# Multi-Object Tracking with Predictive Information Fusion and Adaptive Measurement Noise
## 2 Materials and Methods
### Adaptive Measurement Noise (AMN)  
The classic tracking algorithm ByteTrack utilizes the KF as its motion model. The state vector includes the object's center coordinates \(x_{c}\), \(y_{c}\), aspect ratio \(a\), height \(h\), and their corresponding velocity components, as shown in Equation (6):  
\[x=[x_{c},y_{c},a,h,x_{v},y_{v},a_{v},h_{v}]^{T} \tag{6}\]  
This state vector typically assumes that the object's contour does not undergo significant deformation, meaning that the aspect ratio of the tracked object remains relatively constant. However, this assumption does not always hold. In certain cases, the object may exhibit nonlinear movements, such as a pedestrian falling, a dancer dancing, or an athlete competing. In these scenarios, the traditional state space setup may struggle to effectively handle the object's morphological changes, impacting tracking accuracy. However, our experiments show that directly estimating the width and height of the bounding box yields better tracking of complex motion objects. Therefore, we improved the state space of the Kalman filter by replacing the aspect ratio \(a\) and its velocity components with the width \(w\). At this point, the complete state vector of the Kalman filter is shown in Equation (7):  
\[x=[x_{c},y_{c},w,h,x_{v},y_{v},w_{v},h_{v}]^{T} \tag{7}\]  
The KF consists of prediction and update steps. The prediction steps, as shown in Equations (8) and (9), predict the current state based on historical information, resulting in a priori estimate. Here, \(x_{k}^{-}\) represents the a priori estimate state at the current time, \(x_{k-1}\) is the a posteriori estimate from the previous time step, and \(F\) is the state transition matrix that describes the evolution of the system's state from the previous to the current time step, as shown in Equation (10). Additionally, we calculated the a priori state covariance matrix, where \(P_{k}^{-}\) represents the a priori state covariance matrix at the current time, \(P_{k-1}\) is the a posteriori state covariance matrix from the previous time step, and \(Q\) is the process noise covariance matrix. \[x_{k}^{-}=Fx_{k-1} \tag{8}\]  
\[P_{k}^{-}=FP_{k-1}F^{T}+Q \tag{9}\]  
Figure 4: The figure illustrates the workflows of two different methods. The yellow boxes in (2) represent the detection candidate boxes before NMS is applied. The first row demonstrates the processing flow of the classical NMS: the yellow box in (3) indicates the detection box resulting from applying NMS to the detection candidate boxes in (2). Due to significant overlap between the boxes, (3) ultimately retains only one detection box for the frame. The white boxes in (4) show the object position predicted by the Kalman filter., The green box in (5) represents the result of associating the detection box from (3) with the prediction boxes from (4), successfully associating only one object trajectory. The second row showcases the workflow of our method: the yellow boxes in (6) indicate the detection boxes resulting from our method applied to the detection candidate boxes in (2), successfully detecting two objects. The green boxes in (7) represent the result of associating the detection boxes from (6) with the prediction boxes from (4), successfully associating both objects. \[F= \begin{bmatrix}1&0&0&0&1&0&0&0\\ 0&1&0&0&0&1&0&0\\ 0&0&1&0&0&0&1&0\\ 0&0&0&1&0&0&0&1\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1\end{bmatrix}  \tag{10}\]  
The update steps are shown in Equations (11)-(13). First, we calculated the Kalman gain \(K\). Then, using the observation matrix \(H\) and the observation vector \(z_{k}\), we updated the posteriori estimate \(x_{k}\) and the posteriori estimate covariance \(P_{k}\), where \(R\) is the measurement noise covariance matrix. The observation matrix \(H\) is shown in Equation (14). \[K=\frac{P_{k}^{-}H^{T}}{HP_{k}^{-}H^{T}+R} \tag{11}\]  
\[x_{k}=x_{k}^{-}+K(z_{k}-Hx_{k}^{-}) \tag{12}\]  
\[P_{k}=(I-KH)P_{k}^{-} \tag{13}\]  
\[H= \begin{bmatrix}1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\end{bmatrix}