# Detection and Tracking of Weak Exoatmospheric Target with Elliptical Hough Transform
## 4 Elliptical Hough Transform
### Detection and False Alarm Probabilities  
Next, we analyze the statistical performance of the EHT. The thermal noise-induced false alarm is considered as Rayleigh distributed in amplitude or exponential in power. Assuming a square law cell detector, then the normalized power probability of distribution (pdf) of noise is  
\[f(x)=\mathrm{e}^{-x} \tag{46}\]  
For the sake of simplicity, we assume the exoatmospheric target follows a Swerling II distribution. Note that in reality, it is rarely the case that a reentry vehicle follows a classical Swerling model; instead, it depends on a lot of elements. Herein, since the Swerling II model has the simplest analytical expression of the probability of detection \(P_{\mathrm{d}}\), we use it only to give a clear picture of the statistics of the method. Note that any other target fluctuation models are also applicable to this analysis. The power pdf of the target is  
\[f(x|S)=\exp(-x/(1+S))/(1+S) \tag{47}\]  
where \(S\) is the SNR. It is easy to obtain the \(P_{\mathrm{d}}\) for single \(R-A-E-t\) detection cell as  
\[P_{\mathrm{d}}=\int_{\eta}^{\infty}f(x|S)\,\mathrm{d}x=\exp(-\eta/(1+S)) \tag{48}\]  
where \(\eta\triangleq-\ln(P_{\mathrm{fa}})\) is the primary threshold, with \(P_{\mathrm{fa}}\) the probability of false alarms for noises. Obviously, the larger the value of \(P_{\mathrm{fa}}\) is, the lower the threshold \(\eta\) will be. To guarantee the subsequent Hough detection of a target, the primary threshold \(\eta\) should be set relatively lower. To obtain the final probability of detection \(P_{\mathrm{d}}^{\mathrm{Hough}}\) per Hough parameter space cell, we follow the same way originated from [PERSON] [17]. Let \(n\) be the total number of \(R-A-E-t\) cells that can contribute to a given parameter space cell. There are total \(m\) data points exceeding the primary threshold \(\eta\). Thus, the probability of this event is \(P_{\mathrm{d}}^{k}(1-P_{\mathrm{d}})^{mn_{a}n_{\theta}-n}\), where \(mn_{a}n_{\theta}\) is in essence the maximum accessible number for a given Hough parameter cell. According to [17], we directly give the probability \(P_{\mathrm{d}}^{\mathrm{Hough}}\) that a given cell in parameter space exceeds the secondary threshold \(\eta^{\mathrm{Hough}}\) as follows:  
\[ \begin{array}{ll}P_{\mathrm{d}}^{\mathrm{Hough}}&=P\{c>\eta^{\mathrm{Hough }}\}\\ &=\sum\limits_{n=1}^{m_{a}n_{\theta}}C_{mn_{a}n_{\theta}}^{k}P_{\mathrm{d}}^{n} (1-P_{\mathrm{d}})^{mn_{a}n_{\theta}-n}P\{y=\sum\limits_{i=1}^{n}c_{i}>\eta^{ \mathrm{Hough}}\}\end{array}  \tag{49}\]  
where \(c\) denotes the accumulation power for this given parameter cell and \(c_{i}\) is the power of the \(i\)th data point. Actually, for binary integration (Scheme I), \(c_{i}\equiv 1\), \(1\leq n\leq mn_{a}n_{\theta}\)whereas for plane-constrained integration (Scheme II), \(c_{i}=e^{-q|z(i)|}\), \(1\leq n\leq mn_{a}n_{\theta}\). Obviously, for Scheme I, the final probability of detection is reduced to  
\[P_{\text{d}}^{\text{Hough}}=P\{c>\eta^{\text{Hough}}\}=\sum\limits_{n=[\eta^{ \text{Hough}}]}^{mn_{a}n_{\theta}}C_{mn_{a}n_{\theta}}^{n}P_{\text{d}}^{n}(1-P_{ \text{d}})^{mn_{a}n_{\theta}-n} \tag{50}\]  
where \(\lfloor.\rfloor\) denotes the rounding operation. The \(P_{\text{d}}^{\text{Hough}}\) for Scheme II is obtained as  
\[ \begin{array}{ll}P_{\text{d}}^{\text{Hough}}&=P\{c>\eta^{\text{Hough}}\}\\ &=\sum\limits_{n=1}^{m}C_{mn_{a}n_{\theta}}^{n}P_{\text{d}}^{n}(1-P_{\text{d}} )^{mn_{a}n_{\theta}-n}P\{y=\sum\limits_{i=1}^{n}e^{-q|z_{\text{t}}(i)|}>\eta^{ \text{Hough}}\}\end{array}  \tag{51}\]  
Note that it is difficult to obtain the analytical expression \(P\{y=\sum\limits_{i=1}^{k}e^{-q|\tilde{z}_{\text{t}}(i)|}>\eta^{\text{Hough}}\}\) since the accumulation statistic \(y\) has a sophisticated distribution. For the sake of conciseness, we do not go into the details of the detection and false alarm probabilities in this investigation. Once \(P_{\text{d}}^{\text{Hough}}\) is available, the false alarm probability can be obtained by limiting \(S\to 0\).