## 3 Optimization Algorithms
### Multi-Object Optimization  
Based on the mathematical models established in the previous parts, as well as 2D space coverage \(r\) and average reconstruction quality \(q\), the primary purpose of this subsection is to establish a multi-objective optimization function that takes the characteristics of these two objective functions into account. At the same time, an appropriate multi-objective optimization method is constructed to handle the sensor placement optimization problem. In practical applications, it is often necessary to consider both coverage and reconstruction error. The coverage and reconstruction quality can be calculated by Formulas (18) and (25), respectively. As can be observed, the two evaluation index values, coverage \(r\) and reconstruction quality \(q\), have the same order of magnitude and fall inside the range of \([0,1]\). Thus, a multi-objective optimization function is established to comprehensively evaluate the performance of the system by calculating their weighted sum:  
\[e=\omega r+(1-\omega)q \tag{26}\]  
where the degree to which the coverage is subject to the evaluation criteria is indicated by weighting coefficient \(\omega\in[0,1]\). When \(\omega=1\), the evaluation criterion is to maximize the coverage; when \(\omega=0\), it is to maximize the reconstruction quality, i.e., to minimize the reconstruction error. To sum up, the sensor placement optimization problem in this paper can be abstracted as the following nonlinear optimization problem:  
\[\operatorname*{arg\,max}_{\Pi}G(\Pi),s.t.,\forall C^{p}\in\mathcal{B},W^{p} \in\mathcal{M} \tag{27}\]  
where \(\Pi\)=\(\{c_{x,1},c_{y,1},\)\(\varphi_{1},.......,c_{x,N},c_{y,N},\varphi_{N}\},G=r,q,\) or \(e\), \(N\) is the number of cameras. The above process converts sensor pose parameters into discrete variables through an equidistant sampling of space and angle. Meanwhile, the target space is uniformly sampled into a set of grid points. By applying geometric relations to calculate the coverage and reconstruction error of each sampling point, we thus establish a discrete model of the sensor layout problem. This method can simplify the model, speed up the calculation, and then maintain the accuracy required for the task. In order to tackle the discrete optimization problem (27), this research suggests combining the GA and PSO algorithms. GA has been proven to be an effective method for optimizing sensor configuration and for determining the position of attitude sensors. The optimization process of GA does not rely on gradients or other additional information,and the search results can be used as the near-optimal solution to the problem. Still, these results depend on the initial value (initial population). When the initial value is not correctly selected, the algorithm's performance may be affected. On the other hand, PSO is one of the best-known nature-inspired algorithms, inspired by information loops and social behaviors such as bird flocking and fish flocking. This is a global optimization algorithm, especially suitable for solving problems where the optimal solution is a multi-dimensional space parameter. Each particle in the group represents a solution in the solution space. The algorithm solves the problem intelligently through the movement of individual particles and the exchange of information within the group. Due to its simplicity in implementation and fast convergence speed, PSO is widely used. But it still suffers from problems such as dimensionality explosion, and it can easily fall into a local optimum since it convergences too fast. Through the above analysis, it is shown that both GA and PSO have advantages and disadvantages. Therefore, in this section, a combined algorithm is proposed to merge the advantages of the PSO and the GA. First, the PSO is used, and the result is taken as the initial value of the GA, which can reduce the running time, improve the efficiency of the algorithm, and avoid the algorithm falling into the local optimal solution as far as possible.