Virtual tomography axes

The concept of rotational invariance tells us that it is always possible to reconstruct the scattering vectors parallel to the rotation axis a. Furthermore, the reconstruction of a single q-vector is performed independent from all others. This allows us to reconstruct a scalar value in each individual reconstruction, and all well-understood theory on CT can be applied.

As scattering vectors parallel to a rotation axis can be reconstructed, a straight-forward reconstruction of _dΩ^dσ

pq,rq can be implemented by simply recording CT data sets for multiple rotation axes a_k. Each of the individual CT data sets then contributes scatter-ing vectors q k a_k to the complete reconstruction of ^dσ_dΩ

pq,rq. While this in theory is possible, practical limits are quickly reached. SAXS-CT measurements are very time consuming and only a limited number of projections can be recorded. As an example, it took over seven minutes to record a single projection P_j of the tooth sample. Computed tomography requires several P_j to be recorded for each a_k, which ultimately limits the investigation to a small amount of tomography axes. The main drawback of this method rests on the highly inefficient way of using data. Only qka_k are used in the reconstruc-tions. This is only a tiny fraction of the measured data and a large part of all SAXS patterns would be discarded.

A more efficient way of using data therefore is required in order to facilitate a direct reconstruction using the concept of rotational invariance. Here we introduce the concept of virtual tomography axes as a method to use the recorded data in a highly efficient way.

Instead of directly measuring a large amount of CT data sets, we measure many projec-tions distributed over half the unit sphere, as described in section5.2. The reconstruction ofqka_k is performed not by measuring a CT data set arounda_k, but rather by selecting those P_j that form a CT data set around the virtual tomography axisa_k. This technique allows us to reuse the data contained in a single P_j for multiple a_k. Owing to this, a substantially more efficient usage of the recorded data is achieved.

The basic idea of virtual tomography axes is visualized in figure5.11. SeveralPj recorded for different combinations ϕ, θ are shown. Three different virtual tomography axes, a₁, a₂, a₃, are sketched within the sample. The blue axis,a₁, corresponds to the standard CT case. AllPj that form a CT data set arounda1are marked by blue arrows accordingly, which also indicates the direction ofqka₁. Let us now consider another axis,a₂. First of all, we realize that differentP_j are required to form a CT data set arounda₂. TheseP_j are indicated by green arrows accordingly, which also show the orientation of qk a2 for this set of P_j. One P_j is contained both in the blue and green data sets. However, different

5.4 Reconstruction of the local scattering cross section

Sample

P_j a1

a2

a3

Figure 5.11: Concept of virtual tomography axes. SAXS-projectionsPj of a sample are recorded from all directions. For an arbitrary axis within the sample, a subset of all P_j that resembles a conventional CT measurement around that axis is used to reconstruct the scattering information parallel to that axis. Three different virtual tomography axes aare shown by three different colours. Only scattering data parallel to a is used for the respective tomo-graphic reconstruction. This is indicated by the direction of the arrows on eachP_j. As one P_j is used for multiple subsets the amount of data required for a full reconstruction is reduced substantially.

parts of the scattering information are used for the reconstructions. We are therefore able to use the data contained in a projection P_j for more than one a_k.

Another way to look at this is to consider the following. In order to reconstruct _dΩ^dσ

qprq for a certainq, we look for thoseP_j that record scattering information forq. We uniquely denote the pose of P_j by the direction of the incident beam t_j. In sample coordinates it is given as

t_j R¹p0,0,1q^|, (5.5)

with the rotation matrix R_j¹. As the measured SAXS signal is perpendicular to the incident beam, any P_j for which

qt_j 0 (5.6)

contains scattering data for q. As only a limited amount of P_j can be sampled within reasonable time, it is highly unlikely that the condition of equation 5.6 is exactly met.

We use the fact that the SAXS signal does not exhibit sharp peaks and is rather slowly

α [°]

0 60 120 180

α [°]

0 60 120 180

z x

y

z x

y a)

b)

Pj

Pj a =

( )

a =

( )

1.25 √1

Figure 5.12: Data selection for virtual tomography axes. All projections recorded for the tooth measurement are shown (upper hemisphere for clarity). Sub-sets around two virtual tomography axes a p0,1,0q^| (a) and a 1{?

1.25p0, .5,1q^| (b) are highlighted as filled spheres. Those projections are used to reconstruct scattering vectors parallel to the respective axis a. Only scattering data parallel to a is rotationally invariant and can be used in a simple reconstruction. The azimuthal coordinate α for which the scattering data is parallel to a is plotted againstP_j for both data sets.

5.4 Reconstruction of the local scattering cross section varying, contrary to e.g. crystal diffraction. This approximation holds for nearly all materials investigated with SAXS, and we can therefore relax the condition to

0  |qt_j| !1. (5.7)

This approximation allows us to ensure a sufficient amount ofP_j is used for each recon-struction. As a consequence of equation 5.7, nearly perfectly crystalline objects are not suited for the presented method.

After having identified all tP_j | 0   |qt_j| ! 1u, we need to figure out the coordinates q_r, α of q in the recorded data for every P_j. Owing to the small discrepancy introduced with equation5.7 we use the projection of qonto the detector plane,q pqt_jqt_j. Using the appropriate rotation matrices R_j, this two-dimensional vector can be transformed into laboratory coordinates and polar form to comply with the notation used for the recorded intensity, Ipq_r, αq. Note that the azimuthal coordinate α depends on q and t_j, and therefore both on a_k and P_j. This dependence is illustrated in figure 5.12. All P_j measured for the tooth sample are shown, mirrored to the upper hemisphere for clarity.

The subsets of P_j for a p0,1,0q^| and a 1{?

1.25p0, .5,1q^| are shown as filled circles in a) and b), respectively. The azimuthal coordinate, α, which contains scattering data parallel toais plotted againstP_j. Owing to the way the data are recorded, α is constant for the case of a conventional CT axis, as presented in a). For othera, α depends on P_j, as evident in b).

5.4.3 Quantification of rotational invariance for virtual