## 3 Methodology
### Space Syntax Monitoring Scene Analysis
#### 3.2.1 Convex Space Analysis  
Convex space analysis is an effective method for analyzing architectural structure scenes. By analyzing the connections between relatively independent convex spaces in a monitoring scene, key areas within the scene can be identified, and their corresponding spatial distribution and topological relationship feature values can be calculated. Considering that the number of cameras in a typical video surveillance system is limited and their coverage is relatively simple in two-dimensional space, convex space analysis is generally performed manually, defining the connectivity between different convex spaces. For large-scale and complex 3D environments, it is necessary to introduce intelligent convex space analysis methods.  
After obtaining the convex spaces of the monitoring scene, the corresponding spatial topology diagram needs to be constructed. As shown in Figure 2, an example of a monitoring space CAD diagram and its convex space division with the corresponding topology diagram is provided. There are various ways to divide convex spaces, and the installation and coverage of the surveillance system must take into account semantic factors, such as property ownership relationships. Convex space division and its relationships in complex scenes may also undergo dynamic adjustments over time. For instance, as access control systems change over time, their connectivity characteristics may also evolve. This paper focuses on how to optimize the deployment of video systems based on the division of convex spaces.  
Figure 1: The overall framework of camera placement optimization.  
Figure 2: Convex space analysis and its graph representation. (**a**) Floor CAD, (**b**) segmented convex space, and (**c**) topology graph.  
After partitioning the convex spaces, the following metrics are calculated based on the convex space topological relationship graph:  
* Connectivity Value: This is the count of adjacent nodes to a convex space node as shown in (1). A higher value indicates better connectivity and spatial permeability. \[\text{C}_{i}=\text{k}\] (1)
* Control Value: Assuming that each node in the system has a weight of 1, node A receives a weight from adjacent node B of 1/(B's connectivity value). The sum of the reciprocals of the connectivity values of all nodes directly connected to A is A's control value as shown in (2). The control value reflects the degree of mutual control between nodes. \[\text{C}_{1i}=\sum_{j=1}^{k}\frac{1}{\text{C}_{i}}\] (2)
* Depth Value: If the distance between two adjacent nodes is 1, then the sum of the shortest path distances between node \(i\) to all other nodes is its depth value of node \(i\) as shown in (3). \[d_{i}=\sum_{j=1}^{n}d_{ij}\] (3)
* Average Depth Value: This is the average depth value from a node to all other nodes, representing the accessibility of a particular node to other nodes and identifying key areas in video surveillance as shown in (4). \[D_{i}=\frac{d_{i}}{n-1}\] (4)  
These four values are calculated and normalized into 0 to 1; then, they are averaged and added to create the convex importance value for further analysis.