## 3 Test with simulation
### Influences of freedom limit  
In order to investigate the influences of the image pattern tracking freedom limit \(L\) on the accumulated bias \(\beta_{c}\) and the position error \(\delta_{s}\), 601 synthetic frames with an image pattern of \(d_{\theta}\)=6 are tracked with 600 different freedom limit, i.e from \(L\)=1 to \(L\)=600 with increasing step of 1. The image pattern in the last (601 st) frame is exactly the same as that in the first frame, and the only difference between the two frames is that the random background noise distributions are not identical. This synthetic image group is evaluated by using the image pattern tracking algorithm described in section 2 with an image pattern window of 32\(\times\)32 pixels. The errors of the image pattern tracking are given in Fig. 5.  
Fig. 5 indicates that in the current test case the RMS position error \(\delta_{s}\) has the minimum (\(\delta_{s}\approx 0.2\) pixels) between \(L\)=1 and \(L\)=_20,_ and it rise up to the highest peak of more than 2 pixels at \(L\)=_30._ After the highest peak the RMS position error is significant large, i.e. \(\delta_{s}>0.4\) pixels. The accumulated bias error \(\beta_{c}\) also has a local minimum in the freedom limit range of 1 to 20, and it goes up to the highest peak of more than 5 pixels at \(L\)=_30._ After the highest peak, the accumulated bias error \(\beta_{c}\) has some  
Figure 5: Errors of image pattern tracking for synthetic image pattern of \(d_{\theta}\)=6 pixels.  
local minima that are much smaller than the minimum before the highest peak. The reason for small \(\beta_{c}\) at high \(L\)-number is that the reference pattern is chosen with a large skip, so that relatively few bias errors are accumulated to the end of the image pattern tracking. It can be seen that the local minima of \(\beta_{c}\) after the first high-peak can be smaller than the first local minimum, e.g. at \(L\)\(\approx\)150, 190, 260 etc. The possible reason is that the rotation and the asymmetric shrink of the image pattern result in both positive and negative biases during the tracking, so that the accumulated bias can occasionally be very small at certain \(L\)-numbers. Fig. 5 confirms that the two first high-peaks for \(\beta_{c}\) and \(\delta_{r}\), respectively, appear at the same \(L\) number, i.e. the maximal position error occurs at the maximal accumulate bias. The purpose of current test is to find a way for determining an optimal tracking freedom limit \(L_{opt}\). Obviously, it is reasonable to determine \(L_{opt}\) at minimal RMS position error. Unfortunately, the RMS position error cannot be quantified in real experiments because the true values are unknown. However, Fig. 5 indicates that the position error \(\delta_{r}\) has a similar distribution to the accumulate bias \(\beta_{c}\) before its first high peak that is also the highest peak of \(\beta_{c}\) among all the possible freedom limits. Since accumulate bias \(\beta_{c}\) can be obtained by tracking real image patterns, the distribution of accumulate bias \(\beta_{c}\) before the first high peak may be used to determine the optimal tracking freedom limit \(L_{opt}\).  
Fig. 6 shows the distributions of accumulate bias \(\beta_{c}\) and position error \(\delta_{r}\) before the first high-peak, i.e. \(L\leq 30\). The dash-dot line indicates the minimal position error in the region of \(L>30\). In the current case the position error at minimal \(\beta_{c}\), i.e. \(L\)=6, is very close to the minimal position error at \(L\)=7, and it is much smaller than the minimal position error in the region of \(L>30\).  
Figure 6: Tracking error distributions before the highest peaks for synthetic image pattern of \(d_{0}\)=6 pixels.