# Detection and Tracking of Weak Exoatmospheric Target with Elliptical Hough Transform
## 3 Relationship between Measurements and Elliptical Parameters  
In practice, the measurements are usually available in the radar spherical CS at discrete time instants. Next, we establish the relationship between the radar raw measurements and the above-mentioned elliptical parameters through multiple steps of coordinate transformation. Let \(\mathbf{Z}\triangleq(R,A,E)^{\rm T}\) be the raw radar measurements in the radar-centered spherical CS (range, azimuth, and elevation), and denote \(\mathbf{r}_{\rm ENU}\triangleq\left(x_{\rm ENU},y_{\rm ENU},z_{\rm ENU}\right) ^{\rm T}\) the vector in the radar-centered East-North-Up coordinate system (ENU-CS) [47], then the first-step coordinate transformation is  
\[\left\{ \begin{array}{ll}x_{\rm ENU}=R\cos E\cos A\\ y_{\rm ENU}=R\cos E\sin A\\ z_{\rm ENU}=R\sin E\end{array} \right. \tag{15}\]  
Let \(L\), \(B\), and \(H\) be the known radar longitude, latitude, and altitude, respectively, and denote \(\mathbf{r}_{\rm ECF}=(x_{\rm ECF},y_{\rm ECF},z_{\rm ECF})^{\rm T}\) the vector in the Earth-centered fixed (ECF) CS; then, the coordinate transformation between the ENU-CS and the ECF-CS can be obtained as  
\[\mathbf{r}_{\rm ECF}=\mathbf{T}_{\rm ENU}^{\rm ECF}(\mathbf{r}_{\rm ENU}+ \mathbf{\upxi}_{1}) \tag{16}\]where \(\xi_{1}=(0,0,r_{\rm e}+H)^{\rm T}\) is a known vector. \(\mathbf{T}_{\rm ENU}^{\rm ECF}\) is a \(3\times 3\) orthogonal transformation matrix from the ENU-CS to the ECF-CS, i.e.,  
\[\mathbf{T}_{\rm ENU}^{\rm ECF}=\left[\mathbf{T}_{\rm ECF}^{\rm ENU}\right]^{-1}= \left[\mathbf{T}_{\rm ECF}^{\rm ENU}\right]^{\rm T} \tag{17}\]  
Note that \(\mathbf{T}_{\rm ECF}^{\rm ENU}\) is related to the radar position (height \(H\), longitude \(L\), and latitude \(B\)) and can be expanded by the product of three rotation matrices  
\[ \begin{array}{ll}\mathbf{T}_{\rm ENU}^{\rm ECF}&=\mathbf{R}_{\rm Z}(\pi/2-L) \mathbf{R}_{\rm Y}(-\pi)\mathbf{R}_{\rm X}(\pi/2+B)\\ &=\left[\begin{array}{ccc}-\sin(L)&-\cos(L)\sin(B)&\cos(L)\cos(B)\\ \cos(L)&-\sin(L)\sin(B)&\sin(L)\cos(B)\\ 0&\cos(B)&\sin(B)\end{array}\right]\end{array}  \tag{18}\]  
Note that in Figure 1 the ellipse is with reference to an inertial CS, whereas the measurement is obtained in the non-inertial CS. We should build up the relationship between the non-inertial CS and the inertial CS. The coordinate transformation between the Earth-centered inertial CS (ECI-CS) and the ECF-CS is  
\[\mathbf{r}_{\rm ECI}=\mathbf{T}_{\rm ECF}^{\rm ECI}\mathbf{r}_{\rm ECF}= \mathbf{R}_{\rm Z}(-\omega\,\mathrm{t})\mathbf{r}_{\rm ECF} \tag{19}\]  
where \(\omega=7.292115\) rad/s is the Earth's rotation rate, \(t\) is the time with \(t=0\), indicating the time at thrust-cutoff point A. The expansion of \(\mathbf{T}_{\rm ECF}^{\rm ECI}\) is as  
\[\mathbf{T}_{\rm ECF}^{\rm ECI}=\left[ \begin{array}{ccc}\cos(\omega t)&-\sin (\omega t)&0\\ \sin(\omega t)&\cos(\omega t)&0\\ 0&0&1\end{array} \right] \tag{20}\]  
The coordinate transformation between the ECI-CS and the launching inertial CS (LCI-CS) is  
\[\mathbf{r}_{\rm LCI}=\mathbf{T}_{\rm ECI}^{\rm LCI}(\mathbf{r}_{\rm ECI}-\xi_ {2}) \tag{21}\]  
where \(\xi_{2}\) is the thrust cutoff point described in the ECF-CS, i.e., point A in Figure 1, denoting \(A_{0}\) the launching geodetic azimuth (defined as the angle anticlockwise from the launching direction to the meridian line at the thrust cutoff point), \(L_{\rm C}\) and \(B_{\rm C}\) the longitude and latitude of the thrust cutoff, respectively, then \(\xi_{2}\) has the following expression when restricted to a spherical Earth model  
\[\xi_{2}=\left[ \begin{array}{c}x_{\rm C}\\ y_{\rm C}\\ z_{\rm C}\end{array} \right]=\left[ \begin{array}{c}(r_{\rm e}+H_{\rm C})\cos B _{\rm C}\cos L_{\rm C}\\ (r_{\rm e}+H_{\rm C})\cos B_{\rm C}\sin L_{\rm C}\\ (r_{\rm e}+H_{\rm C})\sin B_{\rm C}\end{array} \right] \tag{22}\]  
The transformation matrix \(\mathbf{T}_{\rm ECI}^{\rm LCI}\) involved in Equation (21) can be expanded as  
\[