# Interpolation Methods with Phase Control for Backprojection of Complex-Valued SAR Data +
## 5 Experimentation
### Results  
In Figure 7 is shown comparison of SAR scenes \(h(\xi,\rho)\) of \(251\times 251\) pixels that have been reconstructed from the acquired raw data \(g(\xi,\tau)\). Here, nearest neighbor interpolation algorithm (7) has been used as a part of GBP to reconstruct SAR scenes, and the sampling rate is \(f_{\rm s}=2f_{\rm max}=0.66\,\)THz. Figure 7a, depicts the result, where the raw data has not been filtered and time-gated. The scene contains noise, which is caused by reflections of the surrounding environment, as well as by the absence of the phase control procedure in the nearest neighbor interpolation algorithm. In Figure 7b, the SAR scene has been reconstructed from the raw data, which has been postprocessed, i.e., filtered in the frequency domain and time-gated. It has been observed that the postprocessing procedure allows us to suppress ringing noise in the SAR scene and improves the visibility of the object under test. Nevertheless, the SAR scene in Figure 7b contains distortions, which requires the use of other interpolation methods to suppress them and, furthermore, an investigation of the appropriate sampling rate. Thus, in the rest of the examples, the acquired raw data will be filtered in the frequency domain and time-gated.  
Figure 8 depicts scenes \(h(\xi,\rho)\) of \(251\times 251\) pixels reconstructed at the Nyquist sampling rate via the GBP algorithm (2), into which nearest neighbor (7), linear (13), cubic (23), and sinc (25) interpolation algorithms have been incorporated, respectively. Furthermore, SAR-scenes reconstructed at the Nyquist rate based on conventional linear, cubic, and sinc interpolators are included for comparison; see Figure 8b-d, respectively. Here, the intensity of SAR scenes is normalized with the peak intensity, similarly as in Figure 2. It has been observed that linear, cubic, and sinc interpolators provide an accurate reconstruction of the mannequin head at the Nyquist rate in comparison with corresponding conventional sinc interpolators that do not include the phase-control procedure (6). When nearest neighbor interpolation is used, strong azimuthal distortions occur and the image cannot be reconstructed accurately. To suppress them, the sampling rate has to be increased.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline The highest frequency processed, \(f_{\rm max}\) & 0.33 THz \\ The lowest frequency processed, \(f_{\rm min}\) & 0.22 THz \\ Number of aperture positions, \(N_{\xi}^{\circ}\) & 344 \\ Number of frequency bins, \(N_{f}\) & 3001 \\ Aperture step, \(\Delta_{\xi}\) & 1 mm \\ Integration angle, \(\phi_{0}\) & \(\approx\)9.8\({}^{\circ}\) \\ Reference range, \(R_{0}\) & 2 m \\ \hline \hline \end{tabular}
\end{table}
Table 4: Measurement Setup Parameters.  
Figure 7: SAR scenes \(h\) of \(251\times 251\) pixels reconstructed with nearest neighbor interpolation for the sampling rate \(f_{\rm s}=2f_{\rm max}=0.66\,\)THz. Here, the raw data \(g(\xi,\tau)\): (**a**) has not been postprocessed; (**b**) has been filtered in the frequency domain and time gated.  
In Figure 9 is depicted the comparison of SAR scenes \(h(\xi,\rho)\), the reconstruction of which has been based on the incorporation of nearest neighbor and sinc interpolators into the GBP algorithm. Here, the results based on the nearest neighbor approach are given for the rate \(f_{\rm s}=16f_{\rm max}=5.28\,\rm THz\) and provide approximately similar accuracy, as the results obtained via the sinc interpolation at the Nyquist rate \(f_{\rm s}=f_{\rm max}=0.33\,\rm THz\). The comparison demonstrates that to obtain an accurate reconstruction of the object under test, even at the Nyquist rate, it is enough to employ one of the proposed extended interpolation algorithms that contain the phase control procedure (6) into the GBP algorithm (2).