# RAMC: A Rotation Adaptive Tracker with Motion Constraint for Satellite Video Single-Object Tracking
## 3 Methodology
### Rotation Branch for Adaptive Angle Estimation  
Object rotation is common in SVs because of the nadir view. It can cause the spatial layout between the object and background to change, challenging the accuracy and robustness of object tracking. Since the log-polar conversion can convert the rotation problem into a potential translation problem, we propose an adaptive angle-estimation method incorporating Fourier-Mellin registration [47]. The rotation branch consists of three main parts: log-polar conversion, feature extraction, and phase correlation, which are introduced briefly. For the log-polar conversion, \(I(u,v)\) and \(I^{\prime}(\rho,\,\theta)\) denote the patches in Cartesian coordinates and log-polar coordinates, respectively. After giving the pivot point \((u_{0},\,v_{0})\) and the reference axis (\(u\) axis), it is obtained that:  
\[\rho=\lg\biggl{(}\sqrt{\left(u-u_{0}\right)^{2}+\left(v-v_{0}\right)^{2}} \biggr{)}, \tag{6}\]  
\[\theta=\arctan\biggl{(}\frac{v-v_{0}}{u-u_{0}}\biggr{)}, \tag{7}\]  
where \(\rho\) is the log distance between the original point \((u,v)\), the pivot point \((u_{0},\,v_{0})\), \(lg\) indicates \(log_{10}\), and \(\theta\) denotes the angle between the reference axis and the line through the pivot and original point. \(I_{2}(u,v)\) is a rotated replica of \(I_{1}(u,v)\) with rotation \(\theta_{0}\). The correspondences in Cartesian coordinates are  
\[I_{2}(u,v)=I_{1}(u\cos\theta_{0}+v\sin\theta_{0},-u\sin\theta_{0}+v\cos\theta_ {0}), \tag{8}\]  
Figure 3: Overall framework of the proposed RAMC algorithm. \(pos(t-1)\) and \(\alpha(t-1)\) denote the position and angle of the object at frame \(t-1\), respectively. The color code is shown for visualizing the optical flow, in which the color denotes the displacement direction, and the saturation represents the displacement magnitude. The appearance and motion lines represent the positioning process of appearance and motion information, respectively. Two frames separated by T are selected as adjacent frames to show the overall tracking framework. It yields more intuitive rotation and translation of the object for visualization. In log-polar coordinates, their correspondences are  
\[{I_{2}}^{\prime}(\rho,\ \theta)={I_{1}}^{\prime}(\rho,\ \theta-\theta_{0}). \tag{9}\]  
It can be seen that the rotation between \(I_{2}\) and \(I_{1}\) is deduced as translation between \({I_{2}}^{\prime}\) and \({I_{1}}^{\prime}\). By calculating the offset \(\theta_{0}\), the angle difference between the two patches can be obtained. For the feature extraction, the HOG feature is sensitive to the rotation of the object and can be applied to discriminate the angle difference [40]. In this study, the HOG feature is extracted in log-polar coordinates and used to calculate the offset \(\theta_{0}\) by phase correlation. For the phase correlation, it can be used to match images translated to each other [47]. Therefore, it is employed to solve \(\theta_{0}\). \(f_{1}(u,\ v)\) and \(f_{2}(u,v)\) denote the two 2D patches. \(f_{2}(u,v)\) has a \((u_{0},\ v_{0})\) displacement from \(f_{1}(u,\ v)\) along the \(u\) axis and \(v\) axis, as follows:  
\[f_{2}(u,\ v)=f_{1}(u-u_{0},\ v-v_{0}). \tag{10}\]  
Their Fourier transforms are related by  
\[F_{2}(\alpha,\ \beta)=e^{-j2\pi(\alpha u_{0}+\beta v_{0})}F_{1}(\alpha,\ \beta), \tag{11}\]  
where \(F_{1}(\alpha,\ \beta)\) and \(F_{2}(\alpha,\ \beta)\) denote the DFT of \(f_{1}(u,\ v)\) and \(f_{2}(u,\ v)\), respectively. The cross-phase spectra of \(F_{1}(\alpha,\ \beta)\) and \(F_{2}(\alpha,\ \beta)\) are defined as:  
\[\widetilde{C}(\alpha,\ \beta)=\frac{F_{1}(\alpha,\ \beta)F_{2}^{*}(\alpha,\ \beta)}{ \left|F_{1}(\alpha,\ \beta)F_{2}^{*}(\alpha,\ \beta)\right|}=e^{j2\pi(\alpha u_{0}+\beta v_{0})}, \tag{12}\]  
where \(e^{j2\pi(\alpha u_{0}+\beta v_{0})}\) is the \(u\) and \(v\) axis translation in the Fourier domain.