# Non-Uniform Fusion Tree Generation in a Dynamic Multi-Sensor System
## 2 Preliminaries: The Fusion Process Model
### The Local Fusion Process Model  
This section presents the time-delay model of the local fusion process at the local fusion nodes. The time-delay model of global fusion process is not included, since it is assumed that the central server is equipped with enough computing power to process all received tracks in a fusion cycle. The time-delay model is used in the NU-FTG and local fusion processes. In the NU-FTG, this model is used to calculate the score of each radar. The NU-FTG uses the calculated score of the radars to elect local fusion nodes. In the local fusion process, this model is used to simulate the time-delay of the fusion process at the local fusion nodes. Tracks are the estimated states of targets, and targets are real objects. The state of a track is estimated from a set of measurements that have originated from the same target. The state variable of a track K is:  
\[\widehat{\text{K}}=\big{[}X,Y,Z,V_{x},V_{y},V_{z}\big{]}. \tag{1}\]  
In Equation (1), \(\widehat{\text{K}}\) is the state estimation of a track K, \([X,Y,Z]\) is the estimated position, and \([V_{x},V_{y},V_{z}]\) is the estimated velocity. Figure 2 describes the fusion process model of the local fusion process. The fusion process consists of alignment, assignment, and update. Every fusion cycle, local fusion nodes start the fusion process to create the local SIAP with the tracks from their child radars and their own tracker. The inputs of the \(n\)th fusion process of a local fusion node which has \(N\) child radars is as follows:  
* \(LTL_{all}^{n}=\{LTL_{i}^{n}\mid i=1,\,2,\,\dots,\,N\}\), is a set of local track lists. The \(LTL_{i}^{n}\), which stands for local-track-list, is a set of local tracks received from radar \(i\) after the \((n-1)\)th fusion process. * \(FTL^{n-1}\), which stands for fused-track-list, is a set of tracks in the local SIAP created at the \((n-1)\)th fusion process. It is empty for the first fusion process. In the alignment, the states of tracks are extrapolated to the current time before using tracks that are created or updated at different times. It is assumed that tracks make the uniform motion of a straight line. The execution time of the alignment process can be modeled as:  
\[T_{alg}=(\left|FTL^{n-1}\right|+\sum_{i=1}^{N}\left|LTL_{i}^{n}\right|) \times\text{C}_{1}. \tag{2}\]In Equation (2), \(C_{1}\) is the execution time of the alignment process for a track, and \(\left|FTL^{n-1}\right|\) and \(\left|LTL_{i}^{n}\right|\) stand for the number of tracks in the fused-track-list and the local-track-list, respectively. In the assignment, local tracks are assigned to the tracks in the local SIAP. A local track is assigned to the one of the tracks, originating from the same target, in the local SIAP. The association cost, which is based on the distance between two tracks, is used to decide whether two tracks originated from the same target or not. The following procedures are repeated for every \(LTL_{i}^{n}\), where \(i=1,\ 2,\ \ldots,\ N\):  
* Association costs of every pair of tracks \((a,b)\) where \(a\in LTL_{i}^{n}\) and \(\ b\in FTL^{n-1}\) are calculated. * Based on the association costs and assignment rules, some of the tracks in \(LTL_{i}^{n}\) are assigned to tracks in \(FTL^{n-1}\). Every track of \(LTL_{i}^{n}\) is assigned by, at most, one of the tracks in \(FTL^{n-1}\), and vice versa. The execution time of the assignment can be modeled as:  
\[T_{seg}\approx\sum_{i=1}^{N}\Bigl{(}\left|LTL_{i}^{n}\right|\times\bigl{(} \left|FTL^{n-1}\right|\times C_{2}+C_{3}\bigr{)}\Bigr{)}. \tag{3}\]  
In Equation (3), \(C_{2}\) is the execution time of the process to obtain the association cost of two tracks, and \(C_{3}\) is the execution time of the process to assign a track in \(TTL_{i}^{n}\) to a track in \(FTL^{n-1}\). In the update, the tracks in the local SIAP are updated with the assigned local tracks; unassigned tracks in the local SIAP are removed, and unassigned local tracks are added to the local SIAP. After this process, \(FTL^{n-1}\) becomes \(FTL^{n}\).