## 2 Problem Formulation
### Reconstruction Error Model  
As shown in Figure 3, a simple rectangular environment with two cameras placed opposite each other at the boundary edges is given to clarify the influence of sensor spatial position uncertainty on reconstruction results. The vision sensor can be considered as an orientation sensor to measure the azimuthal angle to the target point, and the uncertainty range of angle measurement is indicated by a light blue triangle. Based on the measurements of the two sensors, the reconstructed coordinates of the target point can be obtained by solving the triangulation method. The two triangular overlapping regions indicate the reconstruction region of this target point, and the size of the region characterises the size of the uncertainty in the position reconstruction, i.e., the reconstruction error. In this case, the dark blue region on the line connecting the two sensors indicates low reconstruction accuracy, while the smaller blue regions in the upper and lower halves indicate high reconstruction accuracy [20]. It can be seen that the positional uncertainty on the straight line between the two sensors is high due to the geometric properties of the sensors. Therefore, if the target point is located in this region, the reconstruction error is higher. The relationship between the reconstruction error and the camera placement strategy is explained in the preceding qualitative analysis. In order to obtain the reconstruction error model resulting from the geometric features of the sensor placement in greater detail,  
Figure 2: Uniform sampling of a 2D rectangle target space. in [29], [PERSON] defined a geometrical precision factor (GDOP, Geometric Dilution of Precision) to map sensor measurement error to triangulation reconstruction error, which is a measure of positioning uncertainty of each point in the working space caused by the two sensors observing it. The pose of the target object can be defined by the following vector:  
\[x(t)=[ \begin{array}{cc}x(t)&y(t)&\theta(t)\end{array} ]^{\mathsf{T}}\in\mathbb{ R}^{3}. \tag{3}\]  
The visual measurements are represented by vector \(z\in\mathbb{R}^{m}\), which is connected to the state vector using a general nonlinear function:  
\[z=h(x). \tag{4}\]  
In some particular cases, the relationship may be linear or invertible (state \(x\) is determined as an explicit function of measurement \(z\)), and the above measurement Model (4) is applicable more generally. It is noted that each element of the measurements poses a constraint on the estimated state. Therefore, the system state can be solved when sufficient constraints are available. However, it is expected that the measurements may be corrupted by additive errors such as noises and disturbances. We suppose that the visual measurements are disturbed by error \(\delta z(t)\):  
\[z^{\prime}(t)=z(t)+\delta z(t) \tag{5}\]  
where error \(\delta z(t)\) may be a systematic measurement error or a random error. We let \(H=\partial h/\partial x\in\mathbb{R}^{m\times 3}\) be the Jacobian matrix of the measurement model. The relationship between measurement error \(\delta z\) and system state error \(\delta x\) is as follows:  
\[\delta z=H\delta x \tag{6}\]  
In general, attention needs to be paid to how the sensor measurement error is amplified into the pose parameter error to be estimated. In this way, a geometric precision factor can be loosely defined as  
\[\text{GDOP}=\|\delta x\|/\|\delta z\| \tag{7}\]  
where \(\|\delta x\|\) and \(\|\delta z\|\), respectively, represent the measurement of sensor pose error and observation error under any quantitative standard. The mapping between the sensor error and the pose error is in the form of a matrix, and any norm of the matrix can measure the relative magnification of the mapping. The Jacobian determinant is a convenient choice among the available norms because it avoids the calculation of eigenvalues.