## 3 Simulation Model for Surveillance Systems
### Surveillance Area  
Figure 2b depicts the various parameters of an image sensor (shown in small rectangles), an optic lens (shown in ellipses), and a target object (shown in large rectangles). Initially, we assume that the center of the optic lens is located at (0,0) and the plane of the target object is accurately parallel with the image plane (or the optic lens). Then, we consider the level of rotation of the optic lens and the image plane (i.e., the viewing angle of the camera) to calculate the surveillance area in the coordinate system. As shown in Figure 2b, a camera's surveillance area in horizontal aspects lies within the two borderlines represented by \((H_{x_{1}},H_{y_{1}})\) and \((H_{x_{2}},H_{y_{2}})\), respectively. Another line represented by \((H_{x_{0}},H_{y_{0}})\) goes through the surveillance area's center, which divides the surveillance area in half and is also utilized to determine if the target belongs to the surveillance area. Equations (1) and (2) define these three lines: the function of variable \(x\) for the distance from the optic lens to the target, where \(H_{FOV}(x)\) is the FOV (Field of View) for variable \(x\) and \(\theta_{h}\) is the viewing angle of the camera. Here, the FOV is a well-known factor defined as \(H_{FOV}(x)=x\times\frac{L_{sw}}{L_{f}}\), and \(L_{sw}\) and \(L_{f}\) are the width of the image sensor and the focal length, respectively. A further elaboration of Equations (1) and (2) from a geometrical perspective can be found within the confines of Appendix A of this paper. \[H_{x_{i}}= \begin{cases}xcos\theta_{h}&i=0\\ xcos\theta_{h}-\frac{1}{2}H_{FOV}(x)sin\theta_{h}&i=1\\ xcos\theta_{h}+\frac{1}{2}H_{FOV}(x)sin\theta_{h}&i=2\end{cases}  \tag{1}\]  
\[H_{y_{i}}= \begin{cases}xsin\theta_{h}&i=0\\ xsin\theta_{h}+\frac{1}{2}H_{FOV}(x)cos\theta_{h}&i=1\\ xsin\theta_{h}-\frac{1}{2}H_{FOV}(x)cos\theta_{h}&i=2\end{cases}  \tag{2}\]  
In the vertical aspect, the surveillance area is also calculated by (3) and (4), which are similar to (1) and (2); however, several factors, such as the FOV \(H_{FOV}(x)\) and the viewing angle \(\theta_{h}\), should be substituted. That is, \(H_{FOV}(x)\) is replaced with \(V_{FOV}(x)\), defined as \(x\times\frac{L_{sh}}{L_{f}}\), and \(\theta_{v}\) is used for the viewing angle in the vertical aspect instead of \(\theta_{h}\). \[V_{x_{i}}= \begin{cases}xcos\theta_{v}&i=0\\ xcos\theta_{v}-\frac{1}{2}V_{FOV}(x)sin\theta_{v}&i=1\\ xcos\theta_{v}+\frac{1}{2}V_{FOV}(x)sin\theta_{v}&i=2\end{cases}  \tag{3}\]  
\[V_{z_{i}}= \begin{cases}xsin\theta_{v}&i=0\\ xsin\theta_{v}+\frac{1}{2}V_{FOV}(x)cos\theta_{v}&i=1\\ xsin\theta_{v}-\frac{1}{2}V_{FOV}(x)cos\theta_{v}&i=2\end{cases}  \tag{4}\]  
Lastly, since we assumed that the center of the optic lens was located at (0, 0), the surveillance area of the coordinate system should be corrected to the original coordinates by (5). \[\overline{H_{x_{i}}} =H_{x_{i}}+x_{o} \tag{5}\] \[\overline{H_{y_{i}}} =H_{y_{i}}+y_{o}\] \[\overline{V_{x_{i}}} =V_{x_{i}}+x_{o}\] \[\overline{V_{z_{i}}} =V_{z_{i}}+L_{ih}\]  
The next step is to determine the effective surveillance distance of the camera, as depicted in Figure 4. The effective surveillance distance of the camera is defined by (6), which is determined by taking into account the difference between the camera's maximum surveillance distance, denoted by \(D_{max}\), and its distance to the blind zone, as represented by \(D_{bz}\). \[D_{eff}=D_{max}-D_{bz} \tag{6}\]  
\[D_{max}=(L_{ih}-L_{th}^{max})\times tan(\frac{\pi-V_{AOV}}{2}-v_{v}) \tag{7}\]  
As defined in (7), the camera's maximum surveillance distance, denoted by \(D_{max}\), is calculated on the basis of the height of the target object that the camera can detect. Here, \(L_{ih}\) denotes the installation height of the surveillance camera, \(L_{th}^{max}\) denotes the maximum height of the target object, \(V_{AOV}=\arctan(L_{sh}/2L_{f})\) refers to the angle of view of the camera, and \(V_{v}\) represents the viewing angle of the camera.