi-modal experiments that will combine 3D phase-contrast tomography with x-ray fluorescence imaging and x-ray absorption spectroscopy.
Appendix A: Simulations with noise
Figure 7 shows data simulated for a phantom from Fig. 1(a) with a Poisson noise that roughly corresponds to the noise level of the experimental data (see Visualization 1). For a high noise level, the SIRT reconstruction has very low quality, whereas FATS reconstruction is more robust to noise. For noiseless data SIRT gives higher resolution than FATS (Fig. 3). However, in the presence of noise this advantage of SIRT is lost, because of the need for regularization.
Fig. 7. Simulations with noise. (a) Noisy multipoint projection with ultranarrow beams calculated for the phantom ($\omega =0$). (b) Cone beam projection reconstructed in the first step of FATS. (c) Axial slice of the phantom. (d) SIRT. (e) FATS. Bottom images in (c-e) show zoomed areas. (f) Comparison of the one-dimensional profiles.
Appendix B: Spatial resolution
The spatial resolution in tomography with multiple ultranarrow cone beams is fundamentally limited by the size of secondary sources generated at the tips of individual capillaries. In previous work it was shown that the maximal half-pitch resolution that can be obtained with our polycapillary optics is 0.4-0.5 $\mu$m [25], which was in agreement with the inner capillary diameter of 0.7 $\mu$m measured with SEM [26]. In this work, the detector was placed at a relatively small distance, which reduced magnification and the resolution was limited by the Nyquist limit of the detector to approx. 1.5 $\mu$m, as shown in Fig. 8. For comparison, the half-pitch resolution of convention cone beam tomography (point-projection relative to the focal spot of the optics) is $\sim 7$ $\mu$m, which is in accordance with the FWHM of the focal spot ($\sim 12 \mu$m) [21].
Funding
Narodowe Centrum Nauki (DEC-2017/25/B/ST2/00152); Ministerstwo Nauki i Szkolnictwa Wyższego (2019-N17/MNS/000044).
Disclosures
The authors declare no conflicts of interest.
References
1. T. M. Buzug, Computed Tomography. From Photon Statistics to Modern Cone-Beam CT (Springer, 2008).
2. K. Machin and S. Webb, “Cone-beam x-ray microtomography of small specimens,” Phys. Med. Biol. 39(10), 1639–1657 (1994). [CrossRef]
3. M. Müller, I. de Sena Oliveira, S. Allner, S. Ferstl, P. Bidola, K. Mechlem, A. Fehringer, L. Hehn, M. Dierolf, K. Achterhold, B. Gleich, J. U. Hammel, H. Jahn, G. Mayer, and F. Pfeiffer, “Myoanatomy of the velvet worm leg revealed by laboratory-based nanofocus x-ray source tomography,” Proc. Natl. Acad. Sci. 114(47), 12378–12383 (2017). [CrossRef]
4. M. Eckermann, M. Töpperwien, A.-L. Robisch, F. van der Meer, C. Stadelmann, and T. Salditt, “Phase-contrast x-ray tomography of neuronal tissue at laboratory sources with submicron resolution,” J. Med. Imaging 7(1), 013502 (2020). [CrossRef]
5. D. Mosher and S. J. Stephanakis, “X-ray light pipes,” Appl. Phys. Lett. 29(2), 105–107 (1976). [CrossRef]
6. F. Pfeiffer, C. David, M. Burghammer, C. Riekel, and T. Salditt, “Two-dimensional x-ray waveguides and point sources,” Science 297(5579), 230–234 (2002). [CrossRef]
7. D. H. Bilderback, S. A. Hoffman, and D. J. Thiel, “Nanometer spatial resolution achieved in hard x-ray imaging and laue diffraction experiments,” Science 263(5144), 201–203 (1994). [CrossRef]
8. S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat. 19(2), 227–236 (2012). [CrossRef]
9. A. Ruhlandt and T. Salditt, “Time-resolved x-ray phase-contrast tomography of sedimenting micro-spheres,” New J. Phys. 21(4), 043017 (2019). [CrossRef]
10. D. Paganin, Coherent x-ray optics (Oxford University, 2006).
11. I. Lindgren and E. Selin, “A simple model for optimization of conical capillaries in XRF analysis,” X-Ray Spectrom. 22(4), 216–219 (1993). [CrossRef]
12. C. Rau, V. Crecea, W. Liu, C.-P. Richter, K. Peterson, P. Jemian, U. Neuhäusler, G. Schneider, X. Yu, P. Braun, T.-C. Chiang, and I. Robinson, “Synchrotron-based imaging and tomography with hard x-rays,” Nucl. Instrum. Methods Phys. Res., Sect. B 261(1-2), 850–854 (2007). [CrossRef]
13. S. Matsuyama, S. Yasuda, J. Yamada, H. Okada, Y. Kohmura, M. Yabashi, T. Ishikawa, and K. Yamauchi, “50-nm-resolution full-field x-ray microscope without chromatic aberration using total-reflection imaging mirrors,” Sci. Rep. 7(1), 46358 (2017). [CrossRef]
14. R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffiere, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90(14), 144104 (2007). [CrossRef]
15. G. Requena, P. Cloetens, W. Altendorfer, C. Poletti, D. Tolnai, F. Warchomicka, and H. Degischer, “Sub-micrometer synchrotron tomography of multiphase metals using Kirkpatrick–Baez optics,” Scr. Mater. 61(7), 760–763 (2009). [CrossRef]
16. J. Yamada, S. Matsuyama, R. Hirose, Y. Takeda, Y. Kohmura, M. Yabashi, K. Omote, T. Ishikawa, and K. Yamauchi, “Compact full-field hard x-ray microscope based on advanced Kirkpatrick–Baez mirrors,” Optica 7(4), 367–370 (2020). [CrossRef]
17. S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, and A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207(2), 79–96 (2002). [CrossRef]
18. C. K. Egan, S. D. M. Jacques, M. D. Wilson, M. C. Veale, P. Seller, A. M. Beale, R. A. D. Pattrick, P. J. Withers, and R. J. Cernik, “3D chemical imaging in the laboratory by hyperspectral x-ray computed tomography,” Sci. Rep. 5(1), 15979 (2015). [CrossRef]
19. M. Kumakhov and F. Komarov, “Multiple reflection from surface x-ray optics,” Phys. Rep. 191(5), 289–350 (1990). [CrossRef]
20. C. A. MacDonald, “Structured X-Ray Optics for Laboratory-Based Materials Analysis,” Annu. Rev. Mater. Res. 47(1), 115–134 (2017). [CrossRef]
21. P. Korecki, K. M. Sowa, B. R. Jany, and F. Krok, “Defect-assisted hard-x-ray microscopy with capillary optics,” Phys. Rev. Lett. 116(23), 233902 (2016). [CrossRef]
22. K. M. Dabrowski, D. T. Dul, and P. Korecki, “X-ray imaging inside the focal spot of polycapillary optics using the coded aperture concept,” Opt. Express 21(3), 2920–2927 (2013). [CrossRef]
23. P. Korecki, T. P. Roszczynialski, and K. M. Sowa, “Simulation of image formation in x-ray coded aperture microscopy with polycapillary optics,” Opt. Express 23(7), 8749–8761 (2015). [CrossRef]
24. T. Sun, M. Zhang, Z. Liu, Z. Zhang, G. Li, Y. Ma, X. Du, Q. Jia, Y. Chen, Q. Yuan, W. Huang, P. Zhu, and X. Ding, “Focusing synchrotron radiation using a polycapillary half-focusing x-ray lens for imaging,” J. Synchrotron Radiat. 16(1), 116–118 (2009). [CrossRef]
25. K. M. Sowa, M. P. Kujda, and P. Korecki, “Plenoptic x-ray microscopy,” Appl. Phys. Lett. 116(1), 014103 (2020). [CrossRef]
26. K. M. Sowa, B. R. Jany, and P. Korecki, “Multipoint-projection x-ray microscopy,” Optica 5(5), 577–582 (2018). [CrossRef]
27. S. Gondrom, J. Zhou, M. Maisl, H. Reiter, M. Kröning, and W. Arnold, “X-ray computed laminography: An approach of computed tomography for applications with limited access,” Nucl. Eng. Des. 190(1-2), 141–147 (1999). [CrossRef]
28. J. T. Dobbins, “Tomosynthesis imaging: at a translational crossroads,” Med. Phys. 36(6Part1), 1956–1967 (2009). [CrossRef]
29. K. M. Dabrowski, D. T. Dul, A. Wrobel, and P. Korecki, “X-ray microlaminography with polycapillary optics,” Appl. Phys. Lett. 102(22), 224104 (2013). [CrossRef]
30. G. Wang, H. Yu, and Y. Ye, “A scheme for multisource interior tomography,” Med. Phys. 36(8), 3575–3581 (2009). [CrossRef]
31. P. Villanueva-Perez, B. Pedrini, R. Mokso, P. Vagovic, V. A. Guzenko, S. J. Leake, P. R. Willmott, P. Oberta, C. David, H. N. Chapman, and M. Stampanoni, “Hard x-ray multi-projection imaging for single-shot approaches,” Optica 5(12), 1521–1524 (2018). [CrossRef]
32. W. Voegeli, K. Kajiwara, H. Kudo, T. Shirasawa, X. Liang, and W. Yashiro, “Multibeam x-ray optical system for high-speed tomography,” Optica 7(5), 514–517 (2020). [CrossRef]
33. C. K. Hagen, P. R. T. Munro, M. Endrizzi, P. C. Diemoz, and A. Olivo, “Low-dose phase contrast tomography with conventional x-ray sources,” Med. Phys. 41(7), 070701 (2014). [CrossRef]
34. C. K. Hagen, F. A. Vittoria, M. Endrizzi, and A. Olivo, “Theoretical framework for spatial resolution in edge-illumination x-ray tomography,” Phys. Rev. Appl. 10(5), 054050 (2018). [CrossRef]
35. E. S. Dreier, C. Silvestre, J. Kehres, D. Turecek, M. Khalil, J. H. Hemmingsen, O. Hansen, J. Jakubek, R. Feidenhans’l, and U. L. Olsen, “Single-shot, omni-directional x-ray scattering imaging with a laboratory source and single-photon localization,” Opt. Lett. 45(4), 1021–1024 (2020). [CrossRef]
36. W. van Aarle, W. J. Palenstijn, J. D. Beenhouwer, T. Altantzis, S. Bals, K. J. Batenburg, and J. Sijbers, “The ASTRA Toolbox: A platform for advanced algorithm development in electron tomography,” Ultramicroscopy 157, 35–47 (2015). [CrossRef]
37. W. van Aarle, W. J. Palenstijn, J. Cant, E. Janssens, F. Bleichrodt, A. Dabravolski, J. D. Beenhouwer, K. J. Batenburg, and J. Sijbers, “Fast and flexible x-ray tomography using the ASTRA toolbox,” Opt. Express 24(22), 25129–25147 (2016). [CrossRef]
38. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995). [CrossRef]
39. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29(1), 133–146 (1996). [CrossRef]
40. T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy x rays,” Phys. Rev. Lett. 86(25), 5827–5830 (2001). [CrossRef]
41. D. A. Thompson, Y. I. Nesterets, K. M. Pavlov, and T. E. Gureyev, “Fast three-dimensional phase retrieval in propagation-based X-ray tomography,” J. Synchrotron Radiat. 26(3), 825–838 (2019). [CrossRef]
42. M. Du, Y. S. G. Nashed, S. Kandel, D. Gürsoy, and C. Jacobsen, “Three dimensions, two microscopes, one code: Automatic differentiation for x-ray nanotomography beyond the depth of focus limit,” Sci. Adv. 6(13), eaay3700 (2020). [CrossRef]
43. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206(1), 33–40 (2002). [CrossRef]
44. L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. A 1(6), 612–619 (1984). [CrossRef]
45. D. Kazantsev, V. Pickalov, S. Nagella, E. Pasca, and P. J. Withers, “TomoPhantom, a software package to generate 2D-4D analytical phantoms for ct image reconstruction algorithm benchmarks,” SoftwareX 7, 150–155 (2018). [CrossRef]
46. L. Brombal, G. Kallon, J. Jiang, S. Savvidis, P. De Coppi, L. Urbani, E. Forty, R. Chambers, R. Longo, A. Olivo, and M. Endrizzi, “Monochromatic propagation-based phase-contrast microscale computed-tomography system with a rotating-anode source,” Phys. Rev. Appl. 11(3), 034004 (2019). [CrossRef]
47. L. Bauer, M. Lindqvist, F. Förste, U. Lundström, B. Hansson, M. Thiel, S. Bjeoumikhova, D. Grötzsch, W. Malzer, B. Kanngießer, and I. Mantouvalou, “Confocal micro-x-ray fluorescence spectroscopy with a liquid metal jet source,” J. Anal. At. Spectrom. 33(9), 1552–1558 (2018). [CrossRef]
48. R. A. Barrea, R. Huang, S. Cornaby, D. H. Bilderback, and T. C. Irving, “High-flux hard X-ray microbeam using a single-bounce capillary with doubly focused undulator beam,” J. Synchrotron Radiat. 16(1), 76–82 (2009). [CrossRef]
49. R. Vescovi, M. Du, V. de Andrade, W. Scullin, D. Gürsoy, and C. Jacobsen, “Tomosaic: efficient acquisition and reconstruction of teravoxel tomography data using limited-size synchrotron X-ray beams,” J. Synchrotron Radiat. 25(5), 1478–1489 (2018). [CrossRef]
50. A. Wawrzyniak, A. Marendziak, A. Kisiel, P. Borowiec, R. Nietubyc, J. Wiechecki, K. Karas, K. Szamota-Leandersson, M. Zajac, C. Bocchetta, and M. Stankiewicz, “Solaris a new class of low energy and high brightness light source,” Nucl. Instrum. Methods Phys. Res., Sect. B 411, 4–11 (2017). [CrossRef]
51. C. Jacobsen, X-ray Microscopy, Advances in Microscopy and Microanalysis (Cambridge University, 2019).
