## 3 Results
### Validation of the Proposed DBIM-FISTA Algorithm
#### 3.1.1 Reconstructions with CST Data  
\"Case I\" considers the numerical model in Figure 2 immersed inside \"infinite\" 90-10% glycerol-water mixture. A cylindrical blood target centered at \(O_{tg}=(20\) mm, \(20\) mm) with \(\varepsilon_{r}=61.1-28.4j\), radius \(\rho=15\) mm and height \(h=30\) mm is inserted into the [PERSON] head phantom. Eight antennas are placed in an elliptical array configuration with semi-major and semi-minor axes equal to \(100\) mm and \(85\) mm, respectively. Two CST simulations are performed to obtain the scattered field data, with and without the target, to which we refer as WT and NT, respectively. The initial guess of FDTD model for reconstruction is chosen to be similar to the NT case CST model. The reconstruction area inside the brain is a cubical volume with height along the \(Z\)-axis between [10; 70] mm for the 3-D model. The 3-D reconstruction results using the simplified Green's function are shown in Figure 5, where the top shows the reconstructed relative permittivity \(\Re(\varepsilon_{r})\) and the bottom shows the contrast \(\Re(\delta_{\varepsilon})\) due to the target. We have also performed reconstructions using the vectorial Green's function (13), and the reconstruction results of \(\Re(\delta_{\varepsilon})\) are shown in Figure 6 with errors \(RMSE=3.65\), \(e_{max}=20.42\), \(RE=0.27\), which are close to the reconstructed values using the simplified Green's function shown in the first row of Table 3.  
Figure 4: Dielectric constant distribution \(\Re(\varepsilon_{r})\) for our validation models. **Top**: (**a**) y-z slice, (**b**) x-z slice, and (**c**) x-y slice for the numerical head model; **Bottom**: (**d**) y-z slice, (**e**) x-z slice, and (**f**) x-y slice for the experimental phantom. This comparison suggests that the z-only approximation for the Green's function does not affect the accuracy of the results. We thus argue that this formulation is sufficient to achieve similar accuracy with the more complete vector formulation of (13). To re-enforce  
Figure 5: 3-D reconstructions with simplified Green’s functions for “Case I” of Figure 2 at 1.0 GHz. (**a–c**) Relative permittivity \(\Re(\epsilon_{r})\), and (**d–f**) contrast \(\Re(\delta_{\epsilon})\) for the y-z, x-z, and x-y slices, respectively. A cylindrical eight-antenna array centered at the target height is used to produce the data. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Simulation** & **Iteration** & \(RMSE\) & \(\mathbf{e_{max}}\) & \(\mathbf{RE}\) \\
**Name** & & & & \\ \hline Case I, 3-D & 20 & 3.39 & 19.38 & 0.28 \\ Case I, 2-D & 20 & 4.11 & 20.46 & 0.72 \\ Case I, 3-D & 120 & 3.21 & 19.42 & 0.01 \\ Case I, 2-D & 200 & 3.71 & 19.03 & 0.07 \\ Exp, 3-D & 20 & 9.73 & 48.87 & 0.17 \\ Exp, 2-D & 20 & 7.31 & 36.23 & 0.73 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reconstruction errors for the cases considered in Section 3.1.1. Figure 6: 3-D reconstructions, same as Figure 5 but with vectorial Green’s function in (13), for “Case I” of Figure 2 at 1.0 GHz. (**a–c**) Relative permittivity \(\Re(\epsilon_{r})\) for the y-z, x-z, and x-y slices, respectively. A cylindrical eight-antenna array centered at the target height is used to produce the data. this argument, we have performed another comparison of the two formulations for a more challenging imaging scenario in Section 3.2. The target is also detected in 2-D as shown in Figure 7, but the reconstructed values are lower. Inaccuracies in these 2-D and 3-D results can be attributed to the complex brain structure which leads to a highly non-linear scattering problem, and to the mismatch between the CST model producing the data and the FDTD forward model of our DBIM-FISTA algorithm. For example, our FDTD solver models the antennas as point sources to avoid the additional computational complexity and much finer resolution required for modeling the full antenna structure, which leads to a model mismatch from the full antenna CST model.