## 5 Results
### Effect of Incorrect Parameter Estimation  
The effects of incorrect average dielectric property estimation are first analysed using the theoretical PSFs in Figure 3. Figure 3a shows the PSF of the ideal system obtained by numerically integrating Equation (4). Additionally, the location of the strongest response in the image is shown in Figure 3b. The true location of the dielectric point scatterer is at \(T=0.2R\). A number of observations can be made from Figure 3:  
* in general, the maximum amplitude of the PSF is when the effective average dielectric properties, \(\epsilon^{\prime}_{r}\), is equal to the true average dielectric properties, \(\epsilon_{r}\);
* if the effective average dielectric properties are underestimated (i.e., \(\sqrt{\frac{\epsilon^{\prime}_{r}}{\epsilon_{r}}}<1\)), the apparent location of the scatterer moves towards the edge of the imaging domain (closer to \(R\)). This localisation error is due to reflections appearing to come from closer than their true origin and the channels closest to the scatterer are dominant in the coherent summation;
* conversely, if the effective average dielectric properties are overestimated, the apparent location of the scatterer moves towards the centre of the imaging domain (closer to 0). This localisation error is due to reflections appearing to come from further away than their true origin;* the number of sidelobes increases as the estimate of the effective average dielectric properties increases; in other words, there is higher spatial frequency content in PSFs with over-estimated effective average dielectric properties, \(\sqrt{\frac{\varepsilon_{r}}{\varepsilon_{r}}}>1\);
* it can be seen that as the effective average dielectric properties are overestimated, the width of the peak decreases. * the localisation error is greater when underestimating the effective average dielectric properties compared to overestimation. Coronal slices of the experimental PSF at the dielectric point scatterer location are shown in Figure 4. Figure 4a-c are reconstructed with effective average dielectric properties of \(\varepsilon_{r}^{\prime}\in\{1.5,6,13.5\}\) respectively, where fatty breast interior has dielectric properties of \(\varepsilon_{r}=6\). Hence, Figure 4a-c are reconstructed with \(\sqrt{\frac{\varepsilon_{r}^{\prime}}{\varepsilon_{r}}}\in\{0.5,1,1.5\}\) respectively. Comparable trends can be observed in the PSF of the theoretical and experimental systems:  
* the maximum amplitude of the images with incorrectly estimated effective average dielectric properties is much lower than the ideal image, 40% when underestimated and 9% when overestimated;
* the apparent location of the scatterer moves towards the edge of the imaging domain when the effective average dielectric properties are underestimated, i.e., \(\sqrt{\frac{\varepsilon_{r}^{\prime}}{\varepsilon_{r}}}<1\);
* the apparent location of the scatterer moves towards the centre of the imaging domain when the effective average dielectric properties are overestimated, i.e., \(\sqrt{\frac{\varepsilon_{r}^{\prime}}{\varepsilon_{r}}}>1\);
* Figure 4c has more clutter with greater magnitude than Figure 4a. This is similar to the theoretical case where the image reconstructed with overestimated effective average dielectric properties (i.e., \(\sqrt{\frac{\varepsilon_{r}^{\prime}}{\varepsilon_{r}}}>1\)) has higher spatial frequency content. * the area of the response decreases as the estimated effective average dielectric properties increase. * The localisation error when overestimating the properties is less than when underestimating the properties. Figure 3: The theoretical point spread function (PSF) is analysed here. (**a**) shows the one-dimensional PSF for various values of \(\sqrt{\frac{\varepsilon_{r}^{\prime}}{\varepsilon_{r}}}\). Localisation error increases as the difference between \(\varepsilon_{r}^{\prime}\) and \(\varepsilon_{r}\) grows.