## 2 Methodology to Generate a Synthetic Image
### Treatment of Initial Data  
Initially, the starting data is a cloud of LiDAR points. The aspect of the point cloud can be seen in Figure 2, which is shown as the areas with a greater density of points (dark zones) when compared to other areas where there is a smaller density of points. The bottom part of Figure 2 shows a dark zone with a double density of points, which corresponds with the overlap zone for LiDAR flight (right zoom of Figure 2).  
Figure 1: Block diagram for the generation of synthetic images.  
Figure 2: Visual aspect of a LiDAR point cloud using the intensity level to generate the image in grayscale. In the darkest areas, a greater density of points can be seen.  
Usually, in this type of point cloud, the density is calculated in points per square meter, which is an average density for the whole cloud in practice.  
The attributes that define a LiDAR point cloud are described below (this development is similar for any other type of point cloud):  
* Planimetric coordinates \((X,Y)\) in a given reference system;
* Height \((Z)\) in a given reference system;
* Intensity level or another attribute as reflectivity.  
The point cloud could be irregular and, from it, two digital elevation models can be extracted:  
* A digital surface model (or ground, if this is the case) using the planimetric coordinates \((X,\ Y)\) and the height \((Z)\). The mathematical expression is: \[N_{Z}=\{(X_{1},Y_{1},Z_{1}),(X_{2},Y_{2},Z_{2}),\ldots,(X_{n},Y_{n},Z_{n})\}.\] (1)
* A digital elevation model, using the planimetric coordinates \((X,\ Y)\) and the intensity level \((I)\). The mathematical expression is: \[N_{I}=\{(X_{1},Y_{1},I_{1}),(X_{2},Y_{2},I_{2}),\ldots,(X_{n},Y_{n},I_{n})\}.\] (2)  
In both expressions, \(n\) is the number of elements in a point cloud.  
The main target of this procedural step is to obtain a good density and a well-distributed point cloud in order to be similar to a photogram where all pixels have a regular distribution and the distance between two contiguous pixels is always the same. Therefore, all white spaces are filled with new points that have an intensity level similar to their neighbors.  
In Figure 4, we define a reference system \(R_{t}\) with a center \(O_{t}\) and one generic terrain point \(P\) with coordinates \((X_{t},Y_{t},Z_{t})\) referring to \(R_{t}\). We created two pictures \(\left(I_{i},\ I_{j}\right)\) of the same place (containing \(P\)) with two different points of view \(O_{i}\): \((X_{t}^{O_{i}},Y_{t}^{O_{i}},Z_{t}^{O_{i}})\) and \(O_{j}\): \((X_{t}^{O_{j}},Y_{t}^{O_{j}},Z_{t}^{O_{j}})\), which refer to \(R_{t}\). The reflection of the rays in \(P\) are depicted as the orange lines in Figure 4 (\(Ry_{i}\) and \(Ry_{j}\)). These lines intersect \(I_{i}\)_and_\(I_{j}\) in \(P_{i}\) and \(P_{j}\) with coordinates \(\left(X_{t}^{i},Y_{t}^{i},Z_{t}^{i}\right)\) and \(\left(X_{t}^{i},Y_{t}^{i},Z_{t}^{i}\right)\), respectively, over the same reference system \(R_{t}\). For all points \(P\), the coplanarity condition [7] must be fulfilled (\(O_{i}\), \(O_{j}\), \(P_{i}\), \(P_{j}\) and \(P\) are all in the same plane, \(\Pi\)) or Equation (3a) must be satisfied.