## 6 Experiments and Results
The clearly visible effects are also supported by a numeric analysis using standard statistical measures for figure/ground separation, i.e., it is tested whether a cell indicates the presence of the object, i.e., the figure, which is considered as positive response, or not, i.e., the cell indicates the (back-)ground, which is as usual considered to be the negative response in this test. Based on this standard definition of the figure/ground separation test, the related statistical measures can be calculated, i.e., the True Positive Rate (TPR), also known as recall or sensitivity, the True Negative Rate (TNR) also known as specificity or selectivity, the False Positive Rate (FPR), also known as fall-out or the probability of false alarm, and the False negative rate (FNR), also known as miss rate. Figure 10: The target object used in the experiments is a mock-up panel including elements from the context of underwater oil- and gas-production (**top, left**). The DexROV system (**bottom, left**) interacts with the panel (**top, right**), for example, based on machine perception of valve states (**bottom, right**). Table 2 shows the according results. The ground truth of the target object (case (a)) leads obviously to perfect results and it is only included for the sake of completeness. The single scan (case (b)) has a smeared-out response that overestimates the object. There is hence a high recall due to many \"lucky hits\", but especially the probability of false alarms is very high. The incoherent summation (case (c)) provides a somewhat clearer representation of the object, but the probability of false alarms is still very high. The hypothetical sonar with 320 sensors in one device (case (d)) leads to an improvement with respect to the probability of false alarms, but false alarms are still quite probable and the recall significantly drops. Our proposed method (case (e)) performs significantly better than the different alternatives and it provides high recall and selectivity under small fall-outs and miss-rates. Furthermore, it can be noted that it is close to the theoretical best-case of an idealized SAS processing (case (f)). Figure 11: 2D imaging example according to the SAS wavefront reconstruction (see Section 2.2). To further illustrate the robustness of our method, Figure 12 shows a comparison of two different resolutions with a set of two different registration errors on the pixel level. Figure 12a,b shows an image reconstruction using a step-width \(r_{uv}=\frac{2}{\gamma_{0}}\). The second example in Figure 12c,d has a doubled resolution using half the step-width \(r_{uv}\). Concretely, a range of \(1.8~{}\mathrm{m}\times 1.6~{}\mathrm{m}\) on \(240\times 212\) and \(480\times 424\) pixel is used. Similar to Section 3 a pixel error is introduced demonstrating different behavior for different image resolutions. On both resolutions, a set of small errors not exceeding a pixel shift one and a set of larger errors up to a maximum pixel shift two is applied (see Table 3). As can be expected, pixel errors have a more significant effect within a smaller resolution than with higher resolutions. Figure 12c shows that in this resolution, the erroneous superposition of signals still leads to an increase in resolution while the other example shows a degrading effect at a similar level like a single scan (Figure 11b). \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **TPR** & **TNR** & **FPR** & **FNR** \\ \hline (a) ground truth & 100.0 & 100.0 & 0.0 & 0.0 \\ \hline (b) single scan & 90.4 & 26.5 & 73.5 & 9.6 \\ \hline (c) incoherent summation & 92.0 & 63.9 & 36.1 & 8.0 \\ \hline (d) single scan—320 sensors & 82.3 & 89.5 & 10.5 & 17.7 \\ \hline (e) SAS—our method & 94.2 & 94.8 & 5.2 & 5.8 \\ \hline (f) SAS—theoretical best case & 94.9 & 95.2 & 4.8 & 5.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numeric True/False (T/F) Positive/Negative (P/N) rates (R) for the qualitative results shown in Figure 11. Please see the text for a discussion.