## 3 Materials and Methods
### Chained-Tracker Pipeline
#### 3.3.2 Node Chaining  
For convenience, we have used \((B_{t-1})\) to represent \([((B_{t-1}^{i},\ \hat{B}_{t}^{i}))]_{i=1}^{n_{t-1}}\). Two important steps for chaining can be highlighted. Firstly, we have nodes with identity tracks randomly assigned to each of them, and hence, the bounding boxes \(B_{1}^{i}\in B_{1}\) are detected during the initialization phase. Secondly, the two nodes \((F_{t-1},F_{t})\) and \((F_{t},F_{t+1})\) are chained for any node \(t\) provided the condition of adjacency is preserved. The IoU, Kalman filter, and Bipartite matching between the bounding boxes in \(B_{t}\) and \(\hat{B}_{t}\) are calculated as illustrated in Figure 1 where \(\hat{B}_{t}\) is the last box set of \((B_{t-1},\ B_{t})\), and \(B_{t}\) are the earlier box set of \((B_{t},\ \hat{B}_{t+1})\). If the IoU affinity is obtained, the detected \(\hat{B}_{t}\) and \(B_{t}\) boxes are matched by applying the Kuhn-Munkres (KM) algorithm [47]. The tracklet, to which \(\hat{B}_{t}^{i}\) belongs for each of the matching box pair \(\hat{B}_{t}^{i}\) and \(B_{t}^{i}\), is updated by adding \(B_{t}^{i}\). The unparalleled box \(B_{t}^{k}\) is initialized with an identity as a tracklet. Therefore, long paths are generated for the video sequence by using adjacent nodes, and chaining is performed sequentially.  
Two nodes were chained by just IoU matching, which is insufficient to chain two nodes and thus resulted in a greater number of IDs (Identity Switch) than previous methods (Section 4.2). The use of box IoU is the reason behind a lot of ID switch present. These are mostly used for fast camera motion in a crowd. IoU [48] is the prevalent evaluation metric for object detection. We have linked two nodes with IoU using strategies as given by:  
1. Adding Kalman Filter [49] (predicts object location in the next frame) helps to get smooth and reasonable tracklets which causes a decrease in the number of ID switches. We can keep only those detections for tracking purposes whose predicted location is near to previous frame detection. The mathematical equations of the Kalman filter provide an efficient computational means for evaluating the states of a process by helping evaluate the present, past, and future states. It can also help evaluate when the modeled system is unknown.  
Figure 1: Schematic details of the chaining node: the pair of bounding boxes denoted by \((B_{t-1},\ \hat{B}_{t})\) generated, and after that, SCT exploits two random adjacent nodes represented by \((F_{t-1},F_{t})\) and \((F_{t},\ F_{t+1})\). Subsequently, IoU matching, Kalman filtering, and Bipartite matching are used on the common frame for sequencing the two nodes. The long trajectories are generated for the video sequence by using adjacent nodes, and chaining is performed sequentially.  
Using Bipartite Matching [50] (one-to-one mapping), we can assign one identity to one person in the next frame. It ensures one person maps to only one person in the next frame and thus reduces the ID switch. The bipartite matching framework [51] allows us to factorize node similarity in the search for a one-to-one correspondence between nodes in two graphs.