## 2 Geometric Particle Filtering Algorithm
### Establishing a Target Model  
The task of visual tracking is the process of the continuous positioning of the target in the image sequence. Therefore, the poses of two consecutive images are closely related. The state equation and observation equation of conventional PF can be expressed as:  
\[x_{k}=f(x_{k-1})+w_{k} \tag{1}\]  
\[y_{k}=g(x_{k})+\eta_{k} \tag{2}\]  
where the \(f\) is the time-varying nonlinear function of \(x_{k},g\) is the time-varying nonlinear function of \(y_{k}\), \(w_{k}\) is the noise in the state transition process and \(\eta_{k}\) is the noise in the observation process. According to Equation (1), the target state equation conforms to a first-order Markov model. The state at time \(k\) is only related to the state at time \(k\)\(-\)1. According to the state equation and the observation equation, the probability density \(\rho(x_{k}\,\mathrm{\text{\textdagger}}y_{0;\,k})\) of the state at time \(k\) can be derived. Using the 2D affine transformation as the particle state is different to the use of the coordinate difference (\(\Delta p_{x}\), \(\Delta p_{y}\))\({}^{\mathrm{T}}\) used in the 2D image as the state that represents the position transformation of the target in continuous images because the former does not belong to the vector space. It belongs to the surface space with Lie group (affine group) structure. Equations (1) and (2) cannot be directly used for filter estimation. After 2D affine transformation groups as particles are introduced, the corresponding state transition equation and observation equation are expressed as follows:  
\[dX=X\cdot A(X)dt+X\sum_{i=1}^{6}b_{i}(X)E_{i}dw_{i} \tag{3}\]  
\[dy=g(X)dt+d\eta \tag{4}\]  
where matrix \(X\in Aff(2)\) is the particle of the system. The matrix \(A\) is the mapping \(Aff(2)\Rightarrow aff(2)\) from a Lie group to a Lie algebra. The bi is a nonlinear transformation  
Figure 1: Traditional particle filter algorithm flow. \(Aff(2)\Rightarrow R\) from a Lie group to the real numbers. The \(dw_{i}\) is the noise in the Wiener process. The \(E_{i}\) is the basic element of the geometric transformation in Lie algebra. The \(y\) is the observation. The \(g\) is the mapping from \(Aff(2)\) to \(R^{p}\). The \(d\eta\) is the noise in the Wiener process. The specific value of the basic \(E_{i}\) corresponding to a Lie algebra is shown in Equation (5):  
\[E_{1}=\left[ \begin{array}{cc}1&0&0\\ 0&-1&0\\ 0&0&0\end{array} \right],\;E_{2}=\left[ \begin{array}{cc}1&0&0\\ 0&1&0\\ 0&0&0\end{array} \right],\;E_{3}=\left[ \begin{array}{cc}0&-1&0\\ 1&0&0\\ 0&0&0\end{array} \right]E_{4}=\left[ \begin{array}{cc}0&1&0\\ 1&0&0\\ 0&0&0\end{array} \right],\;E_{5}=\left[ \begin{array}{cc}0&0&1\\ 0&0&0\\ 0&0&0\end{array} \right],\;E_{6}=\left[ \begin{array}{cc}0&0&0\\ 0&0&1\\ 0&0&0\end{array} \right] \tag{5}\]  
Image acquisition is carried out frame by frame and is a discretization process. Equations (3) and (4) are continuity equations. In order to avoid the difficulty of integration on manifolds, these equations are exponentially Euler discretized:  
\[X_{k}=X_{k-1}exp(A(x,t)\Delta t+dW_{k}\sqrt{\Delta t}) \tag{6}\]  
\[y_{k}=g(X_{k})+\eta_{k}) \tag{7}\]  
where the is \(dW_{k}\) the noise in the Wiener process on \(aff(2)\), \(dW_{k}=\sum_{i=1}^{6}\varepsilon_{i}E_{i}\), \(\varepsilon_{i}\) is Gaussian noise, \(\varepsilon_{i}=(\varepsilon_{1},\,\varepsilon_{2},\,\ldots,\,\varepsilon_{6} )^{\top}\sim N(0,P)\). The \(g\) is the nonlinear mapping from \(aff(2)\) to \(R^{Ny}\). The \(N_{y}\) represents the number of rows of \(y\). The attitude of the target in the current image is closely related to the attitude of the target in the previous image. The \(A(X,t)\) adopts the self-regression model to achieve better tracking. In PF, the particle change satisfies a first-order Markov model. Finally, the first-order self-regression model is adopted. The state equation is rewritten as:  
\[X_{k}=X_{k-1}exp(A_{k-1}+dW_{k}\sqrt{\Delta t}) \tag{8}\]  
\[A_{k-1}=a\;log(X_{k-2}^{-1}X_{k-1}) \tag{9}\]  
where the \(a\) is the coefficient of the self-regression model.