# Learning Spatial-Temporal Background-Aware Based Tracking
## 3 Spatial-Temporal Regularization Background-Aware Correlation Filter
### Optimization of the Proposed Model  
Correlation filters usually benefit from the frequency domain for high-efficiency computation, and can transform complicated convolution operations into simple element-wise multiplications. Therefore, we convert Equation (2) into the following corresponding frequency domain form:  
\[ \begin{split} E(f,\hat{g})=\frac{1}{2}\big{\|}\hat{g}-\hat{X} \hat{g}\big{\|}_{2}^{2}+\frac{\alpha_{1}}{2}\|f-f_{d-1}\|_{2}^{2}+\frac{\alpha _{2}}{2}\|f\|_{2}^{2}\\ \text{s.t.}\quad\hat{g}=\sqrt{T}(I_{\text{D}}\otimes F\hat{B}^{T })f\end{split}  \tag{3}\]  
The symbol \(\hat{}\) in the equation represents the transformation of the signal in the discrete Fourier transform, for instance \(\hat{\alpha}=\sqrt{T}F\alpha\), \(F\) is the orthonormal \(T\times T\) matrix of complex basis vectors for mapping any T dimensional vectorized signal into the Fourier domain. \(\hat{g}=(\hat{g}_{1},\,\hat{g}_{2},\,\dots,\,\hat{g}_{D})\) denotes auxiliary variables parameter. \(\otimes\) denotes Kronecker product, \(I_{\text{D}}\) is \(D\times D\) identity matrix, \(T\) represents the conjugate transpose operation of a complex vector or matrix. It can be clearly seen that Equation (3) is a convex function, which  
Figure 1: The systematic work-flow of the proposed tracking algorithm. can be optimized by the ADMM method to obtain a local optimal solution. The specific incremental Lagrangian form can be reformulated as the following equation:  
\[ \begin{split} E(f,\hat{g},\hat{\mu})&=\frac{1}{2} \big{\|}\hat{g}-\hat{X}\hat{g}\big{\|}_{2}^{2}+\frac{\alpha_{1}}{2}\|f-f_{t-1} \|_{2}^{2}+\frac{\alpha_{2}}{2}\|f\|_{2}^{2}\\ &+\hat{\rho}^{T}(\hat{g}-\sqrt{T}(I_{D}\otimes FB^{T})f)+\frac{ \mu}{2}\big{\|}\hat{g}-\sqrt{T}(I_{D}\otimes FB^{T})f\big{\|}_{2}^{2}\end{split}  \tag{4}\]  
where is the introduced penalty parameter of the error term, \(\hat{\rho}=\left[\hat{\rho}_{1},\ \hat{\rho}_{2}\,,\,\dots,\,\hat{\rho}_{D}\right]^{T}\) is the introduced Lagrange auxiliary variable of \(DT\times 1\). Employing the ADMM optimization method to solve Equation (4) iteratively, each subproblem \(\hat{g}^{*}\) and \(f^{*}\) has corresponding closed solutions:  
\[ \begin{split} f^{*}&=\underset{f}{\operatorname{ argmin}}\left\{\frac{\alpha_{1}}{2}\|f-f_{t-1}\|_{2}^{2}+\frac{\alpha_{2}}{2}\|f\|_{2}^{2}+ \hat{\rho}^{T}(\hat{g}-\sqrt{T}(I_{D}\otimes FB^{T})f)+\frac{\mu}{2}\Big{\|} \hat{g}-\sqrt{T}(I_{D}\otimes FB^{T})f\Big{\|}_{2}^{2}\right\}\\ &=[\alpha_{1}+\alpha_{2}+\mu T]^{-1}[\alpha_{1}f_{t-1}+\hat{\rho} ^{T}\sqrt{T}(I_{D}\otimes FB^{T})+\mu\hat{g}\sqrt{T}(I_{D}\otimes FB^{T})]\\ &=(\frac{\alpha_{1}+\alpha_{2}}{T}+\mu)^{-1}(\frac{\alpha_{1}f_{t -1}}{T}+\rho+\mu g)\end{split}  \tag{5}\]  
where \(g\) and \(\rho\) can be obtained by the following inverse Fourier transform operations:  
\[\left\{\begin{array}{l} \begin{split} g&=\frac{1}{\sqrt{T}}(I _{D}\otimes BF^{T})\hat{g}\\ \rho&=\frac{1}{\sqrt{T}}(I_{D}\otimes BF^{T})\hat{\rho} \end{split} \end{split}\right. \tag{6}\]  
Subproblem \(g^{*}\):  
\[g^{*}=\underset{\hat{g}}{\operatorname{argmin}}\left\{\frac{1}{2}\big{\|} \hat{g}-\hat{X}\hat{g}\big{\|}_{2}^{2}+\hat{\rho}^{T}(\hat{g}-\sqrt{T}(I_{D} \otimes FB^{T})f+\frac{\mu}{2}\big{\|}\hat{g}-\sqrt{T}(I_{D}\otimes FB^{T})f \big{\|}_{2}^{2}\right\} \tag{7}\]  
The computational complexity of Equation (7) is \(O(T^{3}D^{3})\), we solve the equation at every ADMM iteration will generate heavy computation, which will deteriorate real-time performance.