## 3 The Proposed Tracker
### Context Selection Strategy for Tracking  
Context information of the tracking target has a significant effect on overall performance. For instance, in the scenery with lots of background clutter, selecting optimal context patches is very important for successful tracking. [PERSON] et al. [20] introduced a method based on a context-aware framework to add contextual information to the filter and got competitive results in some tracking challenges. However, the conventional CA tracker [20] simply samples context patches on fixed locations around the target, which causes the problem that these selected negative samples are perhaps non-representative (see Figure 2). The ideal strategy of selecting context patches is that context patches are sampled at locations where the filter response is high and spatially far from the maximum.  
In this paper, we propose a novel context selecting strategy to obtain negative samples. Firstly, we divide the surroundings of the tracked object into four areas, each area is twice the target. Then, patches that have the maximum response value in their respective regions are selected as the context patches for collecting negative samples (as shown in Figure 3).  
The method has two advantages. Firstly the strategy of conventional CA tracker may yield meaningless negative samples. By contrast, sampled negative patches by our proposed strategy are more similar with the target. In other words, these context patches will more probably become background distractors in next frames. Secondly, our method avoids redundancy which led by sampling patches of top m response value directly on the whole neighboring region of target.  
Next, we learn the correlation filter using the target samples and the negative samples based on CA [20]. Our goal is to train a filter with a high response on the target patch and closes to zero response on context patches. The regression function that adds context patches can be written as Equation (6):  
\[\underset{\mathbf{w}}{min}\parallel\mathbf{X_{0}w-y}\parallel^{2}+\lambda_{1} \parallel\mathbf{w}\parallel^{2}+\lambda_{2}\sum_{i}\parallel\mathbf{X_{i}w} \parallel^{2} \tag{6}\]  
where \(\mathbf{X_{0}}\) and \(\mathbf{X_{i}}\) are the corresponding circulant matrices of target and context patches, respectively. The parameter \(\lambda_{2}\) can control context patches to regress to zeros.  
We can stack context image patches under the target patch to establish a new data matrix \(\mathbf{B}\), and concatenate \(\mathbf{y}\) with zeros to construct a new regression function \(\mathbf{y}\). The Equation (6) can be rewritten as below:  
\[\parallel\mathbf{Bw}-\mathbf{y}\parallel^{2}+\lambda_{1}\parallel\mathbf{w} \parallel^{2} \tag{7}\]  
where \(\mathbf{B}=\left[\begin{array}{c}\mathbf{X_{0}}\\ \sqrt{\lambda_{2}}\mathbf{X_{1}}\\ \vdots\\ \sqrt{\lambda_{2}}\mathbf{X_{k}}\end{array}\right]\) and \(\mathbf{\bar{y}}=\left[\begin{array}{c}\mathbf{y}\\ \mathbf{0}\\ \vdots\\ \mathbf{0}\end{array}\right]\).  
Because the Equation (7) is convex, it can be minimized by setting the gradient to zero. The closed-form solution of Equation (7) can be obtained.  
\[\mathbf{w}=\left(\mathbf{B^{T}B}+\lambda_{1}\mathbf{I}\right)^{-1}\mathbf{B^{T }\mathbf{y}} \tag{8}\]  
Figure 3: Sampling procedure of proposed strategy.  
Figure 2: Two strategies of selecting context patches. The green rectangle denotes the target, and red rectangles denote selected context patches.  
The closed-form solution in the Fourier domain can be calculated by applying Equation (3).  
\[\hat{\mathbf{w}}=\frac{\hat{\mathbf{x}}_{0}\odot\hat{\mathbf{y}}}{\hat{\mathbf{x}} _{0}^{*}\odot\hat{\mathbf{x}}_{0}+\lambda_{1}+\lambda_{2}\sum_{i=1}^{k}\hat{ \mathbf{x}}_{0}^{*}\odot\hat{\mathbf{x}}_{i}} \tag{9}\]  
The detection formula is exactly the same as in the standard formulation in Equation (5).