# Dynamic Programming Ring for Point Target Detection
## 4 Simulations and Analysis
### Multi-Target Detection Test  
In the multiple point targets detection test, the size of the simulation images was \(128\times 128\) pixels, the sequence length was 100, and the background noise was \(n_{p}^{(t)}\sim N(0,1)\). In this test, three types of test cases were used, as shown in Figure 7. In each type of test case, the SNR of point target was in the range of 1.5-3.0, and 1000 simulations were performed for each data group. After the target trajectories were generated using the DP/DPR point target detection algorithms described in Section 3.3, the multi-target detection performance of each algorithm was evaluated using the detection probability given by Equation (15) and false alarm rate that is given by:  
\[P_{f}=\frac{\text{Number of non $-$ target ponts in all trajectories}}{\text{Number of pixels in image sequence}}. \tag{16}\]  
First, a few common features of the proposed multi-target detection algorithm were analyzed. Because only one coordinate point was extracted from each frame image in each round, and only one trajectory was retained in each round, the correlation complexity of the generated trajectory was significantly reduced; the algorithm complexity was \(O(N_{\text{tar}})\). As the relative amplitude of the multi-target merit function (Figure 8a) can vary, and when the trajectories of multiple targets intersect, it is difficult to extract a complete target trajectory after one round of the DP/DPR algorithm. However, since median filtering was used to replace the observation data corresponding to the trajectory segment, the trajectory segment could not be extracted repeatedly. In theory, a complete target trajectory can be pieced together after multiple rounds of trajectory detection. In the case of straight line 1, the complete trajectory segment was obtained after three rounds of trajectory detection, as shown in Figure 8b. Target identification errors caused by trajectory intersection could be reduced by trajectory regularization, as shown in Figure 8c. Next, the performances of six DP/DPR detection algorithms in multi-target detection were tested. The multi-target detection performances of the three DP algorithms in the descending order were as follows: MFF-DP, CSO-DP, and CFO-DP, as shown in Figure 9a-c, and for the three DPR algorithms, the descending order was MFF-DPR, CSO-DPR, and CFO-DPR, as shown in Figure 9d-f. Compared with the DP algorithms, the DPR algorithms had significantly higher multi-target detection performance, as shown in Figure 9g-i. Figure 7: **(a)** Test case 1; **(b)** test case 2; **(c)** test case 3. Straight-line trajectory 1 is marked in red, and its initial coordinates were (24,34) and its motion velocity was (  
The detection performance of the multi-target detection algorithms was lower compared to that of the single-target detection algorithms (Figure 6), especially the DP algorithms. The two main reasons for this result were identified through analysis, and they are as follows. First, in the single-target detection algorithms, the only required task after the target points have been obtained is to determine whether the target points are correct. In contrast, in multi-target detection algorithms, it is needed to consider targets with trajectory correlation, but when trajectories are discontinuous, only some trajectory segments can be identified. Second, the mutual influence between multiple target trajectories can affect the detection result. With the increase in SNR, the false alarm probabilities of the six multi-target detection algorithms exhibited a downward trend, as shown in Figure 10. Moreover, a comparison of results presented in Figure 10a-c reveals that the false alarm probabilities exhibited an upward trend as the number of target trajectories increased. Under \(SNR>1.8\), the false alarm probabilities of the DPR multi-target detection algorithms were lower than those of DP multi-target detection algorithms.