## 5 Numerical Examples
One may also note that \(I(x,y)\) fails to reproduce the size (even only roughly) of the targets. This could be expected by its very definition. Indeed, \(I(x,y)\) is an indicator function which should reveal whether there is a target in the scene. We now take a little step ahead and consider a more involved scattering scenario. To this end, we consider the two metallic plates being of different sizes and orientations and located at different  
Figure 8: Normalized (linear scale) multi-frequency reconstruction for the same case as Figure 7 using the same under-sampled observations but five frequencies \((0.6694,0.8182,1.0000,1.2222,1.4938)\) GHz. Figure 7: Normalized linear scale) single-frequency reconstruction for the same case as Figure 6 but with under-sampled data taken at only \(4\times 5\) observation points at \(f=1\) GHz. depths. The first target is a rectangular plate of \(0.30\times 0.15\) m\({}^{2}\) located at \((0.6,0.1,-0.1)\) m. The second one is located at \((-0.5,-0.1,-0.3)\) m, \(0.45\times 0.30\) m\({}^{2}\) in size and rotated of \(-\pi/4\). Figures 9-12 show the corresponding multi-frequency reconstructions for the same measurement configuration as Figure 8, except for \(z_{p}\) that is varied from 0 m to 0.3 m. All the figures appear rather weakly corrupted by aliasing artifacts. However, the two targets are not always detected. For example, by looking at Figure 9 (i.e., the one obtained by setting \(z_{p}=0\) m) only the target located at \(z=-0.1\) m is detected and correctly located. This is because this target is closer to the reference plane located at \(z_{p}\). However, by increasing the depth of the reference plane the second target starts to be detect (see Figure 10 where \(z_{p}=-0.1\) cm) and becomes clearly detected and well localized when \(z_{p}\) reaches \(-0.3\) m. By contrast, when \(z_{p}\) increases, the first target becomes undetected. This example is quite interesting since it proves that targets located at different depth can be actually detected by changing the reference depth \(z_{p}\). What is more, as described in the previous section, this can be achieved with a minor processing cost. Figure 10: The same as Figure 9 but with \(z_{p}=-0.1\) m.  
Figure 9: Normalized (linear scale) multi-frequency reconstruction of two metallic plates located at different depths and having different sizes and orientations. The parameter of configuration are the same as Figure 8. Black solid line denotes actual plates’ shape and position. As a further example, we reconsider the two targets of the latter case embedded within a half-space having relative dielectric permittivity 4 and conductivity 0.02 S/m. In this case, the scattered field data have been generated by a commercial forward solver whereas the parameters of the configuration are maintained the same as in the previous examples. Some reconstructions corresponding to this case are reported in Figures 13 and 14. As can be seen, the two targets are detected and well localized transversely. As to the depth, the same behaviour as above is observed, that is, the reconstruction is stronger for that target which is _closer_ to the reference plane. Of course, since the half-space medium features are generally unknown, the estimated depths will not coincide with the actual ones. Figure 11: The same as Figure 9 but with \(z_{p}=-0.2\) m.  
Figure 12: The same as Figure 9 but with \(z_{p}=-0.3\) m.  
The concluding example is shown in Figures 15 and 16. In particular, in order to check the method against targets which are not plates, in those figures cylindrical targets have been considered. The first target is a cylinder having radius of 0.085 m, length of 0.40 m and is centered at \((-0.35,-0.35,-0.2)\) m; the second cylinder has radius of 0.075 m, length of 0.30 m and is centered at \((0.3,0.3,-0.1)\) m. The location and orientation of the target is reported in Figure 7.0 As can be seen, despite the very few employed measurements, in this case, the method clearly detects the two targets.