# Detection and Tracking of Weak Exoatmospheric Target with Elliptical Hough Transform
## 5 Simulations
### Performance Comparison  
Next, we investigate the detection performance due to the influence of the primary threshold, secondary threshold, radar measurement error, and SNR. This is achieved by massive Monte Carlo runs. In the following simulations, the accumulation time interval is still restricted to [150, 250] s. To assess the performance of the proposed method, 10,000 Monte Carlo numerical simulations were performed in Matlab(r) on a computing platform characterized by a 3.6 GHz Intel Core I7-4790 processor with 32 GB RAM. Figure 10 shows the detection performance concerning the primary threshold \(\eta\) and SNR. Figure 11 shows the detection performance concerning the secondary threshold \(\eta^{\rm Hough}\) and radar measurement errors. Figure 8: Estimated parameters of EHT with different secondary thresholds, via the K-means clustering method. (**a**) \(\eta^{\rm Hough}=10\). (**b**) \(\eta^{\rm Hough}=3\). Figure 9: Detection and tracking results of EHT, with different secondary thresholds. (**a**) \(\eta^{\rm Hough}=10\). (**b**) \(\eta^{\rm Hough}=3\). From Figures 10 and 11, some conclusions are obvious. We briefly summarize them as follows:  
1. The larger the SNR is, the higher the detection probability obtained;
2. The lower the primary threshold is, the higher the detection probability obtained;
3. The higher the secondary threshold is, the higher the detection probability obtained;
4. The higher the radar measurement accuracy is, the higher the detection probability obtained. Note that in Figure 11, it does not mean that the higher the secondary threshold, the better. It can be observed that, when \(\eta^{\rm Hough}\) is set too high to cause no secondary threshold crossings, it may result in missing alarms, and consequently decrease the probability of detection. Alternatively, one can set no secondary threshold and assume that the peak value corresponds to the desired target. However, this may also induce false alarms, since even if there is no target, the detector will still declare a target. In fact, the detection performance relates to many elements, such as the primary threshold \(\eta\), secondary threshold \(\eta^{\rm Hough}\), SNR, resolution cell of discretization \(\Delta a\times\Delta b\times\Delta\theta\), total processing time interval \(N\Delta t\), radar measurement errors, and the estimation accuracy of early-warning information. Most of the statistics do not have analytical expressions and need massive Monte Carlo simulations to give a quantitative evaluation. Table 1 shows the average running time of the EHT method with different parameters. The primary false alarm probability is varied in the range \(P_{\rm f}=\{0.01,0.1,1\}\) and the parameter space quantization in the range \(n_{a}n_{b}n_{\theta}=\left\{200^{3},400^{3},600^{3}\right\}\). The target accumulation time interval is still restricted to [150; 250] s. As seen in Table 1, the computational load of the algorithm does increase as the primary detection threshold decreases and the number  
Figure 11: Detection performance related to measurement error and secondary threshold. Figure 10: Detection performance related to SNR and primary threshold. of quantization units increases. Nonetheless, this processing time is still within the total accumulated time. The total running time will also be greatly reduced if a parallel processing strategy and hardware computational platform are used. Next, we investigate the performance in a very low SNR environment. Figure 12 gives the detected curves in the THIS-CS with an average SNR of 0 dB, which means that the primary threshold is very low, and most of the clutter points will contribute to the Hough space accumulation, while that of the target points will be greatly decreased. The other parameters are Scheme II, Plane-constrained integration, with \(q=0.001\), time interval (150-250) s, primary threshold \(\eta=0\), and secondary threshold \(\eta^{\rm Hough}=10\).