## 3 Simulation Model for Surveillance Systems
### Surveillance Resolution  
In this section, we introduce how to measure the surveillance resolution of the target object in the simulation. The first step to calculate the target's resolution is the sampling distance calculation of \(SD_{h_{i}}\) and \(SD_{v_{i}}\), which are the sampling distances for horizontal and vertical aspects, respectively. (The sampling distance is the distance between the centers of two adjacent pixels in an image, as represented as the length per pixel, which is measured based on the distance from the optic lens to the target object, the width of the image sensor, the focal length, and the number of pixels on the image sensor.) Then, we represent the sampling distance in two ways based on the value of the sampling distance considering the minimum and maximum distance from the optic lens to the target.  
First, the sampling distances \(SD_{h_{i}}\) and \(SD_{v_{i}}\) in the horizontal and vertical aspect are calculated by (11) and (12).  
\[SD_{h_{i}} = \frac{L_{sw}\times d_{i}^{h}}{L_{f}\times im_{w}} \tag{11}\] \[SD_{v_{i}} = \frac{L_{sh}\times d_{i}^{v}}{L_{f}\times im_{h}} \tag{12}\]  
where \(d_{i}^{h}\) is the distance from the optic lens to the target point \(p_{t_{i}}\), and \(im_{w}\) is the number of pixels on the image sensors in the horizontal aspect. Similarly, \(d_{i}^{v}\) and \(im_{h}\) are the distance from the optic lens to the target point and the number of pixels on the image sensors in the vertical domain. As mentioned in Section 3.3, since in most cases, any plane of the target object is not parallel with the image sensor and the optic lens, the sampling distance is not uniform on the target plane, so we can classify it into two types of the sampling distance, i.e., the minimum and maximum values. For example, in Figure 2b, the distance from the optic lens to \(p_{t_{2}}\) is the shortest, so the sampling distance is the highest and \(SD_{h_{2}}\) is the maximum sampling distance \(SD_{h_{i}}^{max}\), whereas \(SD_{h_{1}}\) is the minimum sampling distance of the target object.  
As the sampling distance is available, the target object's number of pixels in the horizontal and vertical aspect is determined using (13). Note that the surveillance resolution in the vertical aspect is also calculated similarly to the horizontal aspect.  
\[SR_{i}=P_{h_{i}}\times P_{v_{i}} \tag{13}\] \[P_{h_{i}}=\frac{L_{tw}}{SD_{h_{i}}^{max}}\] (14) \[P_{v_{i}}=\frac{L_{th}}{SD_{v_{i}}^{max}} \tag{15}\]  
Herein, the surveillance resolution of target \(i\) is denoted by \(SR_{i}\). It is determined by the product of the number of horizontal pixels \(P_{h_{i}}\) and the number of vertical pixels \(P_{v_{i}}\), as represented by (14) and (15), respectively. The quantity \(P_{h_{i}}\) corresponds to the number of pixels that lie in the horizontal domain. It is computed as the ratio of the width \(L_{tw}\) of the target to the maximum horizontal sampling distance, \(SD_{h_{i}}^{max}\). Similarly, \(P_{v_{i}}\) represents the number of pixels that fall in the vertical domain. It is calculated as the quotient of the height, \(L_{th}\), of the target and the maximum vertical sampling distance, \(SD_{v_{i}}^{max}\). Hence, it can be inferred that the surveillance resolution \(SR_{i}\) is an indicator of the maximum number of pixels that can be used by the surveillance camera to represent a target that is modeled as a rectangle.