## 5 Results
### Detailed Analysis  
Table 3 analyses the metrics selected in the previous section on five spherical targets of increasing diameter, \(d\in[5.3,20.2]\) mm, in phantoms with increasing volumes of heterogeneous tissues (10%, 20% and 30% fibroglandular content by volume). It is difficult to determine the true average dielectric properties in heterogeneous breast phantoms. Hence, the accuracy, \(\Delta\varepsilon_{r}\), is not shown because the accuracy, \(\Delta\varepsilon_{r}\), is of limited value when the true average dielectric properties are not known exactly. In the heterogeneous breast phantom (rows 1-5) with 10% glandular structures by volume, the Gaussian Energy, \(\phi_{\rm GSS}^{\rm G}\), and the Central Moment, \(\phi_{\rm ACM}^{\rm S}\), perform very similarly, selecting an average dielectric properties value of within \(\Delta\varepsilon_{r}=0.25\) for all five tumour models. The SCR for the chosen images for all metrics in this case are within 0.3 dB of each other. The Modified Laplacian, \(\phi_{\rm M}^{\rm L}\), selects almost the same images except for the fourth tumour model where the wrong image is selected. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Metric** & \(\Delta\varepsilon_{r}\) & \(\Delta\mathbf{r}\) & \(\mathbf{SMR}(I_{\varepsilon_{r}^{\rm W}})\) & \(\mathbf{SCR}(I_{\varepsilon_{r}^{\rm  
As can be seen in Table 3, the Modified Laplacian, \(\phi^{\rm L}_{\rm M}\), selects an image with a higher SMR and slightly lower SCR than the Gaussian Energy, \(\phi^{\rm G}_{\rm GSS}\), and the Central Moment, \(\phi^{\rm S}_{\rm ACM}\). Figure 5a,b show the images selected by the Gaussian Energy, \(\phi^{\rm G}_{\rm GSS}\), and the Modified Laplacian, \(\phi^{\rm L}_{\rm M}\), respectively. Although the Gaussian Energy, \(\phi^{\rm G}_{\rm GSS}\), weights an image where the tumour target is clearly identifiable in the correct location, the Modified Laplacian, \(\phi^{\rm L}_{\rm M}\), weights an alternative image more highly. The image shown in Figure 5b is reconstructed with lower average dielectric properties and exhibits the characteristics identified earlier, where the response in the image is much closer to the skin. As the volume fraction of glandular tissue increases to 20% and 30%, the quality of the optimal image decreases due to increased reflections from other structures within the breast. In particular, the maximum response within the image is much further from the true tumour model location and the tumour model location is not correctly determined. The Gaussian Energy, \(\phi^{\rm G}_{\rm GSS}\), and the Central Moment, \(\phi^{\rm S}_{\rm ACM}\), again reward similar images for all tumour models. Coronal, sagittal and transverse slices of the reconstructed images of \(T_{4}\) in a phantom with 30% glandular structures by volume are shown in Figure 5c,d, corresponding to the image most rewarded using the Gaussian Energy, \(\phi^{\rm G}_{\rm GSS}\), and the Modified Laplacian, \(\phi^{\rm L}_{\rm M}\), respectively. Although, in this very heterogeneous breast phantom, no image is reconstructed that accurately identifies the tumour target location, the Modified Laplacian, \(\phi^{\rm L}_{\rm M}\), rewards an image reconstructed with lower average dielectric properties with a large apparent response close to the skin. Both the Gaussian Energy, \(\phi^{\rm G}_{\rm GSS}\), and the Modified Laplacian, \(\phi^{\rm L}_{\rm M}\), reward an image with a bright response in this case, although the location of this response is approximately 40 mm away from the true tumour target location. Many FQMs are found to have very similar performance in this work, in homogeneous and heterogeneous breast phantoms. This is expected as most FQMs are designed for the same purpose and it is indicated that FQMs are appropriate fitness functions for estimating average dielectric properties. For example, gradient-based metrics, \(\phi^{\rm G}\):  
* can be calculated easily from the image using simple and well-known kernels in two and three dimensions;