## 2 Trinocular Vision Model
### Three-Dimensional Vision  
The trinocular camera has a better visual matching effect compared with the binocular camera. Assuming that the object is located at a certain point \(P\), the projection point of the target object on its imaging surface in camera 1 is p\({}_{1}\), and the camera coordinate origin is O\({}_{\text{c1}}\); similarly, the corresponding points in camera 2 and camera 3 are set to p\({}_{2}\), O\({}_{\text{c2}}\) and p\({}_{3}\), O\({}_{\text{c3}}\), respectively. Due to camera aberrations, as well as the solution error in the least squares [18] calculation and the noise generated during calibration, the line between the origin and the projection point of the three groups of cameras will be slightly shifted to the real position \(P\) of the target object, meaning that the coordinate position \(P_{1}\) of the target image captured by the binocular vision system composed of camera 1 and camera 2 inevitably cannot coincide with the display position \(P\) of the target object. Similarly, the target image positions \(P_{2}\) and \(P_{3}\) captured by camera 2 and camera 3 as well as camera 1 and camera 3 will also fail to coincide with each other and the target object.  
In order to reduce the gap between the position of the target image obtained by the binocular vision system composed of each group of cameras and the actual position of the target object in the world coordinate system, reduce the impact of subsequent calculations and improve the effect of 3D reconstruction, this paper proposes a joint solution algorithm based on trinocular cameras to realize the joint optimization of coordinate points \(P_{1}\), \(P_{2}\) and \(P_{3}\), reduce the system error and make the obtained coordinate values of the target object more accurate.  
From Figure 1, it can be intuitively seen that the object world coordinate point \(P\) is in the middle of \(P_{1}\), \(P_{2}\) and \(P_{3}\), so it can be considered that the minimum value of the sum of the relative distances between P and these three points is the more accurate real coordinate point, as shown in Equation (1).  
\[F=\text{min}(\|P-P_{1}\|+\|P-P_{2}\|+\|P-P_{3}\|) \tag{1}\]  
Figure 1: Convergent trinocular stereo vision model.  
Cameras 1 and 2 measured point \(P_{1}\) coordinates are (\(X_{1}\), \(Y_{1}\), \(Z_{1}\)); similarly, the coordinates of point \(P_{2}\) and point \(P_{3}\) are (\(X_{2}\), \(Y_{2}\), \(Z_{2}\)) and (\(X_{3}\), \(Y_{3}\), \(Z_{3}\)), and the three coordinate point values are substituted into Formula (1) expansion to obtain  
\[\begin{array}{l}F=\min\left[\left(X-X_{1}\right)^{2}+\left(X-X_{2}\right)^{2 }+\left(X-X_{3}\right)^{2}\right]\\ +\min\left[\left(Y-Y_{1}\right)^{2}+\left(Y-Y_{2}\right)^{2}+\left(Y-Y_{3} \right)^{2}\right]\\ +\min\left[\left(Z-Z_{1}\right)^{2}+\left(Z-Z_{2}\right)^{2}+\left(Z-Z_{3} \right)^{2}\right]\end{array} \tag{2}\]  
Thus, the true coordinates of point \(P\) can be derived by applying the properties of the arithmetic mean, i.e., as shown in Equation (3).  
\[\left\{\begin{array}{l}X=\frac{X_{1}+X_{2}+X_{3}}{3}\\ Y=\frac{Y_{1}+Y_{2}+Y_{3}}{2}\\ Z=\frac{Z_{1}+Z_{2}+Z_{3}}{3}\end{array}\right. \tag{3}\]  
By solving the above equation for the coordinate values of realistic target points, more accurate values can be obtained than those of binocular vision systems.