## Materials and Methods
### Data-Driven Free-Form Modelling  
In the proposed data-driven free-form modelling method, the parts of the point cloud corresponding to the damaged N-E and S-E facades and the base were reconstructed by the creation of a supporting \"dummy\" point cloud. The roof and the N-W and S-W facades, where there was sufficient data for modelling, were preserved and modelled according to the input point cloud. The pipeline of the developed method can be divided into three main phases, as shown in Figure 5.  
The first step of the method (Phase 1) provides that contour lines of the roofs are extracted and regularised through iterative transformations from point cloud to mesh and vice versa. Subsequently, the polysurfaces of the facades and the base of the structure are created from the regularised contour lines (Phase 2). The polysurfaces are converted into a single point cloud and, through a segmentation and merging operation, the \"dummy\" point cloud is created. In Phase 3, the original point cloud and the \"dummy\" point cloud are merged. From the resulting point cloud, the triangular mesh (tri-mesh) model is created, and imperfections are automatically corrected. Finally, the tri-mesh model can be textured using nadir and oblique images or it can be transformed into a quadrangular mesh (quad-mesh) model. The generated mesh model is based on the Poisson surface reconstruction method. In the most general form, proposed by [PERSON] et al. [23], given in input a point set S consisting of samples points s.p with s \(\in\) S and normal s.N facing inwards, which are assumed to be over or near a surface \(\partial M\) of an unknown M solid model; the surface is reconstructed by solving a standard Poisson problem:  
\[\Delta\widetilde{X}=\
abla\cdot\overrightarrow{V} \tag{1}\]  
where \(\overrightarrow{V}\) is the vector field and the gradient of the smoothed indicator function XM of M is equal to the vector field obtained by smoothing the field normal to the surface:  
Figure 4: Dataset of surveyed images. \[\
abla(X_{M}*\widetilde{F})(q_{0})=\int_{\partial M}\widetilde{F}_{p}(q_{0}) \overset{\rightarrow}{N}_{\partial M}(p)dp \tag{2}\]  
where \(\overset{\rightarrow}{N}_{\partial M}(p)\) is the inward surface normal in point \(p\in\partial M\), \(\widetilde{F}(q)\) is a smoothing filter, and \(\widetilde{F}_{p}(q)=\widetilde{F}(q-p)\) is the translation to the point \(p\). The surface integral cannot be solved since we do not know the geometric surface. So, the integral is approximated to the discrete summation. The approximation method consists of using the input point set \(\mathbb{S}\) to divide \(\partial\)M into distinct patches \(P_{s}\subset\partial M\), and then approximate the integral over a patch \(\mathbb{Ps}\) at the value of the sample point s.p, scaled by the area of the patch:  
\[\
abla(X_{M}*\widetilde{F})(q)=\sum_{s\in S}\int_{P_{s}}\widetilde{F}_{p}(q) \overset{\rightarrow}{N}_{\partial M}(p)dp\approx\sum_{s\in S}|P_{s}| \widetilde{F}_{s,p}(q)s.\overset{\rightarrow}{N}\equiv\overset{ \rightarrow}{V}(q) \tag{3}\]  
A graphic representation of the Poisson surface reconstruction method applied to a generic figure is given below (Figure 6). Figure 5: Data-driven free-form modelling pipeline. Figure 6: Illustration of Poisson reconstruction adapted from [PERSON] et al. [23]. The Poisson surface reconstruction method solves the Poisson problem for a watertight surface, so a failure to close the point cloud on the damaged facades and at the base, during processing with a surface reconstruction algorithm, would result in unacceptable artefacts that would be hard to modify automatically during post-processing.