# Robust Long-Term Visual Object Tracking via Low-Rank Sparse Learning for Re-Detection
## 3 Tracking Algorithm Design
### Reliability Estimation  
In the overall tracking framework, reliability estimation of tracking results is crucial for adaptive model updating and target re-detection. Here, we have two schemes to estimate the tracking reliability. The first is the peak sidelobe ratio (PSR) defined by the response map, and the second is the trajectory smoothness degree of the tracker. First, the PSR of the correlation filter response map is defined as  
\[PSR_{t}=\frac{(R_{max}^{t})-\mu_{i}}{\sigma_{t}}, \tag{10}\]  
where \(R_{max}^{t}\) indicates the maximum value of the response map in the \(t\)-th frame, \(\mu_{i}\) and \(\sigma_{i}\) denote the mean and standard deviation of the response map, respectively. The higher the PSR value is, the more reliable the response result will be. Figure 3 reflects the relationship between the PSR value and reliability. Second, the trajectory smoothness degree of the tracker can indicate the reliability of our tracking results as well. It can be defined as follow:  
\[Smooth_{t}=exp(-\frac{1}{2\sigma^{2}}D_{t}^{2}), \tag{11}\]  
where \(D_{t}=\left\|c(B^{t})-c(B^{t-1})\right\|\), \(c(B^{t})\) and \(c(B^{t-1})\) denote the center of the bounding box in the current frame and the center of the bounding box in the previous frame, respectively. \(\sigma=\frac{1}{2}[w(B^{t})+h(B^{t})]\) is the average length of the width \(w(B^{t})\) and the height \(h(B^{t})\) of the bounding box. The closer the value of \(Smooth_{t}\) is to 1, the better reliability of the tracking result. Figure 4 reflects the relationship between the trajectory smoothness degree and reliability. Figure 4: The values of \(Smooth_{t}\) significantly decrease when tracking results become less reliable (the green box represents the ground-truth target location, and the red color denotes the tracker outputs). Figure 3: Peak sidelobe ratio (PSR) values significantly decrease when tracking results become less reliable (the green box represents the ground-truth target location, and the red color denotes the tracker outputs). Thereinto, derived from the response of the current frame, the PSR value reflects the reliability of the result spatially. Unlike the PSR value, the trajectory smoothness degree of the tracker considers the reliability temporally. In this paper, we consider both the PSR value and the trajectory smoothness degree. For the switch strategy between the tracking and re-detection module, when the PSR value is lower than the threshold, or the trajectory smoothness degree is lower than the threshold, re-detection is performed. This strategy can increase the robustness of the algorithm effectively in long-term tracking. As for template updates, we update the tracking model adaptively using the reliability estimation criterion. In the current frame \(t\), if the PSR value and the trajectory smoothness degree are higher than the thresholds, we update the template of the correlation filter \(\hat{I}_{template}^{t}=\hat{I}_{template}^{t-1}+\hat{I}^{t}\), in which \(\eta\) denotes the learning rate. Otherwise, we do not update the model to avoid learning unreliable information. For the re-detection module, when the PSR value and the trajectory smoothness degree are higher than the thresholds, the positive and negative templates are updated. Our proposed tracking approach is summarized in Algorithm 1. ```
Input: Image frame \(I_{0}\), initial target bounding box \(x_{0}\). Output: Position \(p_{t}\), scale \(s_{t}\). 1 for\(t=2\) to \(n\)do
2 Extract the appearance and the motion features in frame \(t\); Compute the correlation response map with Equation (2) and estimate new target position \(p_{t}\) and scale \(s_{t}\); Compute the PSR value and the trajectory smoothness degree with Equation (10) and Equation (11);
3 if\(PSR_{t}<\tau_{1}\cdot M_{PSR}\) or \(Smooth_{t}<\sigma_{1}\)then
4.