# Robust Long-Term Visual Object Tracking via Low-Rank Sparse Learning for Re-Detection
## 3 Tracking Algorithm Design
### Tracking Module  
The DCF-based trackers have been extensively researched, thanks to their excellent performance. The goal of these trackers is to learn a DCF to infer the location of the target in the subsequent frames. Here, we introduce a multi-channel correlation filter \(h\)[40; 50]. \(x\) denotes the training example, which is extracted from the video sequence images. \(y\) is a 2D Gaussian function label, representing the regression target. The filter \(h\) can be achieved by solving a ridge regression problem. \[\varepsilon(h)=\frac{1}{2}\bigg{\|}y-\sum_{l=1}^{d}h_{l}\star x_{l}\bigg{\|}_{2 }^{2}+\frac{\lambda}{2}\sum_{l=1}^{d}\|h_{l}\|^{2}, \tag{1}\]  
where \(x_{l}\in\mathbb{R}^{D}\) expresses the \(l\)th channel of the vectorized image. \(h_{l}\in\mathbb{R}^{D}\) indicates the \(l\)th channel of the filter. \(y\in\mathbb{R}^{D}\) is the desired correlation response. \(\sum_{l=1}^{d}h_{l}\star x_{l}\) denotes the actual correlation response. \(l\) represents the feature channel, \(l\in\{1,2,...,d\}\). The \(\star\) symbol refers to the circular correlation. \(\lambda\) is the weight of the regularization term. The minimizer has a closed-form solution in the frequency domain by using [PERSON]'s formula. It can be expressed as  
\[\hat{h_{l}}=\frac{\hat{x_{l}}\odot\hat{y}}{\sum_{i=1}^{d}\hat{x_{i}}\odot\hat {x_{i}}^{*}+\lambda}. \tag{2}\]  
Here, the hat denotes the discrete Fourier transform (DFT) of a function. \(*\) symbol indicates the complex conjugation. \(\odot\) symbol is element-wise multiplication. As for scale estimation, we estimate the scale \(s\) by a single scale-space correlation filter from the discriminative scale space tracking (DSST) algorithm [13]. During tracking, the visual feature representation has a great influence on tracker results. Most trackers use only the appearance features extracted from a single frame, such as HOG [43], color histogram [15], CN [17] and CNN features [16; 18; 19]. However, the motion features integrate information between a pair of frames and are complementary to the appearance features. In this work, we introduce the motion features into correlation tracking. Figure 1: The framework of our proposed tracker. Our method mainly consists of tracking and re-detection modules. After the tracking module has processed the input frames, we perform a reliability estimation for the tracking result, update the tracking model adaptively, and decide whether re-detection is needed for the current result. If the re-detection process is adopted, we perform a reliability estimation for the re-detection result and decide whether the re-detected result can replace the originally detected result re-detection is needed for the current result. **Motion Descriptors:** Optical flow extends static image features into the temporal domain, representing the motion information between two neighboring images. We can obtain optical flow \((u,v)\) using a gradient-based optical flow algorithm [51], where \(u\), \(v\) represent the horizontal and vertical components of optical flow respectively. Optical flow takes both target motion and camera motion into account, but taking camera motion into account may lead to errors in some cases. In most cases, the camera motion is locally translational and varies smoothly across the image plane, such as zooming, tilting. [PERSON] et al. [46] adopt the motion boundary histograms (MBH) descriptor to extract motion features, which can mitigate the effects of camera motion. The results of their experiment showed that MBH is more discriminative for target motion detection compared to optical flow. Therefore, to keep the information about motion boundaries and remove the constant motion information, we try to adopt the MBH descriptor to extract motion features. Let \(u_{x}\), \(u_{y}\), \(v_{x}\), \(v_{y}\) denote the corresponding horizontal and vertical gradients of optical flow \((u,v)\), respectively.