## 3 Results
### Imaging Performance with Limited Data
#### 3.2.2 Reconstructions of Small Target at an Offset Height Using a Headband
Results from 3-D and 2-D reconstructions at 1.0 GHz are shown in Figures 17 and 18 for Case III.2. Note that we have limited the colorbar's range of values in these images  
Figure 16: 3-D (**a**–**c**) and 2-D (**d**–**f**) reconstructions of Case III.1 at 1.0 GHz: (**a**) y-z slice, (**b**) x-z slice, and (**c**) x-y slice of 3-D reconstructions. 2-D slice reconstructions (**d**) bottom ring, (**e**) top ring, and (**f**) both rings. The headband of Figure 14 is used to obtain the data. Figure 15: True values of \(\Re(\varepsilon_{r})\) of Case III at 1.0 GHz: (**a**) y-z slice, (**b**) x-z slice, and (**c**) x-y slice. to increase the contrast. The 3-D reconstructions show that the target's location and size is detected accurately with the use of the absorbers. The 2-D results, however, fail to distinguish the target from the noise. Similarly, 3-D and 2-D reconstruction results for Case III.3 are presented in Figure 19 and Figure 20. Detection is improved from Case III.1, as the distance from the interface has been increased, but is less accurate than Case III.2 where the absorbing material was added. Similar to the other two cases, the 2-D imaging results in Figure 20 suggest that the algorithm fails to detect the target in all cases. The \(RMSE\), \(RMSE\)-\(T\), \(e_{max}\) and \(RE\) of Case III.1, 2 and 3 are shown in Table 5, where the 3-D reconstructions have a lower \(RMSE\)-\(T\) for all the three cases. Figure 19: 3-D reconstructions of \(\Re(\epsilon_{r})\) for Case III.3 at 1.0 GHz: (**a**) y-z slice, (**b**) x-z slice, and (**c**) x-y slice. The headband of Figure 14 (with an increased distance between the array and the interface with air) is used to obtain the data. Figure 17: 3-D reconstructions of \(\Re(\epsilon_{r})\) for Case III.2 at 1.0 GHz: (**a**) y-z slice, (**b**) x-z slice, and (**c**) x-y slice. The headband of Figure 14 (with the addition of absorbing material) is used to obtain the data. Figure 14.0 2-D Reconstructions of \(\Re(\epsilon_{r})\) of Case III.2 for the x-y slice at 1.0 GHz using data from the: (**a**) bottom ring, (**b**) top ring, and (**c**) both rings. The headband of Figure 14 (with the addition of absorbing material) is used to obtain the data. Reconstructions of other cases with the vectorial Green's function (13) have also been performed and Case III.2 is shown in Figure 14.1 The relative errors are \(RMSE=2.37\), \(RMSE-T=20.89\), \(e_{max}=21.08\), \(RE=0.62\), which again shows similar reconstruction quality with the simplified Green's function even when the antenna array is placed at an offset height. This comparison confirms that the simplified Green's function does not affect the 3-D reconstruction quality significantly, leading to equally accurate results for the scenarios considered in this paper. Finally, we note that we have also performed reconstructions for additional scenarios related to Cases III.1-3, i.e., headbands with radii \(\rho=135\) and \(140\) mm, with and without the absorbing layers. As expected, when the interface distance increases, detection accuracy and reconstruction quality improves. Moreover, the absorbing materials improve reconstructions further. Importantly, the 3-D imaging outperforms its 2-D counterpart in all these cases. Figure 14.2 3-D reconstructions of \(\Re(\epsilon_{r})\) for Case III.2 at 1.0 GHz: (a) y-z slice, (b) x-z slice, and (c) x-y slice, same as Figure 17 but with vectorial Green’s functions in (13).