## 3 Materials and Methods
### 6 DOF Pose Tracking Module  
In this section, the 6 DOF pose tracking method used in the AR system is described. The 6 DOF pose tracking method used is the one proposed by [PERSON] et al. [17], as it meets all the requirements to solve the technical limitations of the method of estimating the pose based on LINEMOD. Basically, this tracking method consists of iteratively solving a nonlinear optimization problem of the parameters that define the rigid transformation of an object between two consecutive frames. Consequently, the first step to take is to determine a simpler representation of the object pose by reducing the number of parameters from 12 (9 for rotation and 3 for translation) to 6 by using a Lie Algebra. More specifically, to represent the rigid movement of an object between two consecutive frames, a twist vector \(\zeta=[t_{x},t_{y},t_{z},w_{x},w_{y},w_{z}]^{T}\in\mathbb{R}^{6}\) is used, whose representation by the Lie algebra is  
\[\widehat{\xi}=\left[ \begin{array}{cccc}0&-w_{x}&w_{y}&t_{x}\\ w_{z}&0&w_{x}&t_{y}\\ -w_{y}&w_{x}&0&t_{z}\\ 0&0&0&0\end{array} \right]\in se(3), \tag{6}\]  
where \(se(3)\) is the Lie algebra associated with the Lie group \(\mathbb{SE}(3)\)[17]. On the other hand, a twist vector can be converted into an element of the Lie group \(\mathbb{SE}(3)\) by an exponential function \(\exp(\cdot)\) using the Rodrigues formula [53]. Once the new parametrization of the pose has been carried out, the next step of the tracking method consists of projecting the 3D model onto the 2D plane of the image captured by the camera using the initially pose estimated (see Equation (4)). The 2D contour of the 3D object projected onto the 2D plane of the image, which we denote as \(C\), divides the image into two disjoint regions: \(\Omega_{f}\in\Omega\) (foreground) and \(\Omega_{b}=\Omega\backslash\Omega_{f}\) (background) (see Figure 6). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Input Images** & **Batch Size** & **Subdivisions** & **Momentum** & **Initial Learning Rate** & **Decay** & **Max Batches** \\ \hline
416 \(\times\) 416 & 64 & 8 & 0.949 & 0.0003 & 0.0005 & 40k \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main parameters used to train YOLOV4. Figure 5: Photorealistic synthetic images generated using BlenderProc4 BOP [18] with some of the 3D models used in our experiments, where other 3D models of different data sets, such as LINEMOD [15] or T-LESS [36], were used as distractors. On the other hand, \(C\) can be represented by a level set function, such that \(C=\{\mathbf{x}|\phi(\mathbf{x})=0\}\), where \(\phi(\mathbf{x})\) is defined as:  
\[\phi(\mathbf{x})=\left\{ \begin{array}{cc}-d(\mathbf{x},C)&\forall\mathbf{x} \in\Omega_{f}\\ d(\mathbf{x},C)&\forall\mathbf{x}\in\Omega_{b}\end{array} \right. \tag{7}\]  
and \(d(\mathbf{x},C)=\min_{c\in C}|c-\mathbf{x}|\). The next step of the method is to determine a function that maximizes the discrepancies between the appearance of the background and the foreground with respect to its pose. Specifically, the functional measures the separation between the background and the foreground with respect to the 2D shape determined by \(\phi\). Each region has its own statistical model of appearance: \(P(\mathbf{y}|M_{b})\) for the background and \(P(\mathbf{y}|M_{f})\) for the foreground, where \(M_{b}\) and \(M_{f}\) are the parameters of the background and foreground models, respectively. Specifically, \(P(\mathbf{y}|M_{f})\) and \(P(\mathbf{y}|M_{b})\) represent the probabilities that a pixel with color \(\mathbf{y}=I_{rgb}(\mathbf{x})\) belongs to the foreground or background region, based on the information obtained from the color histogram.