Research Article: Small-angle X-ray scattering tensor tomography: model of the three-dimensional reciprocal-space map, reconstruction algorithm and angular sampling requirements

Date Published: January 01, 2018

Publisher: International Union of Crystallography

Author(s): Marianne Liebi, Marios Georgiadis, Joachim Kohlbrecher, Mirko Holler, Jörg Raabe, Ivan Usov, Andreas Menzel, Philipp Schneider, Oliver Bunk, Manuel Guizar-Sicairos.


The mathematical framework and reconstruction algorithm for small-angle scattering tensor tomography are introduced in detail, as well as strategies which help to reduce the amount of data and therewith the measurement time required. Experimental validation is provided for the application to trabecular bone.

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Nanoscale features can be probed in a spatially resolved fashion when scanning a sample through a focused X-ray beam and recording a small-angle X-ray scattering (SAXS) pattern at each location (Fratzl et al., 1997 ▸), a technique referred to as scanning SAXS. The real-space resolution is defined by the beam size and the scanning step size, which is typically chosen in the order of several to a few tens of micrometres, but can also be as small as some tens of nanometres or as large as millimetres. For each scanning point, nanoscale features are probed in reciprocal space by the SAXS pattern, typically measuring a spectrum of feature sizes in the range of a few nanometres to a few hundred nanometres. The capability to study the distribution of nanoscale structures over extended sample areas is of particular interest for hierarchically structured materials for which the length scales of interest span many orders of magnitude (Fratzl & Weinkamer, 2007 ▸; Meyers et al., 2008 ▸; Beniash, 2011 ▸; He et al., 2015 ▸; Van Opdenbosch et al., 2016 ▸).

The ab initio characterization of the anisotropy of nano­strucure in a three-dimensional volume requires measurement of the sample from many possible orientations, not only around one rotation axis, but in a grid of two-dimensional orientation angles using two rotation axes. It is necessary to define a coordinate transformation from laboratory coordinates , with z pointing along the direction of the X-ray beam, to sample Cartesian coordinates , as illustrated in Fig. 2 ▸. This transformation is described for the nth sample orientation by a rotation matrix .

For the reconstruction of the three-dimensional reciprocal-space map in one specific q range, an iterative optimization algorithm is used to minimize the error metric between the modelled intensity, as calculated in equation (3), with respect to the measured data , for all scanning positions and sample orientations, Minimizing this error metric is a first-order approximation to a maximum-likelihood estimation for photon-counting Poisson noise (Thibault & Guizar-Sicairos, 2012 ▸). The measured intensity at each point and each orientation is divided by the transmitted intensity to compensate for absorption of the sample (Schroer et al., 2006 ▸). The parameters to optimize are the spherical harmonic coefficients, , and their orientation through and . A binary mask, , is used to denote valid data regions. It is equal to one for all valid data points and can be set to zero for bad detector angular sectors, φ, or where the scattering of the sample is obstructed, for example by the sample mount.

To validate the suitability of a series of spherical harmonics as a model to describe the three-dimensional reciprocal-space map, we used data from a 20 µm-thin section of trabecular bone (sample A). The data were taken with a beam size of 20 × 20 µm at different rotation angles β around the y axis of the beamline coordinate system as illustrated in Fig. 1 ▸(a). Further experimental details can be found in Appendix B. As the lateral resolution matches the thickness, the measurements give an adequate representation of imaging a planar arrangement of voxels and it has been shown that for thin samples a single rotation axis provides sufficient information on the three-dimensional arrangement of nanostructures (Georgiadis et al., 2015 ▸). Image registration (Guizar-Sicairos et al., 2008 ▸) of the transmission images, recorded simultaneously to the SAXS data, was used to assign the scattering from multiple orientations to individual voxels. This procedure is described in more detail by Georgiadis et al. (2015 ▸).

In order to extend this method to volumetric samples we combine SAXS with CT. In standard CT a scalar quantity, such as the sample absorption, is measured for each point within two-dimensional projections and the reconstruction is three dimensional. In such a case it is sufficient to measure projections at different sample orientations around a single rotation axis which is perpendicular to the X-ray beam propagation direction. For the case of SAXS one needs a reconstruction of the three-dimensional reciprocal-space map for each voxel, as described by the six-dimensional function in equation (3). Using the principles of invariant scattering along the direction of sample rotation (Feldkamp et al., 2009 ▸), Schaff et al. (2015 ▸) showed that for an ab initio reconstruction it is sufficient to measure SAXS patterns from all points of the sample, while sampling object orientations in the full steradians. To achieve this in the experiment a second rotation axis was introduced as schematically shown in Fig. 1 ▸(b). Raster scans of the whole sample, here also referred to as projections, are measured at different angles α and different tilt angles of this rotation axis, β. The object rotation matrix in each sample orientation n can be calculated by a rotation around y by an angle followed by a rotation around x by an angle , resulting in Because the sample is not perfectly aligned in the rotation centre of both rotation axes, the translational alignment of the measured projections has to be refined. For this purpose an X-ray absorption tomogram is reconstructed from the measured transmission images at β = 0°, using standard filtered back-projection algorithms. The sample transmission is measured simultaneously to the SAXS pattern using a photo-diode mounted on the beamstop, which blocks the direct unscattered beam and avoids damage to the detector. For each object orientation, (α, β), a projection from this absorption tomogram was calculated and used as a reference for alignment of the measured transmission images at the corresponding orientation. For this an efficient image registration approach based on selective up-sampling of the cross-correlation (Guizar-Sicairos et al., 2008 ▸) was used. If the sample has little contrast in the absorption measurement, alternatively the scattering intensity, averaged over φ in a chosen q range, can be used for this step.

SAXS tensor tomography aims at reconstructing the local three-dimensional reciprocal-space map for each volume element within a three-dimensional sample. This can be achieved through gradient-based optimization. An adequate numerical representation of the three-dimensional reciprocal-space map, for which only a few quantities or coefficients have to be recovered for each voxel, can be critical towards developing an approach that is efficient both in computational and measurement time. Liebi et al. (2015 ▸) introduced this reconstruction approach using spherical harmonics as a base to represent the reciprocal-space map and demonstrated it with a millimetre-sized sample of trabecular bone. The three-dimensional reciprocal-space map comprises information on the main orientation of the nanostructure for different q ranges and also its degree of orientation. The reciprocal-space map could further be used as input for fitting the underlying nanostructure, similarly to what has been done on two-dimensional SAXS data, for example to retrieve size parameters of the mineralized platelets in bone (Fratzl et al., 2005 ▸; Turunen et al., 2016 ▸).