# Sparse Indoor Camera Positioning with Fiducial Markers
## 4 Experiments
### Metagroup Edges Automatic Estimation  
This experiment evaluates our algorithm's capability to automatically find the markers \(C(g^{k},g^{t})\) that have not been moved between groups. The accuracy of this marker labeling is crucial, as it directly impacts the quality and reliability of the linkages within the reconstructed environment.  
We tested each generated dataset against manually annotated ground-truth data, which provided the precise labeling of markers for evaluation purposes. Table 5 presents the results, detailing the output rates of true positives (\(TPR\)), false positives (\(FPR\)), and false negatives (\(FNR\)) for each dataset. These metrics offer a comprehensive assessment of the algorithm's performance across various configurations.  
The results of these evaluations are detailed in Table 5. The high true positive rates (\(TPR\)) across all datasets, reaching 100% in some cases, indicate near-perfect accuracy in identifying markers that have remained stationary between groups. The absence of false positives (\(FPR\)) confirms that no markers were incorrectly classified as common, ensuring that all detected common markers were indeed unchanged. This level of precision highlights our algorithm's effectiveness in consistently identifying valid sets of common markers, which are crucial for accurate environment reconstruction, especially in large and complex settings, like entire buildings.  
\begin{table}
\begin{tabular}{l c c} \hline \hline  & \multicolumn{2}{c}{**Entire Building**} \\ \hline
**Method** & \(\mathbf{E_{t}}\) **(cm)**\(\downarrow\) & \(\mathbf{E_{t}}\) **(\%)**\(\downarrow\) \\ \hline \(E_{rp}\) & 242.35 \(\pm\) 81.16 & 0.892 \\ \(E_{rp}+E_{cc}\) & 221.43 \(\pm\) 78.08 & 0.815 \\ \(E_{rp}+E_{cm}\) & 216.93 \(\pm\) 66.97 & 0.798 \\ \(E_{rp}+E_{cc}+E_{cm}\) & 216.17 \(\pm\) 58.42 & 0.796 \\ \(E_{rp}+E_{cp}\) & 45.92 \(\pm\) 17.00 & 0.169 \\ \(E_{rp}+E_{cp}+E_{cc}\) & 43.83 \(\pm\) 19.90 & 0.161 \\ \(E_{rp}+E_{cp}+E_{cm}\) & 41.68 \(\pm\) 20.81 & 0.153 \\ \hline \(E_{rp}+E_{cp}+E_{cc}+E_{cm}\) & 41.96 \(\pm\) 21.47 & 0.154 \\ \hline \hline \end{tabular}
\end{table}
Table 4: **Results using the entire building dataset**. The term \(E_{rp}\) refers to the baseline, the term \(E_{cp}\) refers to the usage of control points, \(E_{cc}\) refers to the enforcement of cameras being at the same plane, and \(E_{cm}\) applies the same principle to the markers. \(E_{t}\) denotes the translational errors, expressed in cm and as a percentage with respect to the running distance.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{4}{c}{**Markers Linkages Accuracy**} \\ \cline{3-5}
**Dataset** & **Variant** & \%\(TPR\)\(\uparrow\) & \%\(FPR\)\(\downarrow\) & \%\(FNR\)\(\downarrow\) \\ \hline Corridor & A & 100.00 & 0.00 & 0.00 \\  & B & 100.00 & 0.00 & 0.00 \\ Complete Floor & A & 98.00 & 0.00 & 2.00 \\  & B & 98.00 & 0.00 & 2.00 \\ Entire Building & & 99.09 & 0.00 & 0.91 \\ \hline \hline \end{tabular}
\end{table}
Table 5: **Marker linkage accuracy**. Accuracy results of marker linkage across different datasets and algorithm variants, showing the percentages of the true positive rate (\(TPR\)), false positive rate (\(FPR\)), and false negative rate (\(FNR\)).