# Detection and Tracking of Weak Exoatmospheric Target with Elliptical Hough Transform
## 4 Elliptical Hough Transform  
### Data Space and Parameter Space  
Note that before transformation Equation (24), all the data expressed in the LCI-CS are explicitly known, except for some measurement errors and early-warning parameter-related errors. However, the transformation Equation (24) is still ambiguous since \(\theta\) is an undetermined parameter, which must be searched over all possible discrete values, together with the search of parameters \(a\) and \(b\).  
In fact, if we denote  
\[\mathbf{r}_{\mathrm{LCI0}}\triangleq\mathbf{r}_{\mathrm{LCI}}+\boldsymbol{ \upxi}_{3}\triangleq(x_{\mathrm{LCI0}},y_{\mathrm{LCI0}},z_{\mathrm{LCI0}})^{ \mathrm{T}} \tag{34}\]  
as a new vector, then the LCI0-CS can be viewed as the LCI-CS, translating from the cutoff point \(\mathrm{A}\) to the Earth center \(\mathrm{O}_{\mathrm{e}}\). The vector \(\mathbf{r}_{\mathrm{LCI0}}\) is also a known quantity related to the measurements.  
Using \(\mathbf{n}_{\mathrm{LCI0}}\) to represent Equation (2) and expanding the rotation matrix \(\mathbf{R}_{Z}(-\theta/2)\), we obtain the three-parameter elliptical equation as  
\[\begin{array}{l}\mathbf{\Gamma}:\;\frac{\left[\cos(\frac{\theta}{2})x_{ \mathrm{ECIO}}-\sin(\frac{\theta}{2})y_{\mathrm{ECIO}}\right]^{2}+\left[\sin( \frac{\theta}{2})x_{\mathrm{ECIO}}+\cos(\frac{\theta}{2})y_{\mathrm{ECIO}} \right]^{2}}{a^{2}}=1\\ \mathrm{s.t.}\left\{\begin{array}{l}\left|\cos(\frac{\theta}{2})x_{\mathrm{ ECI0}}-\sin(\frac{\theta}{2})y_{\mathrm{ECIO}}\right|\leq(r_{\mathrm{e}}+H_{ \mathrm{C}})\sin(\frac{\theta}{2})\\ \sin(\frac{\theta}{2})x_{\mathrm{ECIO}}+\cos(\frac{\theta}{2})y_{\mathrm{ECIO}} \geq 0\\ z_{\mathrm{ECIO}}=0\end{array}\right.\end{array} \tag{35}\]  
Thus, the EHT maps the points \(\mathbf{r}_{\mathrm{LCI0}}\) in the converted data space into curves in a \(a-b-\theta\) three-parameter space. Note that, when discussed here, \((a,b,\theta)\) are the state variables, whereas \((x_{\mathrm{LCI0}},y_{\mathrm{LCI0}},z_{\mathrm{LCI0}})\) are the parameters. According to Equation (35), it is difficult to judge the curve type in the Hough parameter space. Instead, we use the vector in the THIS-CS to analyze its properties. Using the coordinate relationship  
\[\left\{\begin{array}{l}x=\cos(\frac{\theta}{2})x_{\text{ECIO}}-\sin(\frac{\theta }{2})y_{\text{ECIO}}\\ y=\sin(\frac{\theta}{2})x_{\text{ECIO}}+\cos(\frac{\theta}{2})y_{\text{ECIO}}\\ z=z_{\text{ECIO}}\\ \end{array}\right. \tag{36}\]  
to replace Equation (35), it follows from that Equation (35) can be rewritten as  
\[\frac{a^{2}}{y^{2}}+\frac{b^{2}}{x^{2}}-\frac{a^{2}b^{2}}{y^{2}x^{2}}=0,\ \theta\geq 2 \text{arcsin}\bigg{(}\frac{|x|}{r_{\text{e}}+H_{\text{C}}}\bigg{)} \tag{37}\]  
Thus, for a given value \(\theta\), the vector \(\tau_{\text{THIS}}\) in THIS-CS can be explicitly computed by using Equation (36); then, the data mapping Equation (37) results in a planar quartic curve, instead of an elliptic curve.  
It can be seen that any one of the quartic curves in the Hough parameter space corresponds to the set of all possible elliptical curves in the data space in the THIS-CS through the corresponding data point. If an elliptical curve of points does exist in the THIS-CS, the curve is represented in Hough parameter space as the point of intersection of all of the mapped quartic curves, i.e., in the parameter space, multiple quartic curves will gather at a point, and the parameters at this point are the desirable ones we are searching for.