# High-Speed Tomography--A New Approach to Plasma Bulk Velocity Measurement
## 2 Materials and Methods
### Data Processing
Consequently, the exposure time of the high-speed camera is set arbitrarily as long as it exceeds the required minimum exposure time mentioned above. Then, the delay between trigger edges (one full rotation of the trigger flywheel) should be also taken into account. The maximum rotation speed of the mirror in the proposed setup was around 20,000 revolutions per minute and the diameter of the trigger wheel was 80 mm. This means that the high-speed camera was triggered at 3 ms intervals and therefore every trigger edge was recorded in the above example. However, since each image captured by the high-speed camera consists of several exposures displayed by the image intensifier, a frame rate can be defined for each individual image. This frame rate is determined by the delay between two successive exposures of the image intensifier. In the case of the  
Figure 9: Radon transform of the measurement object defining a projection \(p(s,\theta)\), with \(s\) being the distance from the origin and \(\theta\) being the observation angle. example shown in Figure 10, a delay of 10 \(\upmu\)s corresponds to a frame rate of 100,000 frames per second. This \"individual image frame rate\" is considered by the authors to be more relevant for the proposed setup than the frame rate of the high-speed camera. For further processing, six contours (corresponding to the six viewing directions) of the first exposure are cut out and are shown side by side in \[p(s,\theta)=R\{f(x,y)\}=\int_{g}f(x,y)dl. \tag{1}\]0  
The contours of the measurement object were obtained using a well-known algorithm called marching squares [37]. In the next step, cross-sectional images based on these six directions of view were computed. In Figure 12 the resulting sinogram (a) and cross-sectional image (b) for one exemplary cut plane are shown. The six cut-out contours were scanned simultaneously pixel line by pixel line. The sinogram and the cross-sectional image respectively are the result of the computation from the identical line of all six contours. When all contours are positioned correctly, a clear sine wave should be visible in the sinogram, as marked by the blue sine wave in Figure 12a. In addition, the cross-sectional image can be used for verification. The back-projection of the six viewing directions should overlap as shown in Figure 12b. The computation of such a cross-sectional image was done for every pixel line of the cut-out contours and finally, these images were stacked on top of each other. This ultimately leads  
\[p(s,\theta)=R\{f(x,y)\}=\int_{g}f(x,y)dl. \tag{1}\]1 Cut-out contours from first exposure (marked with green boxes) of \[p(s,\theta)=R\{f(x,y)\}=\int_{g}f(x,y)dl. \tag{1}\]2 The numbers 1–6 correspond to the respective direction of view from \[p(s,\theta)=R\{f(x,y)\}=\int_{g}f(x,y)dl. \tag{1}\]3  
\[p(s,\theta)=R\{f(x,y)\}=\int_{g}f(x,y)dl. \tag{1}\]4 Recorded image with nine exposures of the plasma torch (green: first exposure; yellow: second exposure). The exposure time was 200 ns and the delay between each exposure was set to 10 \(\upmu\)s; thus, the above image represents a time interval of 81.8 \(\upmu\)s. The numbers 1–12 mark the directions of view of the respective exposures. to a three-dimensional reconstruction of the measurement object. The entire procedure was done for each individual exposure (nine, in the case of the example shown in Figure 10) recorded in an image from the high-speed camera. To determine the velocity, all cross-sectional images of one exposure (as shown in Figure 12b), were analyzed and the mean grey value of each cross-section was calculated and plotted afterwards. The procedure was executed for all other (nine, in the case of the discussed example from Figure 10) exposures. Then, the maximum values of each exposure (more precisely of the mean grey value plots) were fitted, thus specifying the horizontal shift in pixel. Finally, the velocity was calculated using the horizontal pixel shift, a pre-determined scale and the known time interval between each exposure (10 \(\upmu\)s in the discussed example).