# Detection and Tracking of Weak Exoatmospheric Target with Elliptical Hough Transform
## 4 Elliptical Hough Transform
### Algorithm Design  
Next, we give full steps of the algorithm design for the presented EHT. The basic idea is illustrated in Figure 3, and it has four main parts. 1. **Primary Threshold Processing**: The radar operates in a TBD mode, searching a limited number of sectors (surveillance region) \(\mathbf{D}\). The search is implemented via a sequential mode, i.e., every resolution cell in the region is scanned only once during a search frame time \(\Delta t\). The radar raw pulse compressed signals (range-azimuth-elevation channels) are through a much lower primary threshold \(\eta\). In this way, we obtain the original measurement points \(\{t,R,A,E\}\), although this dataset contains quite a lot of clutter. 2. **Elliptical Hough Transform**: Firstly, the values of the reference parameters, as well as the rough bound of the elliptic parameters, are obtained from the apriori information. The parameter space is then quantized in the \(a-b-\theta\) dimensions. Any \(R-A-E-t\) cell with a value exceeding \(\eta\) is mapped into parameter space via multiple steps of coordinate transformation, and its power is reset to count 1. When a primary threshold crossing is mapped into the parameter space, its power (note, we use a binary integrator; hence, its power is 1) is added into the \(a-b-\theta\) cells that intersect the corresponding quartic curve in the parameter space. In this way, the accumulator cell at the intersection of multiple quartic curves will have a high value; this is called voting accumulation. 3. **Secondary Threshold Processing**: A secondary threshold can be applied in the parameter space to declare detections of targets of interest. Note that if the secondary threshold is set too low, there might be multiple detection scatters in neighboring parameter cells. In fact, there indeed may be only one target; it is the quantization of parameter space that leads to the false alarm. To reduce false alarms and meanwhile obtain fine estimation accuracy of the desired parameters, a clustering method can serve this purpose. 4. **Inverse Coordinate Transformations**: The detected elliptical parameters \(a-b-\theta\) are back-transformed into the measurement space \(R-A-E-t\) by a series of inverse coordinate transformations, and both detection and tracking results are announced to show the time history and current position of the target. Meanwhile, it is very easy to use the detected parameters to predict the long-term evolutionary behaviors of the trajectory. For instance, the intersection between the detected trajectory and the Earth's surface is the landing point. The detailed algorithm to implement the EHT is listed as follows:  
_Step 1_: Search the surveillance space \(\mathbf{D}\) sequentially and store the \(N\) frame raw measurement data. If the power of one \(R-A-E-t\) cell exceeds the primary threshold \(\eta\), then its position \(\mathbf{Z}\) is recorded. Thus, during the time interval \(N\Delta t\), we obtain the total measurement data \(\mathbf{Z}^{N}\triangleq\left\{\mathbf{Z}_{\mathrm{I:N}}\right\}\), wherein the \(i\)th frame has \(m_{i}\) measurements, and the total number of measurements in the processing period is \(m=\sum\limits_{i=1}^{N}m_{i}\). _Step 2_: Discretize the three-dimensional parameter space \(a\times b\times\theta\in[a_{\mathrm{L}},a_{\mathrm{U}}]\times[b_{\mathrm{L}},b _{\mathrm{U}}]\times[\theta_{\mathrm{L}},\theta_{\mathrm{B}}]\) (the parameter discretization region can be obtained according to the early warning or determined by experience), and obtain \(n_{a}\times n_{b}\times n_{\theta}\) cells. The parameter resolutions are  
\[\left\{ \begin{array}{l}\Delta a=\left(a_{\mathrm{U}}-a_{\mathrm{L}}\right)/ n_{a}\\ \Delta b=\left(b_{\mathrm{U}}-b_{\mathrm{L}}\right)/n_{b}\\ \Delta\theta=\left(\theta_{\mathrm{U}}-\theta_{\mathrm{L}}\right)/n_{\theta} \end{array} \right. \tag{38}\]  
and the parameter values can be obtained as  
\[\left\{