# Small-UAV Radar Imaging System Performance with GPS and CDGPS Based Motion Compensation
## 2 Materials and Methods
### Imaging Approach
The time gating procedure selects the interval (along the fast time) of the radargram, where signals scattered from targets of interest occur. This allows a reduction of environmental clutter and noise effects [35]. Herein, we define a suitable time window around the time where reflection of the air-soil interface occurs. The last processing stage is the focusing. In this stage a focused image of the scene under test, as appearing into the vertical imaging domain, is obtained by solving an inverse scattering problem formulated into the frequency domain. Each trace of the radargram is transformed into the frequency domain by means of the Discrete Fourier Transform (DFT) so to provide the input data to the inversion approach. This latter faces the imaging as an inverse scattering problem by adopting an electromagnetic scattering model based on the following assumptions:  
* The antennas have a broad radiation pattern;
* The targets are in the far-field region with respect to the radar antennas;
* A linear model of the scattering phenomenon is assumed, hence the mutual interactions between the targets can be neglected (the Born approximation [36]);
* The time dependence \(e^{i\omega t}\) is assumed, and, for notation simplicity, it is dropped. Figure 3: Signal Processing Strategy. \(K\times H\) scattering matrix related to the linear operator which maps the space of the unknown vector \(\chi\) into the space of data (measured scattered field) \(E_{s}\). The inverse problem defined by Equation (2) is ill posed, thus a regularization scheme needs in order to obtain a stable and robust solution with respect to noise on data [39]:  
\[\widetilde{\chi}=\sum_{n=1}^{T}\frac{1}{\sigma_{n}}<E_{s},u_{n}>v_{n}, \tag{3}\]  
where \(<,>\) denotes the scalar product in the data space, \(T\) is the truncation threshold, \(\sigma_{n}\) is the set of singular values of the matrix \(\mathcal{L}\) ordered in a decreasing way, \(u_{n}\) and \(v_{n}\) are the sets of the left and right singular vectors. The threshold \(T\leq K\) defines the \"degree of regularization\" of the solution and is chosen as a trade-off between accuracy and resolution requirement from one side (which should push to increase the \(T\) value) and solution stability from the other side (which should push to limit the value of \(T\)) [40]. Therefore, the radar image is obtained according to the evaluation of the contrast function in Equation (3). Since the TSVD inversion algorithm is a computational intensive procedure when large (in terms of the probing wavelength) domain are investigated, the Shift and Zoom concept [24] has been implemented in order to speed up the computational time. The Shift and Zoom approach consists in processing data on partially overlapping intervals and combining the images in such a way to get an overall focused image. Specifically, it is schematically represented in Figure 5 and the main steps may be explained as follows:  
* The measurement acquisition line \(\Gamma\) and the survey area \(D\) are divided into V partially overlapping subdomains \(\Gamma_{i}\) and \(\mathrm{D}_{i}\) with \(i=1,2,\cdots,V\);
* For each subdomain \(\mathrm{D}_{i}\), the tomographic reconstruction \(\widetilde{\chi_{i}}\) is obtained by the TSVD inversion scheme indicated in Equation (3);
* The tomographic image of the overall surveyed area \(D\) can be obtained by combining the \(V\) reconstructions \(\widetilde{\chi_{i}}\) achieved for each subdomain \(D_{i}\). A detailed description of the Shift and Zoom implementation is in [27]. Thanks to MoCo, the relative distance between the radar acquisition measurements and the pixel belonging to each subdomain are equivalent for all subdomain. In this way, the SVD calculation of the matrix \(\mathcal{L}\) have to be evaluated just in a single shot for the first subdomain and the inversion for each subdomain mainly involve matrix times data vector multiplications. By doing so, the computational  
Figure 5: Shift and Zoom Algorithm description. time for the overall reconstruction process decreases drastically.