# Detection and Tracking of Weak Exoatmospheric Target with Elliptical Hough Transform
## 4 Elliptical Hough Transform
### Algorithm Design
\begin{array}{l}a(i)=a_{\mathrm{L}}+(i-1/2)\Delta a,\;i=1,\cdots,n_{a }\\ b(j)=b_{\mathrm{L}}+(j-1/2)\Delta b,\;j=1,\cdots,n_{b}\\ \theta(k)=\theta_{\mathrm{L}}+(k-1/2)\Delta\theta,\;k=1,\cdots,n_{\theta} \end{array} \right. \tag{39}\]  
Set the initial accumulation variable as  
\[c(i,j,k)=0,\;1\leq i\leq n_{a},\;1\leq j\leq n_{b},\;1\leq k\leq n_{\theta},\; i,j,k\in\mathbb{N}\]  
_Step 3_: For every \(\theta(k),1\leq k\leq n_{\theta}\) and every data \(\mathbf{Z}\) total number \(m\)), use the apriori information of early warning \(\hat{A}_{0},\hat{L}_{\mathrm{C}},\hat{B}_{\mathrm{C}},\hat{H}_{\mathrm{C}}\) to calculate the converted data \(\mathbf{r}_{\mathrm{THHS}}\) via Equation (25). Thus, we have  
\[\hat{\mathbf{r}}_{\mathrm{THHS}}(l,k)=\left[\hat{x}(l,k),\hat{y}(l,k),\hat{z} (l,k)\right]^{\mathrm{T}},\;1\leq l\leq m;\;1\leq k\leq n_{\theta} \tag{40}\]  
Figure 3: Flow chart of the Elliptical Hough Transform algorithm. _Step 4_: For every \(\hat{\mathbf{r}}_{\text{THHS}}(l,k)\) and every \(a(i)\), compute the semi-minor axis \(b(i,k,l)\)  
\[b(i,k,l)=a(i)\sqrt{\frac{\hat{x}^{2}(l,k)}{a^{2}(i)-\hat{y}^{2}(l,k)}},1\leq i\leq n _{a},1\leq k\leq n_{\theta},\,1\leq l\leq m \tag{41}\]  
_Step 5.1_: **Scheme I (Binary integration)**: For every \(b(i,k,l)\) and every \(b(j)\), if \(b(i,k,l)\) subjects to  
\[ \begin{split}|b(j)-b(i,k,l)|\leq\Delta b/2\\ \left\{\begin{array}{l}1\leq i\leq n_{a}\\ 1\leq j\leq n_{b}\\ 1\leq k\leq n_{\theta}\\ 1\leq l\leq m\end{array}\right.\end{split}  \tag{42}\]  
Then, the count of the accumulation cell \(c(i,j,k)\) is added to 1, i.e.,  
\[c(i,j,k)=c(i,j,k)+1 \tag{43}\]  
_Step 5.2_: **Scheme II (Plane-constrained integration)**: For every \(b(i,k,l)\) and every \(b(j)\), if \(b(i,k,l)\) subjects to Equation (42), then the accumulation steps change to  
\[c(i,j,k)=c(i,j,k)+e^{-q|\hat{z}(l,k)|} \tag{44}\]  
where \(q\) is a design coefficient. _Step 6_: Set the secondary threshold \(\eta^{\text{Hough}}\) in the Hough parameter space, if \(c(i,j,k)\geq\eta^{\text{Hough}}\), we obtain a detected cell, and the estimation parameters are obtained as \(\hat{a}=a(i)\), \(\hat{b}=b(j)\), and \(\hat{\theta}=\theta(k)\). In certain cases, the detections might be multiple and scatter in several adjacent regions. A k-means clustering method or hierarchical agglomerative clustering method [48] can be employed to combine the neighboring detected cells into one cell. _Step 7_: Suppose ultimately that we have detected \(n\) targets with parameters \([\hat{a}(k),\hat{b}(k),\hat{\theta}(k)],1\leq k\leq n\). Using a series of inverse coordinate transformations indexed by Equation (37) back to Equation (15), we can retrieve the positions of targets of interest in any desirable CS. In this way, the presented EHT method declares the detection and tracking results simultaneously. Furthermore, it is also possible to use the estimated parameters to predict the long-term trajectory of the targets to be detected. Note that, in Equation (44), we have considered the fact that the orbit of an exoatmospheric target in the midcourse usually follows a planar trajectory, hence \(\hat{z}(l,k)\) might be sufficiently small if the target turns out to be the true one. However, for random thermal noises or clutter, they would uniformly intersperse in the surveillance space and might have large values for \(\hat{z}(l,k)\). Therefore, for Scheme II, it indeed has the advantage of improving the accumulation performance for true targets, as well as reducing the false-alarm rate for noises. When in the case \(q\to 0\), \(e^{-q|\hat{z}(l,k)|}\to 1\), Scheme II will reduce to Scheme I, whereas when \(q\rightarrow\infty\) we will have \(e^{-q|\hat{z}(k)|}\to 0\). For the latter case, both targets and noises have no accumulation effect. Because remains a design parameter, in this investigation, we find that when \(q=0.01\sim 0.0001\) it might have better accumulation performance. Now we analyze the computational complexity of the EHT. The points in the measurement space correspond to a quadratic curve in the parameter space.