## 5 Results and Discussion
### Performance Comparison Based on Aberration-Free OV Beams
To concretely demonstrate these differences, as well as the origin of the differences and why this factor significantly changes the TC determination performance, we present some examples in Figure 7. In this figure, the columns from left to right indicate the results using the WTM, MAICM, WT-MAICM, FC-PRCM, and SC-PRCM, and the rows from top to bottom are the inputted OV beams of TC values of 10 (top row), 15 (middle row), and 20 (bottom row). In Figure 7, each row has the same intensity sum map, and the red circle superimposed on the intensity sum map is the closed path extracted by the individual method. From the figure, it can be seen that the WTM, MAICM, and WT-MAICM can properly extract closed paths passing through the ridge elements, thus obtaining precise TC values. On the other hand, for the FC-PRCM and SC-PRCM, the closed path might go through the low-intensity sum regions where the measured phase slopes are invalid. When the TC value of the OV beam varied from 10 to 15 to 20, the effects of the nonuniformity of the intensity sum map along the azimuthal direction increased. As a result, the closed path deviated more from the ridge elements and went through the low-intensity regions, generating a large error in the measured TC values. As for the SC-PRCM, its performance deteriorated more compared with that of the FC-PRCM. This reveals that simply shifting the center position of the concentric perfectly round circles does not improve, but rather reduces the performance. We believe that this is because shifting the center position of perfectly round circles may deviate the centers even further from the real position of the vortex, and thus they become liable to be affected by the invalid slope data in the low-intensity regions. Considering that we chose the generated perfectly round circle with the maximum absolute calculated TC value as the determined closed path, the SC-PRCM mostly output a measured TC value larger than the theoretical value. Moreover, in combination with the poor performance of the FC-PRCM, we found that the unified closed path determination criterion in the PRCM strategy, which is to select the generated perfectly round circle with the maximum absolute calculated TC value as the determined closed path, is not entirely rational, although it is simple. The above finding was also supported by analyzing the errors of the measured TC values. Figure 8 shows the histogram distributions of the absolute error (AE) of TC measurements, wherein the \(x\)-axis is the interval of the AE, which is the absolute difference between the measured TC value and the theoretical TC value, and the \(y\)-axis is the frequency of the 600 measurements. From the distribution, we can view the striking difference between the performances of the WTM, MAICM, and WT-MAICM, and those of the FC-PRCM and SC-PRCM. Almost all of the AEs of the former three methods are concentrated within the value interval of 0 to 0.2; on the contrary, the AEs are distributed in a partly uniform pattern with one-sixth within the value interval of over 0.5 when using the FC-PRCM, while the majority are in the value interval of over 0.5 when using the SC-PRCM. Figure 6: TC determination performances of five closed path determination methods in terms of the WTCRM (bars) and speed in frames per second (fps) (points). Each bar or point is the statistical average of all 600 measurements under the corresponding closed path determination method. This phenomenon profoundly influences the correct rate of TC determination when we shrink the confidence interval of the well-determined TC measurements. For example, when using AE < 0.3 as a criterion for the well-determined TC measurement definition, the WTCMR values of the WTM, MACIM, and WT-MAICM are still over 99%; however, those of the FC-PRCM and SC-PRCM drop to 63.3% and 19.7%, respectively. Figure 8: Absolute error (AE) histograms of 600 TC measurements separately using WTM, MAICM, WT-MAICM, FC-PRCM, and SC-PRCM.