## 3 Preliminaries  
When a vehicle moves between two adjacent points, its path in space can be determined by observing the vehicle's spatial position between adjacent points. In this section, time reconstruction is performed for each pair of adjacent collection points. First, the following definition is given.  
**Definition 1**.: _Adjacent point spatial trajectory: After spatial trajectory reconstruction, for an AVI trajectory that moves between a pair of adjacent points \((C_{A},C_{B})\), the trajectory is represented in space-time trajectory format as_  
\[[(as_{A},at_{A}),(as_{B},at_{B})]\rightarrow[s_{1},s_{2},\ldots,s_{n}] \tag{1}\]  
_where \([(as_{A},at_{A}),(as_{B},at_{B})]\) are the spatial and temporal positions of the vehicle at points \(C_{A}\) and \(C_{B}\), respectively; \([s_{1},s_{2},\ldots,s_{n}]\) is the reconstructed trajectory; \(n\) is the number of reconstructed spatial points; and \(s_{i}\) denotes the \(i\)-th spatial position in the trajectory._  
_In trajectory reconstruction, the AVI records at both ends of the trajectory \((as_{A},at_{A})\) and \((as_{B},at_{B})\) contain time information, while the reconstructed spatial trajectory \([s_{1},s_{2},\ldots,s_{n}]\) only contains spatial information. The purpose of spatiotemporal trajectory reconstruction, which is the focus of this section, is to complete the time information for all spatial points in the reconstructed spatial trajectory \([s_{1},s_{2},\ldots,s_{n}]\). To this end, spatiotemporal trajectory reconstruction can be defined as in Definition 2._  
**Definition 2**.: _Spatiotemporal trajectory reconstruction: For a given pair of adjacent points \((C_{A},C_{B})\) and a pair of adjacent AVI records \([(as_{A},at_{A}),(as_{B},at_{B})]\) between these collection points, the goal of spatiotemporal trajectory reconstruction is to reconstruct the spatiotemporal trajectory between the adjacent points, which can be represented as_  
\[[(as_{A},at_{A}),(as_{B},at_{B})]\rightarrow[(s_{1},t_{1}),(s_{2},t_{2}), \ldots,(s_{n},t_{n})] \tag{2}\]  
_where \(n\) represents the total length of the reconstructed spatiotemporal trajectory; \((s_{i},t_{i})\) represents the spatiotemporal position of the \(i\)-th point; \(s_{i}\) represents the spatial position; and \(t_{i}\) represents the time at which the vehicle passes through this point. The goal of spatiotemporal trajectory reconstruction is illustrated in Figure 3._  
**Definition 3**.: _Historical trajectory: This refers to the time-space normalized result of all trajectories in a trajectory set \(T(A,B)^{k}\) for a given adjacent pair of collection points \((C_{A},C_{B})\). For a specific path \(k\), the historical trajectory set \(T(A,B)^{k}\) can be represented as_  
\[T_{(}A,B)^{k}=Tr_{1},Tr_{2},\ldots,Tr_{N} \tag{3}\]  
_where \(k\) represents the path number, \(N\) represents the total number of neighboring trajectories in the set, \(Tr_{i}\) represents the \(i\)-th trajectory in the set, and each trajectory \(Tr_{i}\) is represented as_  
\[Tr_{i}=[(as_{A},at_{A}),(gs_{1},gt_{1}),(gs_{2},gt_{2}),\ldots,(gs_{m},gt_{m}),(as_{B},at_{B})] \tag{4}\]  
_where \((as_{A},at_{A})\) and \((as_{B},at_{B})\) are the fixed positions and timestamps at \(C_{A}\) and \(C_{B}\), respectively. \((gs_{j},gt_{j})\) represents the \(j\)-th GPS record in the trajectory. The historical trajectory defined in this section is the result of time-space normalization applied to all trajectories in the trajectory set, represented as \(T_{\rm H}\). The method for the time-space normalization of trajectories is described below._  
Figure 3: The goal of AVI trajectory spatial–temporal reconstruction.