## 2 Methods Used in This Study  
### LK Optical Flow Method  
When an observer views a moving object, the human eye's retina captures the light from the brightness changes caused by the object's motion. This allows the eye to track the object's position at any given moment. In computer vision, the optical flow method leverages this principle. It tracks the position of an object moving relative to a stationary background by calculating the pixel changes in a continuous grayscale image due to the object's movement ([PERSON] et al., 2018; [PERSON] et al., 2018). The optical flow method primarily computes the velocity vectors along the x-and y-axes of the image to obtain the displacement information of the moving object. Before applying the optical flow method, certain constraints need to be established. Specifically, in a continuous grayscale image, the brightness of the moving object should remain essentially constant between two adjacent frames, and the object's speed cannot be too high. Equation (1) can be derived from the constant brightness I of the object ([PERSON] et al., 2018).  
\[\mathbf{I}(\mathbf{x},\mathbf{y},\mathbf{t})=\mathbf{I}(\mathbf{x}+\Delta\mathbf{x},\mathbf{y}+\Delta\mathbf{y},\bm {t}+\Delta\mathbf{t}) \tag{1}\]  
In the Equation (1), \(\Delta x\), \(\Delta y\) represent the image in which the coordinate point (\(x\),\(y\)) has elapsed time \(\Delta t\) to move the distance along the \(x\)-axis and \(y\)-axis, respectively. A Taylor series expansion of the right-hand side of the equation to first-order terms yields Equation (2), which is then brought to a simplification to yield Equation (3).  
\[I(x+\Delta x,y+\Delta y,t+\Delta t)=I(x,y,t)+\frac{\partial I}{\partial x} \Delta x+\frac{\partial I}{\partial y}\Delta y+\frac{\partial I}{\partial t}\Delta t \tag{2}\]  
\[\frac{\partial I}{\partial x}\Delta x+\frac{\partial I}{\partial y}\Delta y+ \frac{\partial I}{\partial t}\Delta t=0 \tag{3}\]  
In the Equation (3), both sides of the equation are divided by \(\Delta t\), then Equation (4) can be obtained, where \(I_{x}\)=\(\frac{\partial I}{\partial x}\), \(I_{y}\) = \(\frac{\partial I}{\partial y}\), \(I_{t}\)=\(\frac{\partial I}{\partial t}\), \(u\) = \(\frac{\Delta x}{\Delta t}\), \(v\) = \(\frac{\Delta y}{\Delta t}\).  
\[\mathbf{I}_{x}\mathbf{u}+\mathbf{I}_{y}\mathbf{v}+\mathbf{I}_{t}=\mathbf{0} \tag{4}\]A window is set up with the coordinate point (_x,y_) as the center, and the n pixel points within the window have the same motion, combining these pixel points in Equation (4) can be expressed in a matrix as Equation (5).  
\[\left[\begin{array}{ccc}I_{x_{1}}&I_{y_{1}}\\ I_{x_{2}}&I_{y_{2}}\\.&.\\.&.\\.&.\\ I_{x_{n}}&I_{y_{n}}\end{array}\right]\cdot\left[\begin{array}{c}u\\ v\end{array}\right]=-\left[\begin{array}{c}I_{I1}\\ I_{I2}\\.\\.\\.\\ I_{tn}\end{array}\right] \tag{5}\]  
In Equation (5), the two-dimensional optical flow vectors \(u\) and \(v\) can be found by applying the least squares method.  
\[\left[\begin{array}{c}\mathbf{u}\\ \mathbf{v}\end{array}\right]=\left[\begin{array}{ccc}\sum_{i}\mathbf{I}_{\mathbf{x}i}^ {2}&\sum_{i}\mathbf{I}_{\mathbf{x}i}\mathbf{I}_{\mathbf{y}i}\\ \sum_{i}\mathbf{I}_{\mathbf{x}i}\mathbf{I}_{\mathbf{y}i}&\sum_{i}\mathbf{I}_{\mathbf{y}i}^{2}\end{array} \right]^{-1}\cdot\left[\begin{array}{c}\sum_{i}\mathbf{I}_{\mathbf{x}i}\mathbf{I}_{\mathbf{i }}\\ \sum_{i}\mathbf{I}_{\mathbf{y}i}\mathbf{I}_{\mathbf{i}i}\end{array}\right] \tag{6}\]