Six-dimensional real and reciprocal space small-angle X-ray scattering tomography

Abstract

When used in combination with raster scanning, small-angle X-ray scattering (SAXS) has proven to be a valuable imaging technique of the nanoscale1, for example of bone, teeth and brain matter2,3,4,5. Although two-dimensional projection imaging has been used to characterize various materials successfully, its three-dimensional extension, SAXS computed tomography, poses substantial challenges, which have yet to be overcome. Previous work6,7,8,9,10,11 using SAXS computed tomography was unable to preserve oriented SAXS signals during reconstruction. Here we present a solution to this problem and obtain a complete SAXS computed tomography, which preserves oriented scattering information. By introducing virtual tomography axes, we take advantage of the two-dimensional SAXS information recorded on an area detector and use it to reconstruct the full three-dimensional scattering distribution in reciprocal space for each voxel of the three-dimensional object in real space. The presented method could be of interest for a combined six-dimensional real and reciprocal space characterization of mesoscopic materials with hierarchically structured features with length scales ranging from a few nanometres to a few millimetres—for example, biomaterials such as bone or teeth, or functional materials such as fuel-cell or battery components.

Main

By raster scanning an object through a focused X-ray beam and recording a scattering pattern at each point, which we refer to as a projection in the following, scanning SAXS imaging combines the ability of SAXS to probe the nanoscale with spatial resolution. Changes in nanostructure over an area many orders of magnitude larger can be revealed using this technique1. Because a SAXS pattern is recorded at every scanned point on an area detector (Fig. 1b), it is possible to extract oriented scattering information. This information may be of great importance for the understanding of material properties, and has proven to be very useful for the characterization of highly anisotropic structures, such as collagen3,12,13,14,15,16,17.

Figure 1: Experimental set-up and raster-scanning technique.
figure1

a, Schematic representation of the experimental set-up. The sample is raster scanned through a focused X-ray beam, indicated by the vertical and horizontal arrows, and a diffraction pattern is recorded at each point. In addition to a rotation φ around the tomography axis, the sample is rotated by θ around an additional axis. b, Each pixel of a SAXS projection consists of an entire SAXS pattern rather than just a single value. c, Schematic representation of the virtual-tomography-axis technique. Projections recorded at different views of the sample are used for various virtual tomography axes, represented by the red, green and blue arrows within the sample. Each of these axes is reconstructed independently by using appropriate projections and detector segments, as indicated by arrows of matching colour.

PowerPoint slide

Owing to the penetrating nature of X-rays, scanning SAXS may be used to image thick samples. One drawback of pure projection imaging is that the signal is integrated along the beam direction, and consequently all depth information about the scattering structures is lost. It is possible to recover the exact location of a signal in three dimensions using computed tomography. In the case of SAXS, computed tomography has previously been demonstrated, for example, for polymer fibres, amorphous glass and myelin sheaths6,7,8,9,10,11. Compared with other tomographic techniques such as micro computed tomography or ptychographic tomography, which are either unable to go down to nanometre resolution or are limited to very small sample volumes18,19, SAXS tomography is able to probe structural information at the nanoscale for objects of comparatively much larger dimensions.

However, only a small fraction of the full two-dimensional (2D) SAXS signal has been able to be used for a correct reconstruction thus far, and, in particular, the orientation information contained in 2D SAXS patterns has not been able to be preserved during tomographic SAXS reconstruction. We demonstrate the full potential of SAXS tomography, by deriving a way to combine orientation-sensitive scanning SAXS imaging with computed tomography. Therefore we can obtain the full three-dimensional (3D) scattering distribution in each voxel of the 3D space (hence a six-dimensional (6D) combined real and reciprocal space map of the specimen).

For most standard reconstruction techniques, a correct result is only possible if the signal is rotationally invariant, that is, the sum of all line integrals that form a projection image does not depend on the rotation of the sample. The most prominent example where this condition is satisfied is in the attenuation contrast of, for example, neutron or X-ray tomography as used in many industrial and clinical applications. In the case of SAXS tomography, rotational invariance is not generally present in the signal, except in special cases such as when imaging isotropic scattering powders, colloids or samples with certain symmetries. This lack of invariance restricts the number of applications, and excludes a wide range of materials with a preferred scattering orientation, including biomaterials composed of oriented collagen fibres and a wide range of functional materials with fibrous nanostructures.

In addition to being observed in isotropic samples, rotational invariance also exists in oriented scattering samples if the momentum transfer vectors q, that is, the scattering orientations, are parallel to the rotation axis9. This implies that for an arbitrary sample, a correct reconstruction is guaranteed only for q vectors parallel to the tomography axis; all other scattering orientations cannot be reconstructed. Therefore, for a complete reconstruction of all possible q vectors in three dimensions, additional rotations of the sample are necessary.

One solution to the problem of requiring additional rotations of the sample is to measure and reconstruct the sample from a large number of different rotation axes, each parallel to q vectors in one specific direction. This would be highly inefficient and time consuming and thus infeasible in practice. Instead, we add the additional necessary rotations by introducing a tilt to the conventional rotation axis around the horizontal axis perpendicular to the beam direction (Fig. 1a). By doing so, projections are recorded from all possible sample orientations by combining the tomographic rotation φ with a tilt by an angle θ. For each of the SAXS patterns of one projection, the scattering vectors qd recorded on the detector represent the integration of 2D slices of the 3D q spaces of each voxel along the path of the X-ray beam. For the purpose of tomographic reconstruction, it is thus vital to know which q vectors are probed by each projection. This information is obtained by using a projection-dependent rotation matrix Ai that describes the relationship between the laboratory and sample coordinate systems:

If we define the direction of the X-ray beam to be along the z axis of the laboratory coordinate system, then each projection, indexed i, is specified in the sample coordinate system by its plane normal vector .

Only q vectors parallel to the tomography axis can be reconstructed, and every SAXS projection probes a 2D slice of the 3D q space. Consequentially, all projections with orientation ni perpendicular to an axis t, given in sample coordinates, must necessarily contain q vectors parallel to t and so fulfil the rotational-invariance requirement for a rotation around t (see Methods for a more general discussion on rotational invariance).

We now introduce virtual tomography axes. Instead of physically rotating the sample around an axis t, we select a subset of the full set of available projections that contains all those with q vectors parallel to the desired axis t. This technique is illustrated in Fig. 1c. Several projections are sketched in the sample frame of reference. For three select axes t within the sample, projections with q vectors parallel to t are selected, shown in Fig. 1c by arrows of matching colours. The orientation on the detector that contains the rotationally invariant SAXS data for each axis tand projection is indicated by the respective arrows orientations.

Owing to a finite number of possible recorded projections, it is unlikely that a perfect match between t and the available projections will be found. We use the scalar product |nit| to quantify the mismatch between the projection ni and the virtual tomography axis t. For most materials of interest, the SAXS signal generally varies slowly and does not exhibit sharp azimuthal peaks (contrary to crystal diffraction, for example); consequently, projections for which may be included as well as those for which |ni ∙ t| = 0, to ensure a sufficient number of projections for the tomographic reconstruction. Our method is not suitable for materials such as nearly perfect single-crystalline objects, for which the SAXS signal does not vary slowly.

Given such a subset of possible projections, for each individual projection, we need to calculate the orientation of the recorded scattering vectors qd corresponding to . The correct part of the detector is calculated by projecting t onto the plane specified by ni, which allows us to account for the small discrepancy introduced by considering projections for which : . Using virtual tomography axes, many different axes t, and therefore q vectors, can be reconstructed by selecting the appropriate subsets and corresponding qd vectors. A major advantage of this method is that all the information from every SAXS pattern can be used for a complete reconstruction of the 3D q space in each voxel, rather than just a few qd vectors, as is the case when using the standard tomography axis. Removing the requirement of having to directly measure a large number of tomography axes and instead reusing the same projections for many different virtual tomography axes allows us to efficiently perform a real and reciprocal space resolved 6D SAXS computed tomography characterization.

To demonstrate the usefulness of the method, particularly for the characterization of biomaterials, we performed an experiment to resolve the nanostructures found in a human tooth. Teeth are made of a complex, highly hierarchically structured material that is able to withstand years of cyclic load in the harsh environment of the mouth. Knowledge of the exact structural details of this material at length scales spanning the nanoscale to the macroscale is still missing, and would help to understand the basic structure–function relationships that have been optimized during millions of years of evolution. This understanding is important for the design of tailored synthetic materials that mimic the durability and stiffness of teeth. An overview of four different tissues present in the investigated sample is shown in Fig. 2a. Mean mineral density information, observed along the long axis of the sample and obtained using high-resolution, state-of-the-art absorption tomography, reveals the layers of cementum and of mantle, primary and secondary dentine in the sample. All these tissues are composed of mineralized collagen fibres that give rise to a strong characteristic SAXS signal, owing to the periodic arrangement of mineral platelets along the fibre axis with approximately 67.6-nm spacing3. However, the details of the nano-architecture that is essential for the mechanical performance remains unresolved.

Figure 2: Virtual SAXS patterns extracted from the reconstructed 3D volume.
figure2

a, Average density within the sample obtained using state-of-the-art absorption micro computed tomography. Distinct dental tissues found in the sample are identified around the root canal (r): secondary dentine (s), primary dentine (p), mantle dentine (m) and cementum (c). b, 2D slices along the coordinate axes of the reconstructed 3D q = (qx, qy, qz) space. The logarithmic scattering strength for three points in the centre of the sample, indicated by the blue, green and red squares, is shown (colour coded by the outer ring on the plots) for slices in the qzqy, qzqx and qxqy planes. Distinct collagen peaks can be seen at |q| ≈ 0.9 Å−1, except for slices roughly perpendicular to the fibre orientation. The orientation of the fibres and the intensity of the peaks change noticeably in the different regions of the sample. An animation showing multiple slices is provided in Supplementary Video 1.

PowerPoint slide

Using the SAXS computed tomography technique described above, we reconstructed the full 3D scattering distribution in every voxel for the whole q range, rather than reconstructing just one value as in conventional tomography. From this data, we virtually extract information about what the reciprocal scattering space from each voxel looks like from any given orientation. Planar cuts along the coordinate-system axes of this voxel-wise 3D information are shown in Fig. 2b for three different regions within the sample. The top, blue-ringed slices of Fig. 2b corresponding to the pulp region of the tooth sample, which consists of secondary dentine, have a pronounced vertical fibre orientation, seen as clear collagen peaks in the qy direction. Different orientations and scattering signal strengths are seen in the green- and red-ringed slices, which correspond to regions of the tooth that are composed of primary and mantle dentine.

Given the abundance of reconstructed data, a major difficulty is to extract the information of interest and compress the results into an easily understandable form. One possibility for determining collagen orientation is as follows. As well as diffuse background scattering that stems from the fibre bundles, collagen fibres display distinct peaks around |q| ≈ 0.9 Å−1. Previous studies20 have subtracted the background scattering and isolated the collagen peak by fitting a power-law function with an exponent of about 2.6 (ref.). To extract the orientation of the collagen peaks, we took the ratio of q-vector intensities in the range with (0.88–0.94 Å−1) and without (0.82–0.86 Å−1) collagen peaks. Assuming a constant exponent for the background scattering, the ratio of q-vector intensities deviates from a constant value only in those directions with additional scattering caused by the distinct periodicity of the collagen fibres. From the resulting 3D distribution of ratios, we extracted the mean fibre orientation by ellipsoid fitting and determining the largest principal axis. The resultant orientations for example slices inside the sample are shown in Fig. 3. The fibre orientations are represented by small bars whose colour represents the average scattering intensity in the collagen range 0.88–0.94 Å−1. Their length is scaled with the respective absorption values, so that equally long and vanishingly small bars are obtained inside and outside the sample volume, respectively. A strong change in collagen fibre orientation near the root-canal region (upper side of Fig. 3) of the sample contrasts the gradual increase in scattering strength towards the outer sides of the root (lower side of Fig. 3). The collagen fibre orientation is just one example of many possible parameters that can be extracted from the 6D data. Various other parameters have been extracted from SAXS data previously, such as the local degree of alignment of the mineral platelets contained within the collagen matrix, or their shape and size distribution21.

Figure 3: 3D visualization of collagen fibre orientation within the tooth sample.
figure3

The orientation of the coloured bars indicate the mean orientation of collagen fibres obtained from ellipsoid fits to the ratios of scattering intensities of q ranges with and without collagen peaks. The colour represents the average scattered intensity in the collagen range 0.88–0.94 Å−1. The underlying 3D nanomorphology within the entire sample is revealed. An animation showing all slices in one direction is provided in Supplementary Video 2.

PowerPoint slide

In conclusion, we demonstrated a combined 6D real and reciprocal space SAXS computed tomography technique as a method to simultaneously obtain information about the structural features of a sample from the nanometre to the millimetre scale. Using virtual rotation axes, we take advantage of all data recorded and are able to reconstruct the full 3D scattering distribution in each voxel of the 3D real space of the sample. This vast amount of additional information in each voxel could enable analysis of the 3D nanostructure of materials, spatially resolved in 3D on a large scale. As with all imaging techniques, analysis of the resultant data is a big part of the work. Given the amount of data reconstructed by the presented method, there is a need for data analysis tailored to a specific problem. For the purpose of the investigated collagen sample, we proposed one possible way to extract fibre orientations in three dimensions. Although our approach is a substantial step forward, measurement and reconstruction times were considerable. (A discussion of the prospects of advances in hardware and the subsequent reduction in time needed for an experiment is given in Methods.) Nonetheless, we believe that a combined 6D real and reciprocal space characterization is of great interest for nanostructure characterization of mesoscopic materials and composites with hierarchically structured features ranging from the few-nanometre to the few-millimetre length scale. Possible applications of such a characterization include the study of natural bones and teeth, or man-made functional materials.

Methods

Sample

A human upper lateral incisor was obtained from a pool of teeth collected from patients undergoing routine dental treatment unrelated to this study. The tooth was obtained with written informed consent, following the directives of the Ethical Review Committee of the Charité - Universitätsmedizin Berlin (EA4/002/09). The tooth had no caries and was extracted for periodontal reasons from an anonymous donor. For the SAXS experiment, a cylindrical segment of the tooth about 3 mm in diameter and 4 mm in length was extracted using water-cooled dental drills. The harvested piece of the root was extracted from a region beneath the tooth crown, facing the lip (buccal side of the tooth).

Experiment

The measurements were conducted at the X12SA (cSAXS) beamline of the Swiss Light Source (Paul-Scherrer Institute) with monochromatic 18.6-keV X-rays. A pencil beam of 50 μm × 50 μm at the sample position was used to raster scan the specimen with a matching step size of 50 μm and exposure time of 50 ms per point. In total, scanning SAXS projections were recorded from 288 unique sample orientations. The 288 projections were recorded for 10 different tilt angles θ = 0°, −4°, −12°, −20°, −28°, −36°, −44°, −52°, −60°, −68° of the tomography axis. A varying number (55, 29, 29, 29, 29, 29, 29, 25, 19, 15) of tomography angles φ, evenly distributed over 360°, was used for each of the tilted tomography scans. The θ = −12°, −28°, −44° and −60° tomographic measurements included a φ offset of 6.2° so a more even distribution of the projections could be achieved. Each projection consisted of 59 × 81 SAXS patterns, in the horizontal and vertical directions, respectively, whereby the horizontal axis was scanned in continuous mode. In total, 1,376,352 SAXS patterns were recorded in slightly less than 40 h. The scattering signal was collected using a photon-counting PILATUS 2M detector22 placed 7,363 mm downstream of the sample. Simultaneously, X-ray attenuation data was acquired using a diode directly mounted on the beamstop, which was used for a necessary normalization of the scattering signal10 and registration of the projections. The normalization was done by dividing each recorded SAXS pattern at point p (including all variables describing sample position and rotation) by the relative absorption of the sample :

Reconstruction

Azimuthal integration was performed using pyFAI23 for all recorded SAXS patterns, regrouping the data into 360 azimuthal segments with an extent of 1° each. Following the integration, the data were reduced to an azimuthal range of 180° by averaging, making use of the fact that the data are symmetric with respect to the centre of the diffraction pattern.

The selection of projections for each virtual tomography axis included an allowed deviation of the scalar product |ni ∙ t| < 0.05 throughout the reconstruction. For our 288 recorded projections, this condition ensures that a sufficient number of points are picked, without increasing the introduced error more than necessary.

A virtual tomography axis is expressed in sample coordinates by . The reconstructed virtual tomography axes were chosen for 30 different values of θ from 0° to 90° and a varying number Nφ = 60sin(θπ/180) + 1 of values of φ from 0° to 360°, depending on the current value of θ. This choice of Nφ ensured that the virtual tomography axes were chosen to cover a solid angle of 2π, allowing for a complete reconstruction of all possible orientations.

In total, 1,168 evenly distributed virtual tomography axes in 13 different q-vector intervals ranging from |q| = 0.45 Å−1 to |q| = 1.16 Å−1 were reconstructed; 15,184 individual reconstructions were performed. Each of these consists of a unique set of projections and azimuthal segments. Because only a very small azimuthal part of the detector is used from each SAXS pattern, the radial width of the intervals was chosen to integrate over several pixels to increase photon statistics. The closest azimuthal segment to the projected virtual tomography axis was taken, that is, an azimuthal error of less than 0.5°. Each of the reconstructions was performed using a SART (simultaneous algebraic reconstruction technique) algorithm with total variation regularization, because this reconstruction technique has been shown24 to perform well with strong undersampling and missing angular wedges, as in the present case. The computing time per reconstruction was about 40 s on a computer built for GPU computing (2 × Intel Xeon E5-2643, 4 × Nvidia Tesla Kepler K10, 256 GB RAM). Therefore, the reconstruction time for all reconstructions amounts to slightly over one week. In principle, all reconstructions can be run in parallel because they are completely independent from each other. An optimized, parallel reconstruction would thus allow for a substantial reduction in computing time—potentially to a few hours.

To verify a correct reconstruction result, intensity maps of the reconstructed and reprojected scattering vectors alongside their measured counterparts are shown in Extended Data Fig. 1 for the |q| = 0.88–0.94 Å−1 interval at two select azimuthal scattering orientations of one projection.

The 3D visualization of the reconstruction results was done using ParaView ( http://www.paraview.org).

Absorption computed tomography

The high-resolution micro computed tomography of the sample was acquired with a VersaXRM-500 X-ray microscope, Xradia (Pleasanton), operated at 50 kVp, 4 W and 30 s exposure time. A voxel size of 3.087 μm was achieved.

Coordinate system

To describe the necessary rotations it is important to define coordinate systems, which was done according Extended Data Fig. 2. A fixed laboratory coordinate system is given by xlab, ylab, zlab, with zlab in the beam direction, ylab pointing upwards and xlab completing the right-handed coordinate system. During the acquisition of each projection, the sample is raster scanned along the xlab and ylab axes for a total of n × m SAXS patterns per projection. For the reconstruction, a sample coordinate system is defined by its axes x, y, z, which are equivalent to the laboratory coordinate axes prior to any rotation. The rotations required for 3D SAXS computed tomography are described using Euler angles ψ, θ and φ, which follow a YZY convention. For our purpose, it is convenient to set ψ = 90° so that φ and θ represent the tomographic rotation and its tilt around xlab. For all sample rotations, the corresponding rotation matrix Ai is calculated as a combination of three successive rotations:

The scattering vectors recorded on the detector are described using detector coordinates dx and dy. The relationship between qd and q for each projection is given according to equation (1) as .

Rotational invariance

Following ref. 25a rotational invariance check is performed for a standard, vertical tomography axis as follows. For any horizontal line ylab of a projection P, a value is assigned to each pixel of the detector as the sum of all pixel values for this specific line scan. This function necessarily must be constant under rotation if rotational invariance is given—the information changes its horizontal position, but is not lost. Pixel-wise rotational invariance is then calculated as the ratio of the standard deviation to the mean of over all projections P:

A result for the vertical tomography axis of our measurement is shown in Extended Data Fig. 3. The pixel-wise rotational invariance is shown for two different slices of the sample, marked by the white lines. Because a rotation is only performed around the vertical axis, these slices are independent from each other. Whereas the upper slice (Extended Data Fig. 3b) displays certain rotational invariance for pixels not on the vertical axis, owing to fibre symmetry, the lower slice (Extended Data Fig. 3c) clearly shows that rotational invariance is given only in the vertical orientation.

We want to extend the rotational-invariance check to more than one rotation axis, but a line-wise treatment has proven to be insufficient. Because of additional rotations, different slices of the sample are no longer independent from each other, and the sample needs to be treated as a whole. Rather than summing over one scanning line, we have to sum over all scattering patterns obtained for one projection: . Additionally, for an arbitrary virtual rotation axis t, projections from a tilted rotation axis are included in the reconstruction. From this it follows that the rotationally invariant data are no longer restricted to the vertical direction, but are found in different azimuthal segments on the detector for each projection P. Thus, the scattering patterns need to be rotated accordingly for a proper comparison. This rotation is most easily done with the already azimuthally regrouped data IP(χ, r), with χ the azimuthal and r the radial coordinate. A rotation of the data then simplifies to a projection-dependent offset of χ, and rotational invariance can be checked for any arbitrary set of projections and virtual tomography axis t. Extended Data Figure 4 shows rotational-invariance results for two different virtual tomography axes—(Extended Data Fig. 4a) and (Extended Data Fig. 4b)—for already azimuthally regrouped data. For each pattern, all data has been shifted so that χ = 0° or χ = 180° corresponds to the detector orientation parallel to the chosen virtual rotation axis. Rotational invariance is achieved for any given virtual tomography axis by using data only from projections and orientations with scattering parallel to this axis.

Discussion on acquisition times and resolution

The number of SAXS patterns that can be collected in a given time frame is mostly determined by two key parameters of the X-ray beam used in the experiment—the X-ray energy and flux of the beam. To reduce the exposure time needed per single SAXS pattern, it is necessary to have as many photons contribute to the SAXS signal as possible. We measured the tooth sample presented in this work at 18.6 keV, the highest available photon energy at the cSAXS beamline. At this energy, a large proportion of the X-rays were absorbed in our sample and therefore did not contribute to the SAXS signal. Even a small increase in photon energy would have a very noticeable effect on the transmission, and ultimately the exposure time required for each point. The second major factor for reduction of the scanning time needed is the flux available for the experiment. Even though synchrotron facilities today are already very powerful, their technology is continuously being improved. An example is the upgrade being installed at the European Synchrotron Radiation Facility (ESRF). Planned to be finished by 2018, this upgrade is expected to increase the available photon flux by up to two orders of magnitude. These ongoing developments will enable very rapid data acquisition to be performed in the future, facilitated by recently developed fast-framing pixel detectors with frame rates of several kilohertz26.

A point of concern arising from much stronger sources is the issue of radiation damage. During our experiment we performed several control-scans throughout the measurement and confirmed that there was no change in the signal even after several hours of beam exposure. It is hard to predict at what point radiation damage inflicted by the beam starts to become an issue. There are, however, possibilities to limit these effects, such as active cooling with cryo-jets.

The achievable real-space resolution in SAXS computed tomography is dictated mainly by the investigated sample and number of SAXS patterns (points) available. The total number of projections strongly depends on the number of individual scanning points per projection. From a tomographic point of view, there is little benefit from measuring only a few projections with a large number of points each or, similarly, a large number of projections with only a few points each. Therefore, the number of points per projection cannot be scaled up easily without a massive increase in the acquisition time required. As a consequence, the real-space resolution is strongly limited by the size of the sample. For reference, the real-space resolution of our measurement for a 4-mm object was about 100 μm, because of the continuous acquisition used.

The resolution in reciprocal space has to be discussed for two different cases, namely radial and angular resolution in reciprocal space. The radial resolution is mostly limited by the angular divergence of the beam and the pixel size of the detector. We chose to radially integrate over several detector pixels, which limits our radial reciprocal-space resolution to 0.05 Å−1. In contrast to this, the angular resolution in reciprocal space is largely limited by the number of projections recorded. This number determines the deviation allowed when selecting projections for each virtual tomography axis. In our case, a deviation of the scalar product |ni ∙ t| < 0.05 corresponds to a minimal angular resolution of slightly less than 6°.

Considering all these points, the greatest potential to improve the performance of our method lies in the use of stronger sources and higher photon energy. Our measurement took 40 h. An increase in usable flux by a factor of ten would decrease the time needed for exactly the same measurement by approximately the same factor—to slightly less than 4 h. This reduction would enable SAXS computed tomography to be used for a case study of several samples. A substantial reduction in acquisition time can also be achieved by scanning projections with lower real-space and angular reciprocal-space resolution, owing to the relationships mentioned previously.

Code availability

The code used for azimuthal regrouping is openly available at https://github.com/kif/pyFAI. Additional code used is available upon request.

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Acknowledgements

The SAXS experiments were performed at the cSAXS beamline of the Swiss Light Source (SLS) at the Paul Scherrer Institut (PSI), Villigen, Switzerland. We are grateful for travel support that was granted by the EU access program CALIPSO. We acknowledge financial support through the DFG Cluster of Excellence Munich-Centre for Advanced Photonics (MAP) and the DFG Gottfried Wilhelm Leibniz program. P.Z. is grateful for funding of the DFG (German Research Foundation) through SPP1420. F.S. and C.J. thank the TUM Graduate School for support of their studies. F.P. acknowledges support through the TUM Institute for Advanced Studies (TUM-IAS). We thank A. Fehringer for developing the GPU projectors used for the reconstruction.

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Contributions

F.P., M.B. and F.S. conceived the experiment. M.L. and M.G.-S. developed the sample stage used in the experiment. M.B., C.J., M.L., M.G.-S. and F.S. performed the experiment at cSAXS/PSI. P.Z. prepared the tooth sample. Data analysis was performed by F.S. with input and discussion from M.B., F.P. and P.Z. F.S. wrote the manuscript with contributions from all co-authors.

Corresponding authors

Correspondence to Florian Schaff or Martin Bech or Franz Pfeiffer.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Comparison of measured and reconstructed values for one projection.

Left, reprojected q data for two different orientations (90° and 152°) is compared to the measured data. Right, azimuthal values for both the reconstructed and measured data are given for three select points, indicated by the dashed lines in the left panel. For the chosen |q| range, distinct collagen peaks are reconstructed correctly for points 1, 2 and 3 at around 15°, 0° and 90°, respectively. A good agreement between reconstruction and measurement is seen. Animations of this and further projections showing all q orientations are provided in Supplementary Videos 3, 4, 5.

Extended Data Figure 2 Coordinate system used for the experiment.

The sample orientation is described using three Euler angles θ, φ and ψ. With ψ = 90°, θ represents a tilt of the tomography axis around xlab and φ describes a rotation around this tilted axis. The sample is scanned along xlab and ylab, with a diffraction pattern collected at each point. The detector coordinates are given by dx and dy.

Extended Data Figure 3 Rotational invariance for a standard SAXS computed tomography with a vertical tomography axis.

a, Absorption image acquired from the diode data with two vertical slices marked. b, c, Rotational invariance as defined in the text for both slices. In both cases, rotational invariance is present for all pixels that correspond to scattering orientations parallel to the vertical rotation axis. The collagen fibres in the top part of the sample, shown in b, are mainly vertical. Owing to this symmetry, rotational invariance is also present for pixels not on the vertical axis. Without this symmetry, rotational invariance exists only for the vertical direction, as seen in c. The white bars in the lower half of the images are areas between the individual detector modules.

Extended Data Figure 4 Generalized rotational invariance.

a, b, Rotational invariance shown for different virtual tomography axes: (a) and (b). Radially integrated data are used. Compared to the standard case, shown in Extended Data Fig. 3, a line-wise integration is not possible in the general case. Instead, the standard deviation and mean are calculated from the projection-wise sum of all integrated SAXS patterns. A shift of the azimuthal angle so that the scattering orientation parallel to t is at 0° was applied. As can be seen, rotational invariance is also achieved in the general case for scattering orientations parallel to t.

Supplementary information

41586_2015_BFnature16060_MOESM121_ESM.mp4

All slices from top to bottom are shown. Animation of Figure 2. (MP4 1436 kb)

SAXS-patterns extracted from the reconstructed data at three different points.

All slices from top to bottom are shown. Animation of Figure 2. (MP4 1436 kb)

41586_2015_BFnature16060_MOESM122_ESM.mp4

All slices of the sample from front to back are shown. Animation of Figure 3. (MP4 14801 kb)

Three-dimensional visualization of collagen orientation and scattering strength within the tooth sample.

All slices of the sample from front to back are shown. Animation of Figure 3. (MP4 14801 kb)

41586_2015_BFnature16060_MOESM123_ESM.mp4

Azimuthal scattering orientations over 180° are shown for the projection recorded at θ = 0.0° and φ = 346.9°. Animation of Extended Figure 1. (MP4 1016 kb)

A comparison of measured and reconstructed scattering intensities.

Azimuthal scattering orientations over 180° are shown for the projection recorded at θ = 0.0° and φ = 346.9°. Animation of Extended Figure 1. (MP4 1016 kb)

41586_2015_BFnature16060_MOESM124_ESM.mp4

Azimuthal scattering orientations over 180° are shown for the projection recorded at θ = -52.0° and φ = 216.9°. Animation of Extended Figure 1. (MP4 882 kb)

A comparison of measured and reconstructed scattering intensities.

Azimuthal scattering orientations over 180° are shown for the projection recorded at θ = -52.0° and φ = 216.9°. Animation of Extended Figure 1. (MP4 882 kb)

41586_2015_BFnature16060_MOESM125_ESM.mp4

Azimuthal scattering orientations over 180° are shown for the projection recorded at θ = -68.0° and φ = 120.0°. Animation of Extended Figure 1. (MP4 890 kb)

Comparison of measured and reconstructed scattering intensities.

Azimuthal scattering orientations over 180° are shown for the projection recorded at θ = -68.0° and φ = 120.0°. Animation of Extended Figure 1. (MP4 890 kb)

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Schaff, F., Bech, M., Zaslansky, P. et al. Six-dimensional real and reciprocal space small-angle X-ray scattering tomography. Nature 527, 353–356 (2015). https://doi.org/10.1038/nature16060

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