*2.4. 3D Modeling*

Successful extraction of major rooftop features, as proposed in Sections 2.2 and 2.3, is not sufficient to deliver 3D building modeling. Therefore, the next steps are (i) to produce possible boundary points for all features on the rooftops which are used, and (ii) to create vertical walls connecting rooftops to the ground.

Points on the boundaries of all detected parts on the rooftop are generated by applying rectilinear fitting: A 2D grid is overlaid on the LIDAR points in the x, y plane and each cell of the 2D grid is marked as being occupied and, thus, its boundary represents the shapes of all parts, if there are at least a minimum number of cloud points (based on their density). A robust 2.5D dual contouring method [25] is then utilized to generate facetized, watertight building models (see Figure 3b).

#### *2.5. Evaluation of the Performance Measurements*

The results of 3D building reconstruction of two different segmentation approaches described in Sections 2.2 and 2.3 were evaluated in terms of the geometrical accuracy of the roof polygons and/or final 3D building model. The mean, standard deviation, and Root Mean Square Error (RMSE) of the Euclidean distance (along x, y and z dimension) of each vertex (all the points) of the reconstructed 3D building model and the relative roof polygons and the nearest neighbors of the corresponding reference point were used:

$$RMSE = \sqrt{\sum\_{i=1}^{n} \left(\hat{d}\_i\right)^2}, \text{ where } \hat{d}\_i = \sum\_{i=1}^{n} \frac{\sqrt{\left[p\_i(\mathbf{x}) - r\_i(\mathbf{x})\right]^2 + \left[p\_i(y) - r\_i(y)\right]^2 + \left[p\_i(z) - r\_i(z)\right]^2}}{n} \tag{5}$$

In Equation (5), ˆ *di* is the mean of the Euclidean distance (along x, y and z direction) between the *i*-th point *pi* of each segmentation model and of the corresponding nearest neighbors point *ri* of the reference data set.
