## 2 Methodology to Generate a Synthetic Image
### Calculation of the Initial Parameters  
In this second step, the initial parameters needed to generate the synthetic models are calculated:  
1. The GSD (Ground Sample Distance) is the resolution of the pixel on the ground. The problem is due to the infinite solutions, as, according to the values that we wish to give to the parameters, different images will be obtained. However, only one will be the optimal solution and this is what needs to be found. Given the density of the point cloud \(\delta\), calculated according to Equation (11), the best GSD (the GSD with the least magnitude) that can be found is given by the equation: \[GSD=\frac{1}{\sqrt{\delta}}.\] (12)  
In this way, we have at least one point per pixel. There are other possibilities, but this one is very simple, easy to use and easy to implement. There are other parameters, which can be fixed according to the data that we already have. Moreover, they can be fixed freely (we have developed the methodology using a model of two frames or images). These parameters include:  
1. The resolution of pixel \(r_{p}\);
2. The focus of the cameras \(f_{o}\) and \(f_{1}\);
3. The width and height of the sensor \(W_{m}\) and \(H_{m}\);
4. The width and height of the image \(W\) and \(H\); and
5. The flight height \(h\). Once one or more of them have been fixed, the rest are calculated using Equations (13)-(15):  
\[\frac{W_{m}}{f_{i}}=\frac{W\cdot GSD}{h}\ \Rightarrow GSD=\frac{W_{m}\cdot h }{W\cdot f_{i}}, \tag{13}\]  
\[GSD=\frac{r_{p}\cdot h}{f_{i}}, \tag{14}\]  
\[r_{p}=\frac{W_{m}}{W}=\frac{H_{m}}{H}. \tag{15}\]  
Parameters \(b\)-\(f\) can be fixed by the operators. It is possible to assign any value to these parameters, but normally all restitution software has tolerances for the size of the digital files, digital camera work with a standard focal length, sensors for cameras have typical dimensions and there is a typical flight height. Therefore, we recommend that typical values are used for all these parameters. Using our own past experience, we first fixed the width and height for pictures to have a standard magnitude (not above 10,000 pixels for width or height), before continuing with focal length and flight length (standard values). Another important advantage of this methodology is that we can work with near 100% overlap. Therefore, most of the pixels in the synthetic images are useful for restitution (Figures 6 and 7). It is very difficult to simulate this situation in real photogrammetry. If one camera is used for all frames, it must be moved from one place to another and if two or more cameras are used at once, the same place must be targeted. Therefore, it is not easy to obtain near 100% overlap in photogrammetry (see Figure 4 where the frames are convergent not orthogonal). In this procedure, we can set the target. It is possible to introduce (or fix according to all previous magnitudes) the values of the named parameters of the external orientation (Figure 4) for each of the two photographs from which the images \(I_{0}\) and \(I_{1}\) originate. These values are:  
(a) Coordinates of the projection centers \(O_{0}\) and \(O_{1}\) in the systems \(R_{t}\), \(\left(X_{t}{}^{O_{e}}\), \(Y_{t}{}^{O_{e}}\), \(Z_{t}{}^{O_{e}}\right)\) and \(\left(X_{t}{}^{O_{1}},\ Y_{t}{}^{O_{1}},\ Z_{t}{}^{O_{1}}\right)\), respectively. A value (in the case that it is not fixed according to our needs) can be fixed according to the coating that is wished for and is called \(Re\) (in %) [7; 12]:  
\[ \begin{array}{l}X_{t}{}^{O_{e}}=X_{min}+\frac{X_{max}-X_{min}}{2}-GSD\cdot W.\frac{100-Re}{200}\\ X_{t}{}^{O_{1}}=X_{min}+\frac{X_{max}-X_{min}}{2}+GSD\cdot W.\frac{100-Re}{200} \\ Y_{t}{}^{O_{e}}=Y_{t}{}^{O_{1}}=Y_{min}+\frac{Y_{max}-Y_{min}}{2}\\ Z_{t}{}^{O_{e}}=Z_{t}{}^{O_{1}}=\overline{Z}+h\end{array} . \tag{16}\]  
These values are not random.