## 3 Results and Discussions
For both regions the averaged pixel intensity was calculated and the ratio \(R\) was chosen as a quantitative indicator of the goodness of the target localization:  
\[R=\frac{\langle V_{In}\rangle}{\langle V_{Out}\rangle} \tag{9}\]  
where \(\langle V\rangle\) represents the pixel intensity averaged on all the region \(V\). Ideally, the confocal map should not show any signal intensity apart from the expected location of the target giving an \(R\) value approaching infinity as \(\langle V\rangle\ \approx 0\). In the case of a very poor localization, instead, a considerable amount of signal appears also at positions not coinciding with the target position. In which case \(\langle V_{in}\rangle\) is close to zero and \(\langle V_{out}\rangle\) is a large number, bringing the ratio \(R\) close to zero. The ratio \(R\) was calculated for all the confocal maps performed. There were three antennas grids distributions (\(3\times 3\), \(4\times 4\), and \(5\times 5\), i.e., with nine antennas, 16 antennas, and 25 antennas) and for every distribution the central frequency and the depth of the target was changed, between 500 MHz and 5 GHz and between 1 and 6 cm, respectively. This accounted for 180 confocal images. The ratio \(R\) for all these images is displayed as contour maps in Figure 8 separated by the three antenna distributions. The abscissa corresponds to the depth of the target, while the ordinate corresponds to the central frequency of the pulse reported in log scale. The dark part of the contour map represents a combination of pulse frequency and target depth which gave a low ratio \(R\) and therefore a poor localization of the target in those conditions. On the other side, the bright part of the contour map represents the conditions necessary for a good localization. There is a large variation in the accuracy in the localization of the pulmonary nodule by microwave imaging as given by the large variation in \(R\) which varied between some decimals and few decades as a maximum value. In particular, the \(3\times 3\), with a value of 20, shows the lowest maximum value, while the \(5\times 5\) shows the highest. In general, an \(R\) value above 15 represented a good localization and a value of circa 10-15 a mediocre localization. The larger values in the \(5\times 5\) antenna grid were in agreement with the larger number of antennas in this configuration, which enhanced the localization and therefore suppressed the background. All in all, the best localization appeared at the lowest frequencies and for the deepest targets. However, there are some differences between the antenna distributions. For the \(3\times 3\), (Figure 8c), the highest values of \(R\) were between 0.5 and 0.8 GHz and 5 to 6 cm of depth. Similar for \(5\times 5\), however with higher values of \(R\). For the \(4\times 4\), instead, the highest \(R\) were at 0.9 GHz. This can be explained by the fact that in the \(4\times 4\) the ring artifacts were slightly lower than with the \(3\times 3\) and \(5\times 5\) grids because there was no central antenna located exactly above the target, which were those producing the strongest rings artifacts as previously explained. In all cases, the localization was very poor for shallow targets, below 3 cm, and high frequencies, above 2 GHz. Only exception the confocal map at 0.5 GHz, 1 cm target depth, and \(5\times 5\), which shows an \(R\) value circa 20. This particular map was affected by intense ring artifacts localized around the target location which produced an extremely high value of \(\langle V_{In}\rangle\) compared to \(\langle V_{Out}\rangle\). Clearly, the most important factor in the reconstruction of meaningful confocal maps was the attenuation of the electromagnetic waves at the highest frequencies and the strong interaction between the target and the antennas at the lowest depths. Whereas the number of antennas played a secondary role in improving an already good localization, but did not change either the frequency nor the target depth at which the localization worked.