###### Abstract  
This study presents a comprehensive approach for single sensor placement optimization in two-dimensional and three-dimensional spaces. A traditional exhaustive search technique and a novel method called recursive exhaustive search are used to place a sensor in a way that maximizes the area coverage metric. Exhaustive search provides a baseline by methodically evaluating all potential placements, while recursive exhaustive search innovates by segmenting the search process into more manageable, recursive steps. Our findings highlight the significant impact of two key parameters, the number of evaluations and the rasterization value, on the achieved coverage and computation time. The results show that the right choice of parameters can significantly reduce the computational effort without compromising the quality of the solution. This underlines the critical need for a balanced approach that considers both computational complexity and placement efficacy. We show that exhaustive search is not feasible for three-dimensional environment models and propose to establish a modified exhaustive search method as a ground truth for the single sensor placement problem. We then explore nature-inspired genetic algorithms and the impact of the number of evaluations of the optimization function for these algorithms on both accuracy and computational cost. sensor placement; stochastic optimization; exhaustive search; sensor networks; Internet of Things +
Footnote †: journal: Applied sciences  
0000-0001-5000-0000-0000-0000-0000-0000-00000-0000-0000-0000-0000-0000-0000-  
An optimal placement of visual sensors can improve the performance of video surveillance systems while minimizing the number of sensors required to cover a given area, reducing the overall cost of installation and operation. It may be important to cover certain areas of the environment, such as pathways, areas around exhibited artwork and similar locations. When analyzing human behavior, extending the coverage of visual or isotropic sensors to specific areas--such as in front of product shelves, advertising displays or showcases--allows for a closer examination of people's interest in specific products or advertising types. The challenge of determining the optimal positioning of sensors in an environment arises from the Art Gallery problem (AGP) [4]. This problem has its roots in the concept of visibility, a measure that quantifies the effectiveness of sensors in detecting and observing objects in an environment. Visibility appears as an important aspect in various fields such as robotics, computer vision, optimization and computer graphics. Originally, the AGP aimed to find optimal locations for guards that could monitor a full 360\({}^{\circ}\) area from fixed positions, typically within a polygonal art gallery. In its original form, the AGP assumed that visibility between two points is given if the line segment connecting them does not intersect any obstacles--a principle known as line-of-sight constraint. Subsequent research extended the AGP by incorporating distance and incidence constraints [5], but retained the simplified two-dimensional polygonal approach. In these studies, polygonal representations of the floor plan of the environment were used, which often inadequately capture the complicated spatial arrangements of buildings. In many practical scenarios, understanding the three-dimensional spatial structure of a building is crucial for developing effective monitoring strategies [6]. In three-dimensional environments, research usually relies on elevation maps, which are limited to planar terrains [7] or orthogonal polyhedral environments [8]. Both the two-dimensional and the three-dimensional version of the Art Gallery problem have been shown to be NP-hard. However, while solutions can be found for certain instances of the two-dimensional problem and some three-dimensional scenarios with simple polygons [5; 9; 10], practical implementations are faced with increasing complexity due to the different shapes of real-world polygons.