# High-Speed Tomography--A New Approach to Plasma Bulk Velocity Measurement
## 2 Materials and Methods
### Data Processing  
Before describing the processing of the recorded data, a brief summary of the theoretical background of the computed tomography used for the reconstruction in the present work is given. For a successful reconstruction, it was assumed that the expanded plasma plume is optically thin. Modern computed tomography is based on the radon transform of a two-dimensional function \(f(x,y)\). In order to obtain a 3D reconstruction, all projections have to be calculated first. A projection is defined as an integral over \(f(x,y)\), which is formed by all straight lines \(g\) with a path length \(l\) going through the measurement object at the same angle \(\theta\) and at different distances \(s\) from the origin [31; 32], as illustrated in Figure 9. The entirety of all projections is represented by the function \(p(s,\theta)\). The transformation of \(f(x,y)\) into \(p(s,\theta)\) is carried out using the radon transform, which can be mathematically expressed as  
\[p(s,\theta)=R\{f(x,y)\}=\int_{g}f(x,y)dl. \tag{1}\]  
According to the above, the recorded projections of the object under study from each angle of view \(\theta_{i}\) (with \(i=1\dots 6\)) are the radon transforms \(p(s,\theta_{i})\). The straight line \(g\) can be described as follows, where \(l\) is perpendicular to \(s\)[33; 34],  
\[g=s\binom{\cos\theta}{\sin\theta}+l\binom{-\sin\theta}{\cos\theta}. \tag{2}\]  
The six projections of the measurement object originated from the six deflection mirrors shown in Figure 3. Thus, for a 3D reconstruction the inverse radon transform has to be computed, with  
\[f(x,y)=R^{-1}\{p(s,\theta)\}. \tag{3}\]  
Since digital camera recordings were used, the given function \(f(x,y)\) is discrete. Therefore, the discrete version of the radon transform was applied. In this case each projection-value \(p_{i}\) is multiplied by a weighting factor \(w_{i,j}\) introduced in Equation (4), which represents the influence of a single pixel value on the projection \(i\). \[p_{i}=\sum_{i,j=1}^{N}w_{i,j}\times f_{i}. \tag{4}\]  
Figure 8: Process flow of the experiment. The two-dimensional image \(f(x,y)\) is summarized in the vector \(f_{i}\) with \(N\times N\) elements, which represents the image size. A linear equation system results by summarizing the given projections to the vector \(\overrightarrow{p}\) and the weights to the matrix \(W\), which is formulated as  
\[\overrightarrow{p}=W\times\overrightarrow{f}. \tag{5}\]  
With the calculated values of \(\overrightarrow{p}\) and the known weighting matrix \(W\), \(\overrightarrow{f}\) as a sum of the \(N\times N\) values has to be computed. Due to the six directions of view, the system of equations is underdetermined. To solve the system of linear equations, the ART was used in this work. A detailed description of the ART can be found in [35; 36]. In the following, the data processing procedure is explained, starting from the recorded image to the reconstructed 3D model. The process is described using a recorded image as an example, shown in Figure 10.  
As can be seen in Figure 10, nine exposures were recorded on a single image of the high-speed camera. The temporal sequence of exposures goes from top to bottom of the image, with an interval of 10 \(\upmu\)s between each exposure. The first and the second exposure with their appropriate six viewing directions are marked with green and yellow boxes respectively. Since the proposed setup consists of an image intensifier and a high-speed camera and both devices are triggered by the rotating mirror (see Figure 6), the frame rate of the setup is determined by the rotation speed of the mirror. The exposure time of the high-speed camera has to be long enough to capture all the desired exposures displayed by the image intensifier (see Figure 5). In the case of the example presented in Figure 10, a minimum exposure time of about 82 \(\upmu\)s (8 delays of 10 \(\upmu\)s between 9 exposures of 0.2 \(\upmu\)s each plus a few \(\upmu\)s for camera sensor read out) is required to capture 9 exposures of the measurement object.