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main.tex

@@ -819,6 +819,30 @@ In this paper, we propose Polar R-CNN to address two key issues in anchor-based
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+\section{Coordinate Systems of Polar R-CNN}
+In this section, we introduce the details about the coordinate systems used in our model and transformations between them.
+
+\textbf{Cartesian Coordinate System.} This system is widely used in the anchor-based lane detection methods \cite{laneatt}\cite{clrnet}\cite{srlane}. The origin point of the Cartesian coordinate system is defined as the bottom-left of the whole image (different from the origin point located at the top-left of the image in the image coordinate system). In PolarRCNN, the Coordinate of the local poles $\left\{\boldsymbol{c}^l_i\right\}$, the global pole $\boldsymbol{c}^g$, and the sampled points $\{(x_{1,j}^s,y_{1,j}^s),(x_{2,j}^s,y_{2,j}^s),\cdots,(x_{N,j}^s,y_{N,j}^s)\}_{j=1}^{K}$ of the anchors  are all defined in the Cartesian coordinate system. 
+
+\textbf{Polar coordinate system}. The defination of the local and the global coordinate systems are already defined in the main text. Now the proof of the Eq. \ref{l2g} is given as follows. 
+\begin{align}
+        r_{j}^{g}&=\left\| \overrightarrow{c^gh_{j}^{g}} \right\| =\left\| \overrightarrow{h_{j}^{a}h_{j}^{l}} \right\| =\left\| \overrightarrow{h_{j}^{a}h_{j}^{l}} \right\|\\
+        &=\left\| \overrightarrow{c_{j}^{l}h_{j}^{l}}-\overrightarrow{h_{j}^{a}c_{j}^{l}} \right\| =\left\| \overrightarrow{c_{j}^{l}h_{j}^{l}} \right\| -\left\| \overrightarrow{c_{j}^{l}h_{j}^{a}} \right\|\\
+        &=\left\| \overrightarrow{c_{j}^{l}h_{j}^{l}} \right\| +\frac{\overrightarrow{c_{j}^{l}h_{j}^{a}}\cdot \overrightarrow{c^gc_{j}^{l}}}{\left\| \overrightarrow{c_{j}^{l}h_{j}^{a}} \right\|}\\
+        &=\left\| \overrightarrow{c_{j}^{l}h_{j}^{l}} \right\| +\frac{\overrightarrow{c_{j}^{l}h_{j}^{a}}}{\left\| \overrightarrow{c_{j}^{l}h_{j}^{a}} \right\|}\cdot \overrightarrow{c^gc_{j}^{l}}\\
+        &=r_{j}^{l}+\left[ \cos \theta _j;\sin \theta _j \right] ^T\left( \boldsymbol{c}_{j}^{l}-\boldsymbol{c}^g \right)
+\end{align}
+
+\begin{align}
+        &\overrightarrow{c^gp_{i,j}^{s}}\cdot \overrightarrow{c^gh_{j}^{g}}=\overrightarrow{c^gh_{j}^{g}}\cdot \overrightarrow{c^gh_{j}^{g}}\\
+        &\Rightarrow \overrightarrow{c^gp_{i,j}^{s}}\cdot \overrightarrow{c^gh_{j}^{g}}=\left\| \overrightarrow{c^gh_{j}^{g}} \right\| \left\| \overrightarrow{c^gh_{j}^{g}} \right\| \\
+        &\Rightarrow \frac{\overrightarrow{c^gh_{j}^{g}}}{\left\| \overrightarrow{c^gh_{j}^{g}} \right\|}\cdot \overrightarrow{c^gp_{i,j}^{s}}=\left\| \overrightarrow{c^gh_{j}^{g}} \right\| \\
+        &\Rightarrow \left[ \cos \theta _j;\sin \theta _j \right] ^T\left( \boldsymbol{p}_{i,j}^{s}-\boldsymbol{c}^g \right) =r_{j}^{g}\\
+        &\Rightarrow x_{i,j}^{s}\cos \theta _j+y_{i,j}^{s}\sin \theta _j=r_{j}^{g}+\left[ \cos \theta _j;\sin \theta _j \right] ^T\boldsymbol{c}^g\\
+        &\Rightarrow x_{i,j}^{s}=\frac{r_{j}^{g}+\left[ \cos \theta _j;\sin \theta _j \right] ^T\boldsymbol{c}^g}{\cos \theta _j}-y_{i,j}^{s}\tan \theta _j
+\end{align}
+
+
 \section{The Design Principles of the One-to-one classification Head}
 Two fundamental prerequisites of the NMS-free framework lie in the label assignment strategies and the head structures.