## 3 The CFSA Tracker
### Tracking with Kernelized Correlation Filter  
The proposed CFSA tracker is based on KCF tracker proposed in [18]. In this subsection, we review the main idea of KCF tracker and integrate object proposals into KCF tracker. As a discriminative method, KCF trains a linear model \(\mathrm{f}(\mathrm{z})=\mathrm{w}^{\mathrm{T}}\mathrm{z}\) to estimate the probability that the image patch \(\mathrm{z}\) is the target being tracked by solving a ridge regression problem,  
\[\min_{\mathrm{w}}\sum_{i}\lvert f(\mathbf{x}_{i})-y_{i}\rvert^{2}+\lambda\ \parallel \mathrm{w}\parallel^{2} \tag{4}\]  
where \(\mathrm{w}\) is the model parameter matrix, \(\mathbf{x}_{i}\) is an image patch of size \(\mathrm{W}\times\mathrm{H}\), \(\mathrm{i}\in\{0,1,\,\ldots,\,W{-}1\}\times\{0,1,\,\ldots,H{-}1\}\), \(\mathrm{y}_{\mathrm{i}}\) is the regression target and \(\lambda\) is a regularization parameter used to control over-fitting. Mapping the inputs \(\mathbf{x}_{i}\) of the linear ridge regression to a non-linear feature space \(\phi(\mathrm{x})\) with kernel trick that is defined by the kernel \(\kappa(\mathrm{x},\ \mathrm{x}^{\prime})\ =\ \left\langle\phi(\mathrm{x}),\phi( \mathrm{x}^{\prime})\right\rangle\) gets the non-linear ridge regression,  
Figure 2: Examples of object proposals on image sequence _Singer 1_. (**a**) Top five object proposals generated by EdgeBoxes. (**b**) Top five object proposals generated by modified EdgeBoxes. which can be resolved as \(\mathrm{f(z)}~{}=~{}\mathrm{w}^{\mathrm{T}}\mathrm{z}~{}=~{}\sum_{i=1}^{n}\alpha_{i} \kappa(z,x_{i})\) and \(\kappa(z,x_{i})\) denotes the dot-products of \(x_{i}\) and \(z\). The coefficient \(\alpha\) can be expressed as:  
\[\hat{\alpha}^{*}=\frac{\hat{y}}{\hat{k}^{\mathbf{xx}}+\lambda} \tag{5}\]  
where \(\hat{k}^{\mathrm{xx}}\) is a general kernel correlation, \(\hat{\alpha}^{*}\) is the complex-conjugate of \(\hat{\alpha}\), and \(\hat{y}\) is the Discrete Fourier Transform (DFT) of \(\mathbf{y}\). The training label \(\mathbf{y}\) is a Gaussian function, which decays smoothly from the value of one for the centered target to zero for other shifted samples. In order to integrate multiple features, the tracker needs to support multiple channels. In the Fourier domain, this can be achieved by summing the result for each channel with linearity of the DFT. 31 gradient orientation bins for HOG variant and 11 dimensions color naming are concatenated into a vector \(x~{}=~{}[x_{1},~{}x_{2},~{}\ldots,~{}x_{c}]\), \(c=42\). In this paper, we adopt Gaussian kernel \(\kappa(x,~{}x^{\prime})=\exp{(-\frac{1}{\sigma^{2}}\parallel x-x^{\prime} \parallel^{2})}\), multichannel features and circulant matrix trick,  
\[\mathbf{k}^{\mathbf{xx}^{\prime}}=\exp\left(-\frac{1}{\sigma^{2}}\left( \parallel x\parallel^{2}+\parallel x^{\prime}\parallel^{2}-2\mathcal{F}^{-1} \left(\sum_{C}\hat{x}_{c}^{*}\odot\hat{x}^{\prime}_{c}\right)\right)\right) \tag{6}\]  
where \(\mathcal{F}^{-1}\) denotes the inverse Discrete Fourier Transform (IDFT). The full kernel correlation can be computed in only \(\mathcal{O}(\mathrm{nlog}~{}\mathrm{n})\) time using only a few DFT and IDFT. In the tracking stage, the size of interesting image patch z is cropped as the same with x, and all responses for candidate image patches are calculated by:  
\[\hat{\mathbf{f}}(\mathbf{z})=\left(\hat{k}^{\widetilde{\mathbf{x}}}\right)^{ *}\odot\hat{\alpha} \tag{7}\]  
where \(\widetilde{\mathbf{x}}\) denotes the target appearance to be learnt in the model. And the target position is detected by finding the coordinate at \(\mathrm{f}(z)\) which has the maximum value. Subsequently, candidate object proposals are integrated into KCF tracker to estimate the position of the target object. While initialing, the target ground truth of size \(\mathrm{w}_{1}\times\mathrm{h}_{1}\) centered at position \(\ell_{1}\) is given in the initial frame of a video.