# Unmanned Aerial Vehicle (UAV) Robot Microwave Imaging Based on Multi-Path Scattering Model
## 3 Imaging Method  
In this study, the passive imaging of the UAV robot using the satellite communication systems is developed from the field imaging algorithm [18]. The basic idea of this algorithm is to obtain the target equation from the measurement field through the inverse Fourier transform. The target equation is a function of the target position. To realize the target imaging, the scattering fields from multiple observing angles and frequencies need to be measured.  
In this study, both transmitter and receiver work in \"work-stop-work\" mode which is applied to observe the vegetation scene as shown in Figure 2, and the field imaging algorithm suitable for this observing system is adopted to process the data. The UAV and satellite move in two straight lines parallel to the edge of the target area, respectively. The received signal of the UAV robot can be expressed as  
\[E\left(\vec{k}\right)=\sum_{n=1}^{N}e^{\vec{k}\cdot\vec{r}_{n}}F_{n}\left( \vec{k}\right), \tag{12}\]  
where \(\vec{k}=\vec{k}_{i}-\vec{k}_{s}\). \(\vec{k}_{i}\) and \(\vec{k}_{s}\) are the wave vectors of incident and scattered wave. \(N\) denotes the total number of scatterers. \(F_{n}\left(\vec{k}\right)\) is the scattering amplitude of the nth scatterer. It is known that the received signal is an equation about the wave number and the receiving antenna position. Here, the orbit altitude of the transmitter of the transmitting radar remains unchanged.  
In the actual calculation of the target function \(F_{n}\left(\vec{k}\right)\), the received signal \(E\left(\vec{k}\right)\) is not directly transformed by the inverse Fourier transform but by the conjugate multiplication and integration of received signal and the reference signal. The target of this study is virtual vegetation. The echo signal comes from the coherent scattering model of vegetation. The field imaging algorithm is to conjugate and multiply the echo signal at frequency \(k_{m}\) and position \(x_{n}\) with the reference signal, and then add and sum the echo signal at all frequencies and receiving antenna positions. Furthermore, the observation area is divided into \(K\times L\) pixel units according to the requirement of imaging equality as shown in Figure 3. Each unit has a pixel center, which can be employed for the coordinate position of reference phase. The scattering intensity of pixel unit at \(\vec{r}_{0}\) can be expressed as  
\[E_{est}^{s}(\vec{r}_{0})=\frac{1}{N_{k}N_{x}}\sum_{m=1}^{N_{k}}\sum_{n=1}^{N_{ x}}E\left(k_{m},x_{n}^{t},x_{n}^{r}\right)E_{0}(k_{m})^{*}e^{-ik_{m}(|\vec{R}_{t} (x_{n}^{t})-\vec{r}_{0}|+|\overline{R}_{r}(x_{n}^{t})-\vec{r}_{0}|)}, \tag{13}\]  
where \(N_{x}\) and \(N_{k}\) represent the number of spatial sampling points of receiving antenna and discrete frequency points, respectively. The variable \(\overline{R}_{t}(x_{n}^{t})=x_{n}^{t}\hat{x}+y_{n}^{t}\hat{y}+z_{n}^{t}\hat{z}\) and \(\overline{R}_{r}(x_{n}^{r})=x_{n}^{r}\hat{x}+y_{n}^{t}\hat{y}+z_{n}^{r}\hat{z}\) represent the position vectors of transmitting and receiving antennas in the global coordinates. The position of the pixel center (\(k\), \(l\)) is \(\vec{r}_{0}=x_{k}\hat{x}+y_{l}\hat{y}+0\hat{z}\) in the global coordinates. \(E_{0}(k_{m})\) is the reference signal. \(E(k_{m},x_{n}^{t},x_{n}^{r})\) can be calculated by Equation (1). If we chose the different position \(\vec{r}_{0}\) in Figure 3, Equation (13) can then be mapped into the imaging plane to reconstruct the multi-path scattering map.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Start frequency & 6 GHz \\ Frequency bandwidth & 300 MHz \\ Frequency sampling numbers & 50 \\ Flight height of the transmitter & 800 km \\ Flight height of the receiver & 8 km \\ Space sampling interval of transmitter & 500 m \\ Space sampling interval of receiver & 5 m \\ Space sampling numbers & 201 \\ Polarization & HH \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the observation geometry.