Piecewise reconstruction of the anisotropic dark-field signal

A grating alignment with s¹ k a allows to reconstruct the anisotropic dark-field signal df_recp,rq in direction of a reasonably well. Based on this result, we can use the same argumentation as discussed for SAXS-CT in section 5.4.2 here: As a specific scattering orientation is probed with a single CT measurement, multiple CT measurements can be combined to fully reconstruct a three-dimensional function that describes the scattering behaviour in each voxel. For SAXS, this was ruled out as much of the data would be thrown away in the process. Contrary to SAXS, a single dark-field image can be mea-sured relatively fast, albeit only for one single s at a time. Due to this, a piecewise reconstruction ofdfrecp,rq is possible in GBI. The limiting factor is the amount of scat-tering directions that can be measured. The basic idea of such an experiment is shown in figure7.3. A GBI setup with s¹ parallel to the rotation axis has to be used to achieve quasi-rotational invariance of the dark-field signal for a single CT measurement. The en-tire experiment consists of several individual CT measurements. By changing the position

70 mm

~

Figure 7.4: Sample mounting for non-iterative directional dark-field tomogra-phy. The sample is positioned inside a hollow sphere, which allows for an arbitrary positioning of the sample on top of the rotation stage.

of the sample on top of the rotation axis, as indicated in the inset, different orientations of s¹ in terms of sample coordinates, and therefore scattering orientations _k are probed.

A full reconstruction ofdfrecp,rqis possible by combining the results for a larger number of _k.

7.2.1 Experimental realization

The main purpose of this project was to provide a straight-forward, easy to realize imple-mentation of directional dark-field tomography. With this in mind, expensive additional hardware and complex reconstruction algorithms were avoided in the experimental real-ization. Filtered back-projection is the reconstruction algorithm of choice, as it is widely used and easy to implement. However, FBP comes with stringent sampling requirements.

In particular, the algorithm relies on evenly sampled data without missing wedges. This requirement rules out static sample mounting, as for some _k, projections would have to be taken along the axis of the sample mount. To solve this issue, an auxiliary sample mounting in form of a hollow sphere was introduced. A photograph of the wooden ”TUM”

mounted inside the sphere is given in figure 7.4. Such a sample positioning holds several advantages. First of all, and most importantly, the sample can be positioned arbitrarily on top of the rotation stage, given the perfect symmetry of a sphere. Secondly, as the sphere is made of a thin shell of plastic it does not significantly attenuate X-rays. Lastly, owing to its smooth surface and homogeneous material the sphere does not contribute to the dark-field signal. Even though the last two effects do not play a role after CT, a strong background stemming from the sphere would reduce the dynamic range of a measurement noticeably.

In total, seven complete CT data sets of the wooden ”TUM” sample were measured with

7.2 Piecewise reconstruction of the anisotropic dark-field signal

a) b) c) d)

e) f) g) h)

Figure 7.5:3-D renderings of dark-field reconstructions with different sensitiv-ity directions. a)-g) Volume renderings of the seven reconstructed and reg-istered dark-field volumes. The sensitivity direction during each measurement is indicated for each volume by the arrows. h) Volume rendering of the average scattering data derived from all seven registered dark-field reconstructions.

seven unique sensitivity orientations _k, k 1, . . . ,7. As the sample rotation between each measurement was performed by hand, an exact positioning of the sample was not possible. We will see that a perfect distribution of k is not required for a good recon-structed of the fibre orientations. The _k were chosen to approximately sample three orthogonal directions, i.e. the coordinate system axes, as well as the four corresponding space diagonals. Since all seven measurements probe a different scattering orientation, the resulting datasets are completely independent from one another. The attenuation and dark-field volumes, V_k,att and V_k,df, were reconstructed using FBP for all seven datasets.

Given that attenuation and dark-field data are acquired simultaneously in the GBI setup, V_k,att andV_k,df are intrinsically registered for a single measurement. However, the different measurements are rotated and translated with respect to each other. Let us define V_1,att as the reference dataset. The remaining six datasets were registered to V1,att using the isotropic attenuation signalV_k,att;k 2, . . . ,7. There exist several powerful commercially available softwares to perform volume registration in three dimensions. Here, this step was performed usingAvizo Fire 8.0.1 - FEI, Hillsboro, Oregon, USA.

As the registration parameters obtained from attenuation data can directly be used for the dark-field data, allV_k,df were aligned with the help ofV_k,att. Volume renderings of the seven reconstructed and registeredVk,df are shown in figure 7.5 a)-g). Within each panel, the approximate orientation of the probed sensitivity orientation _k is indicated by an

arrow. The complementarity of the information recorded in the individual volumes can be seen clearly. Depending on the relative orientation of the wooden fibres and reconstructed scattering component, different parts of the sample are visible in different volumes. Given that _k are distributed approximately evenly in space, a non-biased average volume, Vdf¯, was calculated as:

Vdf¯ 1