# Dynamic Programming Ring for Point Target Detection
## 4 Simulations and Analysis
### Single-Target Detection Test  
The images used in the simulation had a size of \(128\times 128\) pixels; the maximum sequence length was 100, and the background noise was \(n_{\mathbf{p}}^{(t)}\sim N(0,1)\). Three types of trajectories were tested in the single-point target detection experiment, as shown in Figure 5. The SNRs of the point targets on each trajectory were in the range of 1.5-3.0. For each group of test data, 1000 rounds of simulations were conducted, and the average detection probability was calculated by:  
\[P_{d}=\frac{\text{Number of correctly detected targets}}{\text{Total number of real targets}} \tag{15}\]  
According to the point target detection algorithm described in Section 3.3, only one round of the DP/DPR signal accumulation was needed, and only a single-pixel coordinate needed to be extracted from a single image frame. The single-point target detection process was simplified as follows. Suppose that the maximum value of the merit function \(I_{\mathbf{p}}^{(t)}\) at time \(t\) corresponds to the coordinate of \(\mathbf{p}_{c}^{(t)}=\text{argmax}_{\mathbf{p}\in\Omega}\Big{\{}I_{\mathbf{p}}^{(t)}\Big{\}}\), and the theoretical coordinate of the point target is \(\mathbf{p}_{r}^{(t)}\), if \(\|\mathbf{p}_{c}^{(t)}-\mathbf{p}_{r}^{(t)}\|_{\infty}\leq 1\); then, it can be deemed that the point target has been detected at the time \(t\). To compare the performances of the DP and DPR, we used a concept of detection probability increment \(\Delta P_{d}\), which is defined as the difference in the detection probability value between the DPR and DP (\(\Delta P_{d}=P_{\text{d,DPR}}-P_{\text{d,DP}}\)). Figure 4: (**a**) The two trajectories obtained from the coordinate correlation intersect at time \(t_{c}\); (**b,c**) two possible target trajectories; each color represents a target identification. The performances of the MFF-DP, CFO-DP, and CSO-DP algorithms in detecting a single-point target were tested at first. In the three test cases, different numbers of image frames (20, 50, and 100) were used. The upper limit of the motion speed of a target was \(v_{\text{max}}=2\text{pixel}/\text{frame}\) in test case 1, and 1 pixel/frame in test cases 2 and 3. With the increase in SNR or the number of data frames, the detection performances of the three algorithms improved gradually. The results in Figure 6a,b show that the smaller the value of \(v_{\text{max}}\) was, the smaller the number of state transition searches, the lower the noise interference, and the higher the detection probability was. The results show that the state transition model of the MFF-DP reduced the target state estimation error compared with the first-order DP and also avoided the direct involvement of the observation data in the state transition decision process, which is one of the main drawbacks of the traditional second-order DP, thus reducing the influence of noise on the point target state estimation. When \(SNR<2.0\), the detection performance of the MFF-DP was superior to that of the traditional DPs. The descending order of the algorithms regarding the detection performance was as follows: MFF-DP, CSO-DP, and CFO-DP. Particularly, at \(SNR=1.5\), the highest detection probability of the CFO-DP algorithm was only about 20%, which was far lower than that of the MFF-DP algorithm, which was 45%. The highest detection probability of the CSO-DP algorithm was between those of the two above-mentioned algorithms, and it was about 30%. Above the level of 2.0, the increase in SNR would lead to higher credibility of the original data of a point target, as well as higher efficiency of point target signal accumulation. Particularly, when the number of data frames was very small (\(N=20\)), the detection probability of the CSO-DP algorithm was higher than that of the MFF-DP algorithm, as shown in Figure 6a. Next, the performances of the MFF-DPR, CFO-DP, and CSO-DP algorithms in detecting a single-point target were tested.