#### 5.2. Initial Evaluation

To determine the most suitable FQMs, all FQMs were evaluated for the 5 targets in

Figure 2b using the dielectrically homogeneous phantom. The mean value of the evaluation criteria (

$\Delta {\epsilon}_{r}$,

$\Delta \mathbf{r}$,

$\mathrm{SMR}({I}_{{\epsilon}_{r}^{{B}^{\prime}}})$ and

$\mathrm{SCR}({I}_{{\epsilon}_{r}^{{B}^{\prime}}})$) is shown in

Table 2. Metrics are listed in order of rank for each method of action, and this rank is shown. Also shown is a global rank which is useful for comparing the different methods of action.

Of all the metrics analysed, two metrics perform very well as fitness functions: the Central Moment, ${\varphi}_{\mathrm{ACM}}^{\mathrm{S}}$, and the Gaussian Energy, ${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$. The Central Moment, ${\varphi}_{\mathrm{ACM}}^{\mathrm{S}}$, rewards images that are closest to the average dielectric properties, being on average within $\Delta {\epsilon}_{r}=0.7$ of the known value of ${\epsilon}_{r}=6$. However, the Gaussian Energy, ${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$, rewards images that are of a high quality, rewarding images that have the best localisation error, $\Delta \mathbf{r}$, and the best clutter suppression, $\mathrm{SMR}({I}_{{\epsilon}_{r}^{{B}^{\prime}}})$ and $\mathrm{SCR}({I}_{{\epsilon}_{r}^{{B}^{\prime}}})$. Other metrics based on the gradient, ${\varphi}^{\mathrm{G}}$, or statistics, ${\varphi}^{\mathrm{S}}$, of the image also reward images of high quality; ten of the top eleven metrics use these methods of action.

All metrics based on the Laplacian of the image, ${\varphi}^{\mathrm{L}}$, perform very similarly, selecting images with almost the same accuracy, $\Delta {\epsilon}_{r}$, localisation error, $\Delta \mathbf{r}$, and clutter suppression, $\mathrm{SMR}({I}_{{\epsilon}_{r}^{{B}^{\prime}}})$ and $\mathrm{SCR}({I}_{{\epsilon}_{r}^{{B}^{\prime}}})$. Five more metrics perform very similarly to metrics based on the Laplacian, ${\varphi}^{\mathrm{L}}$: three based on the gradient of the image, ${\varphi}^{\mathrm{G}}$, the Tenengrad mean, ${\varphi}_{\mathrm{M}}^{\mathrm{T}}$, the Squared Gradient, ${\varphi}_{\mathrm{DMS}}^{\mathrm{G}}$, and the Gradient Energy, ${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$; and two based on wavelet decomposition of the image, ${\varphi}^{\mathrm{W}}$, the Detail Variance, ${\varphi}_{\mathrm{V}}^{\mathrm{W}}$, and the Absolute Detail Sum, ${\varphi}_{\mathrm{AS}}^{\mathrm{W}}$.

The two metrics based on the Fourier transform,

${\varphi}^{\mathrm{F}}$, do not perform well as fitness functions in these scenarios, identifying images with poor clutter suppression,

$\mathrm{SMR}({I}_{{\epsilon}_{r}^{{B}^{\prime}}})$ and

$\mathrm{SCR}({I}_{{\epsilon}_{r}^{{B}^{\prime}}})$. Additionally, the Fourier-based metrics,

${\varphi}^{\mathrm{F}}$, select images with localisation errors that are, on average, greater than 10 mm,

$\Delta \mathbf{r}>10\mathrm{m}\mathrm{m}$. As shown in

Figure 4, images generated with underestimated effective average dielectric properties,

$\sqrt{\frac{{\epsilon}_{r}^{\prime}}{{\epsilon}_{r}}}<1$, are characterised by large responses much closer to the skin than the true scatterer location. Metrics based on the Fourier transform,

${\varphi}^{\mathrm{F}}$, reward these images resulting in poor performance [

44]. Metrics based on the Fourier transform,

${\varphi}^{\mathrm{F}}$, were first proposed for low-contrast images where they can be more effective [

71], whereas the contrast for microwave radar images is higher. Although the AC–DC Reduced Ratio,

${\varphi}_{\mathrm{RR}}^{\mathrm{F}}$, performed better in noisy images than the AC–DC Ratio,

${\varphi}_{\mathrm{R}}^{\mathrm{F}}$, in experimental images, that was not found for microwave radar images.

The Detail–Coarse Ratio, ${\varphi}_{\mathrm{R}}^{\mathrm{W}}$, fails to reward any correct image. Similarly to metrics based on the Fourier transform, ${\varphi}^{\mathrm{F}}$, the Detail–Coarse Ratio, ${\varphi}_{\mathrm{R}}^{\mathrm{W}}$ heavily rewards images generated with underestimated effective average dielectric properties, $\sqrt{\frac{{\epsilon}_{r}^{\prime}}{{\epsilon}_{r}}}<1$, such that it always selects images generated with effective average dielectric properties of free-space, ${\epsilon}_{r}^{\prime}=1$.

Three suitable fitness functions were selected for further analysis in the subsequent sections: the Gaussian Energy, ${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$; the Modified Laplacian, ${\varphi}_{\mathrm{M}}^{\mathrm{L}}$; the Central Moment, ${\varphi}_{\mathrm{ACM}}^{\mathrm{S}}$. The three FQMs metrics use three different methods of action based on the image gradient, the image Laplacian and statistic of the image respectively.

#### 5.3. Detailed Analysis

Table 3 analyses the metrics selected in the previous section on five spherical targets of increasing diameter,

$d\in [5.3,20.2]\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{m}$, 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,

$\Delta {\epsilon}_{r}$, is not shown because the accuracy,

$\Delta {\epsilon}_{r}$, 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,

${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$, and the Central Moment,

${\varphi}_{\mathrm{ACM}}^{\mathrm{S}}$, perform very similarly, selecting an average dielectric properties value of within

$\Delta {\epsilon}_{r}=0.25$ 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,

${\varphi}_{\mathrm{M}}^{\mathrm{L}}$, selects almost the same images except for the fourth tumour model where the wrong image is selected. As can be seen in

Table 3, the Modified Laplacian,

${\varphi}_{\mathrm{M}}^{\mathrm{L}}$, selects an image with a higher SMR and slightly lower SCR than the Gaussian Energy,

${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$, and the Central Moment,

${\varphi}_{\mathrm{ACM}}^{\mathrm{S}}$.

Figure 5a,b show the images selected by the Gaussian Energy,

${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$, and the Modified Laplacian,

${\varphi}_{\mathrm{M}}^{\mathrm{L}}$, respectively. Although the Gaussian Energy,

${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$, weights an image where the tumour target is clearly identifiable in the correct location, the Modified Laplacian,

${\varphi}_{\mathrm{M}}^{\mathrm{L}}$, 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, ${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$, and the Central Moment, ${\varphi}_{\mathrm{ACM}}^{\mathrm{S}}$, 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,

${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$, and the Modified Laplacian,

${\varphi}_{\mathrm{M}}^{\mathrm{L}}$, respectively. Although, in this very heterogeneous breast phantom, no image is reconstructed that accurately identifies the tumour target location, the Modified Laplacian,

${\varphi}_{\mathrm{M}}^{\mathrm{L}}$, rewards an image reconstructed with lower average dielectric properties with a large apparent response close to the skin. Both the Gaussian Energy,

${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$, and the Modified Laplacian,

${\varphi}_{\mathrm{M}}^{\mathrm{L}}$, 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, ${\varphi}^{\mathrm{G}}$:

can be calculated easily from the image using simple and well-known kernels in two and three dimensions;

have a well-understood method of action as differentiation is analogous to high-pass filtering;

and identify the optimal image in heterogeneous phantoms with different tumour sizes.

Additionally, the Gaussian Energy, ${\varphi}_{\mathrm{GSS}}^{\mathrm{G}}$, is shown in this work to be a suitable cost function in three-dimensional images with realistic artefact removal.