## 2 Methodology to Generate a Synthetic Image
### Reading the Point Cloud, Calculating the Pixel Coordinates and Generating Synthetic Pixel  
This step is carried out for each point that belongs to the point cloud. The process is laid out in the following sections. Figure 4: Spatial reference systems in the two frames. 3.1 Calculation of the Digital Value of the Pixels to be Represented in Each of the Images \(I_{0}\) and \(I_{1}\)  
Given a point \(P_{t}\in N\) whose attribute \(I\) is \(I_{t}\), the digital value \(VD_{t}\) will be in each of the two images given by Equation (17) without having to consider that \(VD_{i}=VD_{t}\) and \(I_{i}=I_{t}\):  
\[ \begin{array}{ccc}VD_{i}=\frac{255\left[I_{t}-I_{inf}\right]}{3\sigma_{1}}& If\ I_{inf}\leq I_{i}\leq I_{sup}\\ VD_{i}=0& If\ I_{i}\leq I_{inf}\\ VD_{i}=255& If\ I_{i}\geq I_{sup}\end{array} . \tag{17}\]  
At this point, we have to calculate the equations of the rays denoted by \(Ry_{0}\) and \(Ry_{1}\). Ray \(Ry_{0}\) passes through points \(P_{t}\) and \(O_{o}\), while ray \(Ry_{1}\) passes through points \(P_{t}\) and \(O_{1}\). Using Equations (18) and (19), we obtain Equations (20) and (21), which are the equations of lines \(Ry_{0}\) and \(Ry_{1}\), respectively:  
\[\frac{(X-X_{t})}{X_{t}-X_{t}^{O_{i}}}=\frac{(Y-Y_{t})}{Y_{t}-Y_{t}^{O_{i}}}= \frac{(Z-Z_{t})}{Z_{t}-Z_{t}^{O_{i}}}=l, \tag{18}\]  
\[\frac{(X-X_{t})}{X_{t}-X_{t}^{O_{i}}}=\frac{(Y-Y_{t})}{Y_{t}-Y_{t}^{O_{i}}}= \frac{(Z-Z_{t})}{Z_{t}-Z_{t}^{O_{i}}}=k, \tag{19}\]  
\[\frac{(X-X_{t})}{X_{t}-X_{t}^{O_{0}}}=\frac{(Y-Y_{t})}{Y_{t}-Y_{t}^{O_{o}}}= \frac{(Z-Z_{t})}{Z_{t}-Z_{t}^{O_{o}}}=l, \tag{20}\]  
\[\frac{(X-X_{t})}{X_{t}-X_{t}^{O_{1}}}=\frac{(Y-Y_{t})}{Y_{t}-Y_{t}^{O_{1}}}= \frac{(Z-Z_{t})}{Z_{t}-Z_{t}^{O_{1}}}=k. \tag{21}\]  
The calculation of the equations of the planes denoted by \(I_{0}\) and \(I_{1}\) is shown in Figure 5. In this figure, the plane \(I_{0}\) passes through points \(p:(0,\ 0,\ -f_{0})\), \(p^{\prime}:(0,\ 1,\ -f_{0})\) and \(p^{\prime\prime}:(1,\ 0,-f_{0})\) in the system of reference \(R_{0}\), while the plane \(I_{1}\) passes through points \(q:(0,\ 0,-f_{1})\), \(q^{\prime}:(0,\ 1,-f_{1})\) and \(q^{\prime\prime}:(1,\ 0,-f_{1})\) in the system of reference \(R_{1}\) (we checked that points \(p,\ \ p^{\prime},\ \ p^{\prime\prime}\) and points \(q,\ \ q^{\prime},\ \ q^{\prime\prime}\) are not aligned). Figure 5: Transformations to pass from system \(R_{t}\) and the systems \(R_{i}\) and \(R_{j}\). Equation (3a) can be applied to planes \(I_{0}\) and \(I_{1}\) to analytically obtain the equations of the planes that refer to the systems of reference \(R_{0}\) and \(R_{1}\), respectively. However, the equations of those planes referring to the system of reference \(R_{t}\) are required. Looking at plane \(I_{0}\) (a similar reasoning as for plane \(I_{1}\)) defined by points \(p,\ \ p^{\prime},\ \ p^{\prime\prime}\) in the system of reference \(R_{0}\) and applying a spatial transformation of seven parameters (where: \(\lambda=1\)), the rotations are \(\omega_{0},\ \ \varphi_{0},\ \kappa_{0}\) and the translation is \(\left(X_{t}{}^{\circ}{}_{\circ},\ \ Y_{t}{}^{\circ}{}_{\circ},\ Z_{t}{}^{\circ}{}_{ \circ}\right)\) (Figure 5). Following this, Equation (22) is applied:  
\[\left[ \begin{array}{c}Xt\\ Yt\\ Zt\end{array} \right]=\lambda\cdot R_{\omega,\varphi,\mathcal{K}^{*}}\cdot \left[ \begin{array}{c}X\\ Y\\ Z\end{array} \right]+\ \left[ \begin{array}{c}X_{0}\\ Y_{0}\\ Z_{0}\end{array} \right], \tag{22}\]  
where \(R_{\omega,\varphi,\mathcal{K}}\) is the so-called rotation matrix [13]; \(\lambda\) is the scaling factor/custom scale factor; \(\left[ \begin{array}{c}X_{0}\\ Y_{0}\\ Z_{0}\end{array} \right]\) is a vector that shows the translation; and \(\left[ \begin{array}{c}Xt\\ Yt\\ Zt\end{array} \right]\) are the transformed coordinates of point \(P\) of the coordinates \(\left[ \begin{array}{c}X\\ Y\\ Z\end{array} \right]\) in the initial system. The result is:  
\[\left[ \begin{array}{c}Xp_{t}\\ Yp_{t}\\ Zp_{t}\end{array} \right]=R_{\omega_{0},\varphi_{0},\kappa_{0}}\cdot\left[ \begin{array}{c}0\\ 0\\ -f_{0}\end{array}