# Incremental Canonical Correlation Analysis
## 4 Experiments
### Settings  
**Implementation Details. ** For completeness, following the experimental settings [41], we divide an image into left and right patches. Since occlusion and illumination affects the tracking results, we at the meantime extend the method of [45] to the case of two views. In detail, we directly extract the projection matrices of two views as the bases, which serve as the templates. To integrate two views of templates into the particle filtering framework, we utilize the reconstruction errors to determine the best candidate as the position inference. Different from [45], the updating reconstructed coefficients rely on the templates from two views rather than independent views. For clarity, we give the concrete details as Equation (10):  
\[\min_{a,c\in U_{1},c\in U_{2}}\frac{1}{2}\|x-D_{1}\alpha-e_{1}\|_{F}^{2}+ \frac{1}{2}\|y-D_{2}\alpha-e_{2}\|_{F}^{2}+\mu_{1}\|e_{1}\|_{1}+\mu_{2}\|e_{2} \|_{2}+\lambda\|\alpha\|_{0} \tag{10}\]where \(x\) and \(y\) denote the left and right patches, respectively, \(D_{1}\) and \(D_{2}\) denote the projection matrices learned by ICCA, \(\alpha\) is the sparse coefficient vector, \(e_{1}\) and \(e_{2}\) denote the noises, and \(\lambda\), \(\mu_{1}\), and \(\mu_{2}\) denote the regularization parameters. According to [45], we ignore the same solution steps such as the errors \(e_{1}\) and \(e_{2}\), and only describe the reconstructed coefficient update rule. Since the iterative shrinkage technique requires the partial derivative about the coefficient, the first step is to calculate the partial derivative of Equation (10) with respect to \(\alpha\). Let \(G_{\alpha}\) is the objective function about \(\alpha\) after simplification:  
\[G_{\alpha}=\frac{1}{2}\|D_{1}^{T}(x-e_{1})-D_{1}^{T}D_{1}\alpha\|_{\rm F}^{2}+ \frac{1}{2}\|D_{2}^{T}(y-e_{2})-D_{2}^{T}D_{2}\alpha\|_{\rm F}^{2} \tag{11}\]  
Then the gradient about \(\alpha\) is:  
\[\frac{\partial G_{\alpha}}{\partial\alpha}={(D_{1}^{T}D_{1}+D_{2}^{T}D_{2})}^ {-1}(D_{1}^{T}(x-e_{1})-D_{2}^{T}(y-e_{2})) \tag{12}\]  
Similarly, the gradient about \(e_{1}\) and \(e_{2}\) is \(e_{1}-(x-D_{1}\alpha)\) and \(e_{2}-(y-D_{2}\alpha)\). With the three gradients, \(\alpha\), \(e_{1}\), and \(e_{2}\) can be obtained by combination with APG (accelerated proximal gradient). In visual tracking with a particle filter, one particle with maximum probability must be chosen as the target location in the next-coming frame. According to Bayesian theory, the maximum probability of particles is expressed as follows:  
\[p(X_{i},Y_{i}|S_{m})=\exp(-J(\alpha^{*},e_{1}^{*},e_{2}^{*})) \tag{13}\]  
where \(J(\alpha,e_{1},e_{2})=\frac{1}{2}\|x-D_{1}\alpha-e_{1}\|_{\rm F}^{2}+\frac{1 }{2}\|y-D_{2}\alpha-e_{2}\|_{\rm F}^{2}+\mu_{1}\|e_{1}\|_{1}+\mu_{2}\|e_{2}\|_ {1}\), \(\alpha^{*}\), \(e_{1}^{*}\), \(e_{2}^{*}\) are the optimal solution of Equation (10). \(S_{m}\) represents the particle state in the \(m\)-th frame. These states are composed of six affine parameters and are estimated with the maximum posteriori probability. The maximum probability particle is calculated and treated as the target location in the next frame. In order to ensure the fairness of comparison and the reliability of the experiment, the running environment of all video frames is performed on the same workstation. In order to fully mine multimode features, the ICCA tracker uses two parts of each particle evenly divided horizontally or vertically as two view features. In the experiment, the regularization parameters of ICCA tracker are set: \(\lambda\) is selected from {0.005, 0.05, 0.06, 0.25, 0.1}, \(\mu_{1}\) is selected from {0.06, 0.08, 0.09, 0.1, 0.15, 0.25, 0.35, 0.3}, and \(\mu_{2}\) is selected from {0.06, 0.08, 0.09, 0.1, 0.15, 0.25, 0.35, 0.3}. **Datasets.