## 1 Introduction
However, there is also a more realistic probabilistic sensing model in which the coverage in the sensor's sensing range is described as a probability function [23]. Each point in the space is assigned a probability, indicating the probability for the point to be covered. Sometimes, some variables in the layout problem are binary (such as whether or not the camera is placed), and some variables are integers (such as the number of cameras). This kind of problem can be treated as an MIP problem. There is also a large class of discrete optimization algorithms called heuristic algorithms, which are usually used in combinatorial optimization. Although these algorithms cannot guarantee optimal solutions, they can produce a satisfactory outcome within reasonable time. Heuristic algorithms include Simulated Annealing (SA) [24], Genetic Algorithm (GA) [25; 26], Particle Swarm Optimization (PSO) [27; 28] and so on. The current literature on sensor coverage rarely considers both reconstruction error and coverage; instead, the majority of studies concentrate on the 1-coverage problem for a target point, which states that it is sufficient to require that the object point be covered by a single sensor. However, in practical applications, it is sometimes necessary to make the target area covered by multiple sensors. For reconstruction of spatial points by multi-camera systems, it is usually required that the spatial points are captured by multiple cameras at the same time, i.e., the requirement of at least two coverage points must be satisfied. By striking a balance between coverage and reconstruction error, the system is able to capture scene information and provide high-quality reconstruction. Therefore, in this paper, we consider the reconstruction error constraints of the visual tracking task and comprehensively consider the compromise balance of maximizing coverage and minimizing reconstruction error as the objective function. The significant contributions of this paper are as follows:  
* Detailed mathematical models of the sensor placement optimization problem are built. * A search approach for dimensionality reduction is proposed. * A multi-objective optimization function is proposed that considers reconstruction error in addition to coverage. * A combination of PSO and GA algorithms is used to avoid the local optima while increasing optimization speed. * A sensor coverage method for large-range target space with obstacles is discussed. The rest of the paper is structured as follows: in Section 2, we provide a mathematical formulation about the sensor placement problem and also a few basic models, including sensor model, space model, coverage model, and reconstruction error model and propose a dimensionality reduction search method. Next, in Section 3, a multi-objective optimization function is established to consider the space coverage and reconstruction error for 2D spaces, followed by a suitable multi-objective optimization algorithm. In Section 4, extensive comparison experiments are carried out to evaluate the given camera position optimization algorithms. Finally, Section 5 summarizes the conclusions and proposes plans for future research.