## 4 Numerical Results  
The numerical examples refer to a rectangular investigation domain \(\Omega\), as shown in Figure 1, whose parameters are reported in Tables 1 and 2. The measurement line is centered with respect to the dielectric rectangle. The data were simulated using the same model developed to perform the inversion and then corrupted by additive Gaussian noise. The signal-to-noise ratio \(SNR=20\) dB has been adopted. First, to appreciate the role of the measurement configuration, the behavior of the singular values for different values of the geometrical and measurement configuration parameters is shown. Two different bandwidths and two different distances between the measurement line and dielectric have been considered. Note that Table 1 refers to the frequency \(f\) related to the wavenumber \(k=2\pi f/c_{0}\) where \(c_{0}=3\cdot 10^{8}\) m/s. Matrix \(\mathbf{G}\) and its TSVD were calculated for the geometry and parameters defined in Tables 1 and 2. The singular values for the two configurations specified in Table 1 are shown in Figure 3, where the blue line refers to configuration 1, the green line to configuration 2, and the behavior of the singular values obtained when \(f_{max}=5\) GHz and \(D=0.10\) m is shown by the red line. It can be seen that a wider bandwidth and the enlargement of the angle of view (obtained by moving the measurement line closer to the dielectric while keeping its length fixed) provide better behavior of the singular values, which decay after a larger index. This result is expected and is well known. In fact, in the presence of noise in the data, the truncation of the singular values below \(-SNR\) is mandatory to obtain stable inversion [25]. \begin{table}
\begin{tabular}{l l l l} \hline
**Quantity** & **Symbol** & **Config. 1** & **Config. 2** \\ \hline Measurement line distance from & & & \\ the dielectric & & & \\ Measurement line extent & \(L\) & & 200 cm & 200 cm \\ Number of observation points & \(N_{o}\) & 101 & 101 \\ Minimum frequency & \(f_{min}\) (\(k_{min}\)) & 4 GHz (83.78 m\({}^{-1}\)) & 4 GHz (83.78 m\({}^{-1}\)) \\ (vacuum) & & & \\ Maximum frequency & \(f_{max}\) (\(k_{max}\)) & 7 GHz (146.61 m\({}^{-1}\)) & 5 GHz (104.72 m\({}^{-1}\)) \\ (vacuum) & & & \\ Number of frequencies & \(N_{k}\) & 101 & 101 \\ \hline \end{tabular}
\end{table} 
Table 1: Measurement configurations (see Figure 1). \begin{table}
\begin{tabular}{l l l} \hline
**Quantity** & **Symbol** & **Value** \\ \hline \(\Omega\) size along \(x\) & \(W\) & 0.40 m \\ \(\Omega\) size along \(y\) & \(P\) & 0.25 m \\ Dielectric contrast & \(\chi\) & 3 \\ bars\({}^{\prime}\) conductivity & \(\sigma\) & \(10^{7}\) S/m \\ bars\({}^{\prime}\) radius & \(a\) & \(8\cdot 10^{-3}\) m \\ bars\({}^{\prime}\)_candidate_ number & \(N_{b}\) & 294 (see Figure 4) \\ \hline \end{tabular}
\end{table} 
Table 2: Bar and dielectric parameters. To illustrate how the algorithm works, four deployments of the bars have been considered among the \(N_{b}=294\) positions depicted in Figure 4. The different deployments, shown in Figures 5-8 represent cases in which the bars are regularly spaced and cases in which some bars are out of place or completely lacking. Based on the noise level of the data, a truncation value of \(-20\) dB was adopted in the TSVD procedure. Concerning Figure 3, this truncation level corresponds to retaining the first 112 singular values in the \"worst\" configuration (green line) and the first 241 in the \"best\" one (blue line). It should be emphasized that the number of singular values above the threshold does not depend on the _actual_ deployment of the bars inside the structure. The corresponding results in terms of detection and localization are shown in Figures 5-8. In all figures, the bars are numbered from the bottom left to the top right. Each figure shows a pictorial view of the reconstruction, obtained after thresholding, at the top, and the reconstructed function \(\gamma(n)\) at the bottom.