## 3 Target Tracking  
### KCF Tracker  
The KCF tracker [42; 57] that is used in this paper considers the process of sample training as a Ridge Regression problem, which is also a regular minimization problem with a closed solution. Figure 8: Architecture of the gimbal control system in elevation channel. 1 for the preset angle mode, 2 for the stabilize mode, 3 for the search mode, and 4 for the tracking mode. Figure 7: Diagram of the system operation modes. Consider a \(n\times 1\) vector \(x=\left[ \begin{array}{cccc}x_{1}&x_{2}&\cdots&x_{n}\end{array} \right]^{T}\) as the base sample, which represents a patch with the target of interest. A small translation of this vector is given as:  
\[Px=\left[ \begin{array}{cccc}x_{n}&x_{1}&\cdots&x_{n-1}\end{array} \right]^{T} \tag{33}\]  
where \(P\) is the permutation matrix:  
\[P=\left[ \begin{array}{cccc}0&0&0&\cdots&1\\ 1&0&0&\cdots&0\\ 0&1&0&\cdots&0\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ 0&0&\cdots&1&0\end{array} \right] \tag{34}\]  
and \(u\) shifts can be made to achieve a larger translation by using the matrix power \(P^{u}x\). By cyclic shifting operations, we can use these vectors to constitute a circulant matrix as:  
\[X=C(x)=\left[ \begin{array}{c}\left(P^{0}x\right)^{T}\\ \left(P^{1}x\right)^{T}\\ \left(P^{2}x\right)^{T}\\ \vdots\\ \left(P^{u-1}x\right)^{T}\end{array} \right]=\left[ \begin{array}{cccc}x_{1} &x_{2}&x_{3}&\cdots&x_{n}\\ x_{n}&x_{1}&x_{2}&\cdots&x_{n-1}\\ x_{n-1}&x_{n}&x_{1}&\cdots&x_{n-2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{2}&x_{3}&x_{4}&\cdots&x_{1}\end{array} \right] \tag{35}\]  
and it is useful that all circulant matrices are diagonalized by the Discrete Fourier Transform (DFT), regardless of the base sample \(x\), which can be expressed as:  
\[X=Fdiag(\hat{x})F^{H} \tag{36}\]  
where \(\hat{x}\) is the DFT of base sample, \(\hat{x}=F(x)\), and \(F\) is a constant matrix known as the DFT matrix that does not depend on \(x\). Based on Ridge Regression, the goal of training is to find a function \(f(z)\) that minimizes the squared error over samples \(x_{i}\) and their regression targets \(y_{i}\), as shown below:  
\[E=\underset{w}{\min}\sum_{i}\left(f(x_{i})-y_{i}\right)^{2}+\lambda\|w\|^{2} \tag{37}\]  
where the regularization parameter \(\lambda\) is used to control over fitting, as in the Support Vector Machines (SVM) [57] and \(w\) represents the filter coefficients. Consider a linear regression function \(f(z)=w^{T}z\), the minimizer has a closed-form solution given as:  
\[w=\left(X^{T}X+\lambda I\right)^{-1}X^{T}y \tag{38}\]  
where \(X\) is the circulant matrix with one sample per row \(x_{i}\), each element of \(y\) is a regression target \(y_{i}\), and \(I\) is an identity matrix. By utilizing the diagonalization of the circulant matrices, Equation (38) can be expressed in Fourier domain as:  
\[\hat{w}=\frac{\hat{x}^{*}\odot\hat{y}}{\hat{x}^{*}\odot\hat{x}+\lambda} \tag{39}\]  
where \(\hat{w}\), \(\hat{x}\), \(\hat{y}\) are the DFT of \(w\), \(x\), \(y\), respectively, and \(\hat{x}^{*}\) is the complex-conjugate. In addition, \(w\) can be easily recovered in the spatial domain with Inverse Discrete Fourier Transform (IDFT). When regression function \(f(z)\) is nonlinear, the kernel trick is used to map the inputs of a linear problem to a nonlinear and high-dimensional feature space \(\varphi(x)\):  
\[w=\sum_{i}\alpha_{i}\varphi(x_{i}) \tag{40}\]  
Then, the variables under optimization are \(\alpha\) instead of \(w\).