## 3 Optimization Algorithms
### Reconstruction Error  
A two-camera system triangulation scheme is described in Figure 4 to explain the reconstruction principle of VSNs in a 2D space. As a kind of directional sensor, the camera can measure the target point's azimuth angle with respect to itself. Thus, the target point must be the intersection of two rays with known azimuth angles starting from the two cameras. With the two angle constraints, the target point is reconstructed by a triangulation method. As shown in Figure 4, the coordinates of the two visual sensors are \((x_{1},y_{1})\) and \((x_{2},y_{2})\), and the azimuth angles of the target point relative to themselves are \(\theta_{1}\) and \(\theta_{2}\). Here, angles \(\theta_{1}\) and \(\theta_{2}\) are defined as the counterclockwise angle from the positive semi-axis of the \(x\) axis, and the ranges are in \([0,2\pi]\). We define \(r_{1}\) and \(r_{2}\) as the distance vectors of the two vision sensors relative to the target point, and their modulus lengths are \(|r_{1}|\)=\(r_{1}\) and\(|r_{2}|=r_{2}\). According to the derivation process in reference [29], the intersection meets the following constraints:  
\[\left\{ \begin{array}{l}\sin(\theta_{1})(x_{1}-x)-\cos(\theta_{1})(y_{1}-y)=0\\ \sin(\theta_{2})(x_{2}-x)-\cos(\theta_{2})(y_{2}-y)=0.\end{array} \right. \tag{19}\]  
The measurement model is an implicit function as Equation (11) with definition \(x=[ \begin{array}{cc}x&y\end{array} ]^{\mathrm{T}}\in\mathbb{R}^{2}\) and \(z=[ \begin{array}{cc}\theta_{1}&\theta_{2}\end{array} ]^{\mathrm{T}}\in \mathbb{R}^{2}\). Then, the Jacobian matrix is square, and the Jacobian determinant is calculated as  
\[\left\|H\right\|=\left\| \begin{array}{cc}-s_{1}&c_{1}\\ -s_{2}&c_{2}\end{array} \right\|/\left\| \begin{array}{cc}c_{1}\Delta x_{1}+s _{1}\Delta y_{1}&0\\ 0&c_{2}\Delta x_{2}+s_{2}\Delta y_{2}\end{array} \right\| \tag{20}\]  
where \(\Delta x_{1}=x_{1}-x,\;\Delta y_{1}=y_{1}-x,c_{1}=\cos(\theta_{1}),\;s_{1}= \sin(\theta_{1})\). Then, the determinant has the form of  
\[\left\|H\right\|=\sin(\theta_{2}-\theta_{1})/\left\| \begin{array}{cc}r_{1} &0\\ 0&r_{2}\end{array} \right\|=\sin(\theta)/r_{1}r_{2} \tag{21}\]  
where \(\theta\) is defined as convergence angle and \(\theta=\theta_{1}-\theta_{2}\). Therefore, the GDOP is calculated as  
\[\mathrm{GDOP}=\left\|H^{-1}\right\|=\frac{r_{1}r_{2}}{\sin(\theta)}. \tag{22}\]  
Based on the above derivation, GDOP is equal to the product of the distance from the two vision sensors to the target point and the cosecant value of convergence angle \(\theta\). When the target point is on the line connecting the two sensors (\(\theta=0\)), the GDOP tends to infinity; when the convergence angle between the two sensors becomes larger, the GDOP decreases. GDOP reaches its minimum when the convergence angle of the two sensors is a right angle, and the target is close to the sensor. Based on Equation (22), GDOP measures the degree to which the sensor measurement error is amplified into the reconstruction error during the reconstruction process. Therefore, the quality of the reconstruction result can be represented by the reciprocal of GDOP. To confine the quality function to \([0,1]\), a suitable scale factor, \(d_{scale}\), is chosen, and sensors \(i\) and \(i^{\prime}\) to target location \(j\)'s reconstruction quality function can be obtained as follows:  
\[q_{ii^{\prime}j}=d_{scale}\frac{\sin(\theta)}{|r_{i}|r_{j^{\prime}}}v_{ij}v_{ ij}. \tag{23}\]  
Figure 4: Schematic diagram of the principle of 2D spatial reconstruction. The expression of \(v_{ij},v_{fj}\) is shown in Equation (14), and these values guarantee the reconstruction quality to be zero when sample point \(j\) is not 2-coverage. It is worth mentioning that the above quality function can be extended based on the basic model employed. For example, if a probability coverage model rather than a binary coverage model is used, the coverage probability can multiply. In addition, more than two cameras may cover the target point at the same time.