# Calibration Method of Projectional Geometry for X-ray C-arm Fluoroscopy Using Sinogram Data
## 5 Discussion  
We conducted phantom experiments and analyzed the experimental results using three different methods to evaluate our calibration method. First, we assessed the accuracy of the calibration method using a variety of data range conditions applied to the calibration algorithm. Calibrations were performed using different ranges of sinogram data, and the results are shown in Figure 8. The results show that a wider range of sinogram angles could guarantee improved accuracy compared to a narrower range of data or a single image from one angle. Although a broader range of data yields higher accuracy, it also necessitates  
Figure 10: Graph of the projection mean errors according to the angle range with an angle interval of 0.4 degrees; the x-axis represents the rotation angle of each slice in the sinogram, and the y-axis represents the projection mean error of the eight corner points [mm]. Figure 9: Graph of mean errors (orange bars) and computational time (blue plot) according to the different angular intervals applied to the calibration algorithm with the same range of the applied data; the angular interval ranges from 0.4 degrees to 3.2 degrees; the x-axis represents the angular interval of the image used in the calculation. The y-axis on the left gives the error, and that on the right gives the computational time. more computational time. As shown in the blue plot of Figure 8, by increasing the range of the data, the computational time also increases significantly. However, the accuracy with different data ranges shows that there are no significant differences in accuracy if the data range is wider than \(-40\) to \(40\) degrees. Therefore, when considering the required computational time, the range between \(-40\) and \(40\) degrees is an optimized range of data for calibration. Second, we assessed how much the data interval affected the accuracy. The calibration algorithm's accuracy was tested with intervals ranging from \(0.4\) degrees to \(3.2\) degrees over the same angular range. Figure 9 depicts the evaluation results, which show that the interval of the sinogram data applied to the algorithm affects the accuracy less than the range of the sinogram data. As shown in Figure 9, the results show no trend of decreasing accuracy as the data interval is increased. The computational time, on the other hand, can be reduced by reducing the amount of data and increasing the interval of the data using the same angular range. Although the data size is reduced linearly by increasing the interval of the data linearly, the computational time is not reduced linearly. This is because the data loading time is involved in the computational time. By reducing the data size (by increasing the interval), the percentage of the loading time relative to the entire computational time is increased. Consequently, the computational time can be reduced by selecting a larger interval, as shown in Figure 9. Given the accuracy and computational time constraints, an interval of \(1.6\) degrees can be considered an optimized interval for the data used by the calibration algorithm. Finally, the robustness of the algorithm was evaluated based on the results with various data ranges by projecting 3D fiducial markers of the phantom to each projectional image of the sinogram and comparing the error between the projected coordinates and the center of the fiducial marker region in each image slice of the sinogram. As shown in Figure 10, the condition that applies a wider range of sinogram data to the algorithm outperforms that with a narrower range. Furthermore, the shape of the plot is similar to the parabolic shape of the quadratic polynomial with a minimum value near \(0\), and errors increase as the rotation angle of the C-arm gantry increases in both directions within the range. The accuracy of the calibration presented errors below \(1\) mm within the method's range, whereas the errors are much larger outside of the range, particularly for the ranges of \(-20\) to \(20\) degrees and \(-30\) to \(30\) degrees.