# Dynamic Programming Ring for Point Target Detection
## 4 Simulations and Analysis
### Single-Target Detection Test
When \(SNR<2.0\), the DPR algorithms achieved significantly higher detection performances than their DP counterparts, as shown in Figure 6g-i. The performances of the DPR algorithms regarding the detection probability \(P_{d}\) (Figure 6d-f) can be mainly attributed to the state transition model, target motion speed limit, and the number of image data frames. Similar to the MFF-DP algorithm, the MFF-DPR algorithm had the best detection performance when \(SNR<2.0\). However, under the conditions of \(SNR>2.0\), the detection probability of the CSO-DP algorithm could be higher than that of the MFF-DPR algorithm, especially when the number of data frames is small (\(N=20\)), as shown in Figure 6d. Since the number of single-step state searches determines the upper detection performance limit of the DP/DPR, and this number is approximately proportional to the square  
Figure 5: **(a)** Test case 1; parameters of the straight-line trajectory were as follows: initial coordinates were (5, 10) and motion velocity was (1.2, 1.1) pixel/frame; **(b)** test case 2; parameters of the straight-line trajectory were as follows: initial coordinates were (30, 20) and motion velocity was (0.6, 0.8) pixel/frame; **(c)** test case 3; parameters of the arc trajectory were as follows: motion speed was 1 pixel/frame, arc center coordinates were (64, 64), and arc diameter was 20 pixels. of the motion speed limit \(v_{\text{max}}\) of a target, the detection performance will be low if \(v_{\text{max}}\) is high. When \(v_{\text{max}}=2\) pixel/frame and \(SNR=1.5\), even the MFF-DPR algorithm, which is currently the algorithm with the highest performance, could achieve only a detection probability \(P_{d}\) of less than 40%, as shown in Figure 6d. For the long-distance point target detection, it is necessary to eliminate the influence of the detection platform motion or to improve the image frame rate. The motion speeds of the straight-line and curve trajectories were both \(v_{2}=1\) pixel/ frame. However, compared with the straight-line trajectory, the prediction error of the curve trajectory was larger; therefore, the merit function of the curve trajectory diffused faster. In addition, both the detection probability \(P_{d}\) (Figure 6f) and the detection probability increment \(\Delta P_{d}\) (Figure 6i) of the curve trajectories were lower than those of the straight-line trajectories (Figure 6e,h). As the number of image frames increased, the peak value of \(\Delta P_{d}\) tended to move toward the low end of SNR, as shown in Figure 6g-i. However, with the increase in the number of image frames, the growth rate of detection probability gradually decreased, as shown in Figure 6d-f, indicating that blindly increasing the number of image frames could not significantly improve the detection performance. Therefore, it is necessary to select an appropriate number of image frames according to the target SNR so as to reduce unnecessary calculation while ensuring a reasonably high detection probability. Finally, the single-frame processing time was used as an evaluation measure of the operating efficiency of the algorithms, and the obtained results are given in Table 1. The main affecting factor of the single-frame processing time was the number of single-step state searches, i.e., the length of the velocity search list. As the CFO-DP and DPR are the first-order algorithms, they require fewer single-step state searches and have higher operating efficiency than the CSO-DP and DPR algorithms, as well as the MFF-DP and DPR algorithms. The data structure of DPR defines that DPR algorithms require about twice longer processing time than the corresponding DP algorithms. Among the six algorithms, the MFF-DPR algorithm exhibits good detection performance but requires a long processing time. When the image size was \(128\times 128\) pixels and \(v_{\text{max}}=2\) pixel/frame, the MFF-DPR algorithm needed about 0.5 s to process a single frame, which made it difficult to realize real-time processing.