## 3 Proposed Method
### Translation Estimation  
The kernelized correlation filter (KCF) tracker [21] is a representative tracking algorithm based on correlation filters; it is superior in terms of performance and speed. The correlation filter tracker aims to calculate a filter \(h\) that minimizes the square error of sample data and regression data. The KCF tracker calculates filter \(h\) between sample data \(x_{i}\) and regression data \(y_{i}\) with:  
\[\min_{\text{h}}\sum_{i=1}^{n}\left(h^{T}x_{i}-y_{i}\right)^{2}+\lambda\left\| \text{h}\right\|^{2} \tag{5}\]  
where \(\lambda\) is the regularization parameter. In the KCF tracker, we employ the kernel trick [38] for the non-linear regression function. Thus, non-linear filters could be as fast as linear correlation filters. The kernelized version of ridge regression is defined [21] as:  
\[\alpha=(\text{K}+\lambda I)^{-1}y. \tag{6}\]  
where \(K\) is the kernel matrix, \(I\) is the identity matrix and \(\alpha\) is the represented vector [20; 21] of filter \(h\). The \(n\times n\) kernel matrix can be expressed in the circulant matrix [21] as follows. \[\text{C(x)}=\left[ \begin{array}{cccc}x_{1}&x_{2}&x_{3}&\cdots&x_{n}\\ x_{n}&x_{1}&x_{2}&\cdots&x_{n-1}\\ x_{n-1}&x_{n}&x_{1}&\cdots&x_{n-2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{2}&x_{3}&x_{4}&\cdots&x_{1}\end{array} \right] \tag{7}\]  
The circular structure can express the same signal \(x_{n}\) according to \(n\) shift due to periodic characteristics. In Equation (7), the first row \(\{x_{1},\cdots,x_{n}\}\) is the base samples, and cyclically-shifted rows are virtual samples. Figure 3: The whole block and its partial blocks from Tiger2 in OTB-100. (**a**) 1 st frame without occlusion; (**b**) 95 th frame with slight occlusion; (**c**) 115 th frame with slight occlusion; (**d**) 256 th frame with large occlusion. The kernel matrix can be diagonalized by discrete Fourier transform (DFT), and the kernel ridge regression solution is defined [21] by:  
\[\alpha=F^{-1}\left(\frac{F(y)}{F(\kappa)+\lambda}\right). \tag{8}\]  
Equation (8) is a closed-form solution, which is very efficient; it uses only fast Fourier transform (FFT) and element-wise operation [20]. The following equation [21] calculates \(\kappa\),  
\[\kappa^{xx^{\prime}}=\exp\left(-\frac{1}{\sigma^{2}}\left(\left\|x\right\|^{2 }+\left\|x^{\prime}\right\|^{2}\right)-2F^{-1}\left(\sum_{c}F(x)\odot F^{*}(x ^{\prime})\right)\right) \tag{9}\]  
where \(\odot\) indicates the element-wise product and \(\kappa^{xx^{\prime}}\) is the kernel correlation of \(x\); it can be computed quickly with FFT [20]. \(F\) means the fast Fourier transform, and \(F^{-1}\) means the inverse transform. In this paper, we perform the kernel ridge regression solution for all blocks using the equation below. \[R=F^{-1}(F(\hat{\alpha})\cdot F(\kappa^{xz})) \tag{10}\]  
where \(\hat{\alpha}\) means the update model of \(\alpha\), \(R\) means the response map and \(R_{0}\) is for the whole block. \(R_{1}\), \(R_{2}\), \(R_{3}\) and \(R_{4}\) are the response maps for partial blocks, respectively. Furthermore, we perform the weighting for each response map and pick the suitable response map by:  
\[k^{*}=\operatorname*{arg\,max}_{k\in\{0,1,2,3,4\}}\;(\max(R_{k}\cdot B_{k}^{w} )). \tag{11}\]  
The index for all blocks is \(k\); \(k^{*}\) means the index of the picked response that includes the highest value of all response maps. As we mentioned before, \(B_{k}^{w}\) is the weighting for blocks. The position of the highest value in the picked response map means the translated position of the object. Then, we can calculate the center position for the picked block with \(\Delta x\), \(\Delta y\) by:  
\[\hat{B}_{k}(x_{c},y_{c})=B_{k^{*}}(x_{c}+\Delta x,\;y_{c}+\Delta y). \tag{12}\]  
We recalculate the central position for the new whole block using \(\hat{B}_{k}(x_{c},y_{c})\). \[\hat{W}(x_{c},y_{c})=