## 2 Materials and Methods
### The DBIM-FISTA Algorithm
d\mathbf{r}^{\prime}.\]  
where \(J_{x}\), \(J_{y}\) and \(J_{z}\) are the \(x\)-, \(y\)- and \(z\)-components of the source current density. When an antenna due to a single polarization point source current \(I_{z}\) (or \(I_{x}\) or \(I_{y}\)) inside a voxel with length \(\Delta h\) is used,  
\[J_{i}(\mathbf{r})=I_{i}\Delta h\delta(\mathbf{r})\ \text{and}\ \bar{\mathbf{G}}_{ \mathbf{b}}=\{G_{ij}\}, \tag{6}\]  
where \(i,j=x,y\) or \(z\), \(I_{i}\) is the transmitting current at the antenna, and \(\delta(\mathbf{r})\) is the Dirac delta function. In this case, the background dyadic Green's function is linked to the background electric field as [25; 47],  
\[\bar{\mathbf{G}}_{\mathbf{b}}(\mathbf{r}_{0},\mathbf{r})=\frac{j}{\omega_{0}\mu_{0} \Delta h} \begin{bmatrix}\frac{1}{I_{z}}E_{x}(\mathbf{r},\mathbf{r}_{0})&\frac{1}{I_{z }}E_{y}(\mathbf{r},\mathbf{r}_{0})&\frac{1}{I_{z}}E_{z}(\mathbf{r},\mathbf{r}_{0})\\ \frac{1}{I_{z}}E_{x}(\mathbf{r},\mathbf{r}_{0})&\frac{1}{I_{z}}E_{y}(\mathbf{r},\mathbf{r}_{0} )&\frac{1}{I_{z}}E_{z}(\mathbf{r},\mathbf{r}_{0})\end{bmatrix} . \tag{7}\]  
Note that the current terms in (7) correspond to excitation currents for the transmitting antennas. Hence, only the last row of the matrix is non-zero for a \(z\)-polarized antenna, leading to a Green's function that is reduced to [25; 47],  
\[\bar{\mathbf{G}}_{\mathbf{b}}(\mathbf{r}_{0},\mathbf{r})=\frac{j}{\omega_{0}\mu_{0} \Delta h} \begin{bmatrix}0&0&0\\ 0&0&0\\ \frac{1}{I_{z}}E_{x}(\mathbf{r},\mathbf{r}_{0})&\frac{1}{I_{z}}E_{y}(\mathbf{r},\mathbf{r}_{0} )&\frac{1}{I_{z}}E_{z}(\mathbf{r},\mathbf{r}_{0})\end{bmatrix} . \tag{8}\]  
Using (8), the vector product of \(\bar{\mathbf{G}}_{\mathbf{b}}\) and \(\bar{\mathbf{E}}_{\mathbf{b}}\) in (3) is computed as,  
\[\bar{\mathbf{G}}_{\mathbf{b}}(\mathbf{r}_{n},\mathbf{r})\bar{\mathbf{E}}_{\mathbf{b}}( \mathbf{r},\mathbf{r}_{m})=\frac{j}{\omega_{0}\mu_{0}I_{z}\Delta h} \begin{bmatrix}0 \\ 0\\ \sum_{i=x,y,z}E_{i}(\mathbf{r},\mathbf{r}_{n})E_{i}(\mathbf{r},\mathbf{r}_{m})\end{bmatrix} . \tag{9}\]  
As our antenna records the \(z\)-component of the electric field, a scalar approximation of the scattered field  
\[\bar{\mathbf{E}}_{s}=\bar{\mathbf{E}}_{s}\cdot\hat{z}=E_{z}^{s} \tag{10}\]  
in the LHS of (3) can be used [25; 47; 48]. Thus (3) is further reduced to a scalar equation as,  
\[E_{z}^{s}=k_{0}^{2}\Delta h^{3}\sum_{r\in V}[\bar{\mathbf{G}}_{\mathbf{b}}(\mathbf{r }_{n},\mathbf{r})\bar{\mathbf{E}}_{\mathbf{b}}(\mathbf{r},\mathbf{r}_{m})\delta_{\varepsilon} (\mathbf{r})]\cdot\hat{z}, \tag{11}\]  
where the \(z\)-component of product (9) is computed as,  
\[\bar{\mathbf{G}}_{\mathbf{b}}^{z}(\mathbf{r}_{n},\mathbf{r})\bar{\mathbf{E}}_{\mathbf{b}}( \mathbf{r},\mathbf{r}_{m})=\frac{j}{\omega_{0}\mu_{0}I_{z}\Delta h}\sum_{i=x,y,z}E_{i}( \mathbf{r},\mathbf{r}_{n})E_{i}(\mathbf{r},\mathbf{r}_{m}) \tag{12}\]and  
\[\bar{\mathbf{G}}_{\mathbf{b}}^{\mathbf{z}}(\mathbf{r_{n}},\mathbf{r})=\frac{j}{\omega_{0} \mu_{0}\Delta hI_{z}}\big{[}E_{x}(\mathbf{r},\mathbf{r_{n}})\quad E_{y}(\mathbf{r},\mathbf{r_{n} })\quad E_{z}(\mathbf{r},\mathbf{r_{n}})\big{]}. \tag{13}\]  
If we assume that the cross-products of \(x\)- and \(y\)-components are negligible [25] when a \(z\)-polarized antenna is used, we can simplify the Green's function further to account only for the \(z\)-directed components as  
\[G_{b}(\mathbf{r_{n}},\mathbf{r})=\frac{j}{\omega_{0}\mu_{0}I_{z}\Delta h}E_{z}(\mathbf{r}, \mathbf{r_{n}}). \tag{14}\]  
We must emphasize that the simplification of assuming a \(z\)-polarized source is only done by our imaging algorithm, and is not used to generate the data in CST, which models the exact antenna used in our experiments. Moreover, the assumption that only the \(z\)-component of the Green's function is non-zero is employed only in the inversion of the linear matrix at each DBIM iteration. The 3D FDTD forward solver used by the imaging algorithm can calculate all three field components. This allows comparing the \(z\)-only approximation of the Green's function with the vector formulation in (13).