# 4D Building Reconstruction with Machine Learning and Historical Maps
## 5 Results
### Inferring Building Heights from a Historical Map--Trento Case Study  
The building footprints digitised in the four historical maps (1851, 1887, 1908, and 1936) of Trento and their predictors (Section 4.2) were used to infer building heights using the two outperforming predictor methods: Random Forest and Catboost. The actual topographic database was used to learn heights, although some modern buildings were removed beforehand from the input data to limit gross errors. Evaluation metrics to check predicted heights in the historical maps were computed on a smaller test set, including only some buildings still existing, having the same shape and recognisable in the different maps. Tables 11 and 12 show the height difference error calculated in this case, presenting the Random Forest and Catboost results, which demonstrated to be the best performing algorithms (Section 5.1). Again, the Random Forest confirmed to outperform the other algorithms. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Regressor** & **RMSE TEST (m)** & **MAE TEST (m)** & **R\({}^{2}\) TEST** & **Median (m)** & **St. Dev. (m)** \\ \hline Linear & 8.52 & 6.81 & 0.04 & \(-0.75\) & 8.51 \\ Random Forest & 4.52 & 3.54 & 0.49 & 0.00 & 2.72 \\ CatBoost & 4.64 & 3.66 & 0.52 & 0.00 & 2.49 \\ Support Vector & 5.15 & 3.49 & 0.41 & 0.00 & 4.78 \\ Multilayer Perceptron & 4.83 & 3.79 & 0.48 & \(-0.02\) & 3.91 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Accuracy evaluation of the regressor methods on the Bologna dataset after data augmentation. Figure 10: 3D view of the inferred building heights (orange) in Trento with respect to the ground truth data (white) after data augmentation. The visual inspection shows a relevant reduction of the errors on towers and churches. As a further check of the prediction's quality, single-view metrology (Section 4.4.2) was applied to derive some ground-truth heights for disappeared buildings. Vanishing lines and cross-ratio invariants were used with some historical photos where a known height was measurable (Figure 11). Table 13 summarises the evaluation results adopting the presented procedure. Visual results of the 4D LoD1 reconstruction of Trento are presented in Figures 12 and 13. An example of texture mapping with a historical photo is shown in Figure 14. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Historical Map-Year** & **RMSE (m)** & **MAE (m)** & \(R^{2}\) & **Min Error (m)** & **Max Error (m)** & **Median (m)** & **St. Dev. (m)** \\ \hline
1851 & 0.96 & 1.13 & 0.97 & 2.95 & 1.75 & 1.09 & 0.74 \\
1887 & 2.86 & 2.25 & 0.92 & 0.07 & 6.35 & 1.52 & 1.77 \\
1908 & 1.80 & 1.50 & 0.96 & \(-\)2.78 & 3.91 & 1.18 & 0.99 \\
1936 & 1.67 & 1.40 & 0.97 & \(-\)2.62 & 3.20 & 1.14 & 0.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Metrics evaluation of the Random Forest performance on the four historical datasets of Trento, considering twenty unaltered buildings digitised in all the maps as the test set. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Dataset** & **RMSE (m)** & **MAE (m)** & **Median (m)** & **St. Dev. (m)** \\ \hline Trento & 1.41 & 1.28 & \(-\)0.83 & 1.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Evaluation on eight disappeared buildings visible in historical photos: heights predicted with Random Forest were compared with single-view-metrology heights and metrics derived. Figure 11: Two examples of single-view-metrology applied to historical photos in Trento to determine heights of buildings not present anymore in the actual topographic database. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Historical Map-Year** & **RMSE (m)** & **MAE (m)** & \(R^{2}\) & **Min Error (m)** & **Max Error (m)** & **Median (m)** & **St. Dev.