## 3 Material and Methods
### Time-Synchronization and Camera Calibration  
Accurately projecting the 3D ground truth trajectories obtained from the VICON motion capture system into the simultaneously captured 2D images was a crucial requirement for this study. This task involved addressing two major challenges. The first challenge was synchronizing the images from the camera system with the VICON motion capture system. Open Broadcaster Software (OBS version 29.0.2) [47], an open-source software, was employed to synchronize the three differently oriented 2D cameras. OBS allowed for the simultaneous recording of multiple cameras using a single system, minimizing, or eliminating time lapses between different recordings.  
To ensure accurate synchronization between OBS and the VICON motion capture systems, a trigger system was developed using Arduino Uno and VICON Lock Studio in Python [48]. The VICON Lock Studio, an integrated hardware for synchronizing analog or digital devices, delivered a digital signal as soon as the VICON system started recording. Arduino Uno detected this signal and triggered the hotkeys to initiate the OBS recordings. To further ensure precise synchronization, the code recorded the time stamp simultaneously for both the VICON motion capture system and the 2D camera recordings. A schematic of the trigger system is presented in Figure 2.  
The second challenge involved formulating an accurate transformation matrix for converting 3D coordinates to 2D projection values. The calibration method suggested by [PERSON] [49] was employed to calculate the required transformation matrix for each respective synchronized 2D image. An 800 \(\times\) 600 mm checkerboard (11 columns, 8 rows spaced at 60 mm) was used during the calibration process. The 2D projection for all images was determined by the following Equation (1):  
\[s\begin{bmatrix}u\\ v\\ 1\end{bmatrix}=\begin{bmatrix}fx&0&cx\\ 0&fy&cy\\ 0&0&1\end{bmatrix}\begin{bmatrix}r11&r12&r13&tx\\ r21&r22&r23&ty\\ r31&r32&r33&tz\end{bmatrix}\begin{bmatrix}Xw\\ Yw\\ Zw\\ 1\end{bmatrix} \tag{1}\]  
In the equation, (\(u\), \(v\)) represents the projected points, (\(Xw\), \(Yw\), \(Zw\)) represents the 3D ground truth data from the VICON system, and \(s\) represents the scaling factor of the captured images. The first part of the transformation matrix comprising \(fx\), \(fv\), \(cx\), and \(cy\) formulates the camera intrinsic matrix and denotes horizontal and vertical focal lengths and camera principal point coordinates, respectively. The second part of the equation comprised \(rij\), and \(t\) formulates the camera extrinsic matrix. These parameters denoted the relative rotation and translation of the images with respect to the desired global origin.  
To acquire the desired calibration matrix, 50-100 images were captured in different orientations and positions from all the synchronized three differently oriented (0, 45, and 135 degrees) 2D cameras and one additional camera centered at the global origin of the VICON motion capture system. These images were then optimized to obtain proper intrinsic and extrinsic matrices for 2D transformation. The optimization was carried out using the software Calib.io (version v1.6.2a. Calib.io ApS, Svendborg, Denmark) [50].