# Detection and Tracking of Weak Exoatmospheric Target with Elliptical Hough Transform
## 3 Relationship between Measurements and Elliptical Parameters
\begin{array}{ll}\mathbf{T}_{\rm ECI}^{\rm LCI}&=\mathbf{R}_{\rm Y}(-\pi/2- A_{0})\mathbf{R}_{\rm X}(B_{\rm C})\mathbf{R}_{\rm Z}(L_{\rm C}-\pi/2)\\ &=\left[\begin{array}{ccc}-\sin(A_{0})\sin(L_{\rm C})&0&0\\ 0&\cos(B_{\rm C})\sin(L_{\rm C})&0\\ 0&0&-\cos(B_{\rm C})\sin(A_{0})\end{array}\right]\end{array}  \tag{23}\]  
The last step of coordinate transformation from the LCI-CS to the THIS-CS defined in Figure 1 is  
\[\mathbf{r}_{\rm THIS}=\mathbf{T}_{\rm LCI}^{\rm THIS}(\mathbf{r}_{\rm LCI}+ \xi_{3})=\mathbf{R}_{\rm Z}(-\theta/2)(\mathbf{r}_{\rm LCI}+\xi_{3}) \tag{24}\]  
where \(\xi_{3}=(0,r_{\rm e}+H_{\rm C},0)^{\rm T}\). Note that the difference between the LCI-CS and THIS-CS is the translation of the origin by range \(r_{\rm e}+H_{\rm C}\) and the rotation of the CS by angle \(\theta/2\). Figure 2 gives the flow chart of the above-mentioned coordinate transformations, which is relatively complicated. The rudimentary operations are translations and rotations. make an exhaustive search. Since higher search dimensionality (7-D) requires much more processing load and data storage, and it is not easy to implement in real time, one must resort to some heuristic approaches to reduce the search dimensionality in the first place. These parameters can be divided roughly into two groups: elliptical parameters \(a,b,\theta\) and reference parameters \(A_{0},L_{C},B_{C},H_{C}\). Elliptical parameters are those, in essence, related to the thrust-cutoff parameters \(r_{k},V_{k},\theta_{k}\) and are the parameters we must search via a voting scheme in the parameter space, whereas the reference parameters \(A_{0},L_{C},B_{C},H_{C}\) are not essential and can be roughly obtained by an early-warning radar (or by an early-warning satellite) as apriori. If the early-warning information is not available, some heuristic methods can also be developed to estimate these parameters. For instance, \(A_{0}\), in essence, determines that \(z=0\) in the THIS-CS. Considering the factor of noise, we can search \(A_{0}\) by minimizing the sum of the absolute value \(z\). Furthermore, note that the thrust-cutoff point \(\mathrm{A}(L_{C},B_{C},H_{C})\) is only a reference point, which is not unique and dependent on the altitude \(H_{\mathrm{C}}\) at the thrust cutoff point. For a common practice, \(H_{\mathrm{C}}=80\) km is a reasonable reference value. Thus, the final reference parameters might be reduced to two-dimensions, i.e., \(L_{\mathrm{C}}\) and \(B_{\mathrm{C}}\). If the early-warning information is not available, three independent standard polynomial Hough transforms can be used to detect in the separate \(R-t\), \(A-t\), and \(E-t\) planes. The detection results are then combined and transformed to the geodetic CS; the extrapolated value at altitude \(H_{\mathrm{C}}=80\) km is the requisite reference thrust cutoff point. In the following sections, for the sake of conciseness and due to space limitations, we assume that the early-warning information \(\{A_{0},L_{\mathrm{C}},B_{C},H_{\mathrm{C}}\}=\{A_{0},\boldsymbol{\upxi}_{2}\}\) is available, but with some estimation errors. Thus, the main work is focused on the search scheme for the elliptical parameters \(a,b\), and \(\theta\). Actually, according to Equations (13) and (14), it can be seen that \(a,b\), and \(\theta\) are not independent. Known any of the two, then another parameter is obtained. However, it is difficult to use Equations (13) and (14), since in most cases \(\theta\) is usually applied at first. Equations (13) and (14) can be used as constraints to greatly reduce the huge search space.