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\hyphenation{op-tical net-works semi-conduc-tor IEEE-Xpolare}
% updated with editorial comments 8/9/2021

\begin{document}

\title{Polar R-CNN:\@ End-to-End Lane Detection with Fewer Anchors}

\author{Shengqi Wang, Junmin Liu, Xiangyong Cao, Zengjie Song, and Kai Sun\\

\thanks{This work was supported in part by the National Nature Science Foundation of China (Grant Nos. 62276208, 12326607) and in part by the Natural Science Basic Research Program of Shaanxi Province (Grant No. 2024JC-JCQN-02).}%
\thanks{S. Wang, J. Liu, Z. Song and K. Sun are with the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China.}
\thanks{X. Cao is with the School of Computer Science and Technology and the Ministry of Education Key Lab for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an 710049, China.}
}



%\thanks{Manuscript received April 19, 2021; revised August 16, 2021.}}

% The paper headers
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\markboth{S. Wang \MakeLowercase{\textit{et al.}}: Polar R-CNN:\@ End-to-End Lane Detection with Fewer Anchors}%
{S. Wang \MakeLowercase{\textit{et al.}}: Polar R-CNN:\@ End-to-End Lane Detection with Fewer Anchors}

% \IEEEpubid{0000--0000/00\$00.00~\copyright~2021 IEEE}
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\maketitle

\begin{abstract}
    Lane detection is a critical and challenging task in autonomous driving, particularly in real-world scenarios where traffic lanes are often slender, lengthy, and partially obscured by other vehicles, complicating detection efforts. Existing anchor-based methods typically rely on prior Lane anchors to extract features and refine lane location and shape. Though achieving high performance, manually setting prior anchors is cumbersome, and ensuring adequate coverage across diverse datasets often requires a large number of dense anchors. Additionally, non-maximum suppression is used to suppress redundant predictions, which complicates real-world deployment and may fail in dense scenarios. In this study, we introduce Polar R-CNN, a NMS-free anchor-based method for lane detection. By incorporating both local and global polar coordinate systems, Polar R-CNN enables flexible anchor proposals and significantly reduces the number of anchors required without compromising performance. Additionally, we introduce a heuristic GNN-based NMS-free head that supports an end-to-end paradigm, making the model more deployment-friendly and enhancing performance in dense scenarios. Our method achieves competitive results on five popular lane detection benchmarks—Tusimple, CULane, LLAMAS, CurveLanes, and DL-Rail—while maintaining a lightweight design and straightforward structure. Our source code are available at \href{https://github.com/ShqWW/PolarRCNN}{\textit{https://github.com/ShqWW/PolarRCNN}}.
\end{abstract}
\begin{IEEEkeywords}
Lane detection, NMS-free, Graph neural network, Polar coordinate system.
\end{IEEEkeywords}

\section{Introduction}
\IEEEPARstart{L}{ane} detection is a significant problem in computer vision and autonomous driving, forming the basis for accurately perceiving the driving environment in intelligent driving systems. While extensive research has been conducted in ideal environments, it remains a challenging task in adverse scenarios such as night driving, glare, crowd, and rainy conditions, where lanes may be occluded or damaged. Moreover, the slender shapes,  complex topologies of lanes and the global property add to the complexity of detection challenges. An effective lane detection method should take into account both global high-level semantic features and local low-level features to address these varied conditions and ensure robust performance in real-time applications such as autonomous driving.

Traditional methods predominantly concentrate on handcrafted local feature extraction and lane shape modeling. Techniques such as the Canny edge detector\cite{cannyedge}, Hough transform\cite{houghtransform}, and deformable templates for lane fitting\cite{kluge1995deformable} have been extensively utilized. Nevertheless, these approaches often encounter limitations in practical settings, particularly when low-level and local features lack clarity and distinctiveness.

In recent years, fueled by advancements in deep learning and the availability of large datasets, significant strides have been made in lane detection. Deep models, including convolutional neural networks (CNNs) and transformer-based architectures, have propelled progress in this domain. Previous approaches often treated lane detection as a segmentation task, which, despite its simplicity, involved time-consuming computations. Some methods relied on parameter-based models, directly outputting lane curve parameters instead of pixel locations. These models offer end-to-end solutions, but the curve parameter sensitivity to lane shape compromises robustness. 

\begin{figure}[t]
        \centering
        \def\subwidth{0.24\textwidth}
        \def\imgwidth{\linewidth}
        \def\imgheight{0.5625\linewidth}

        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/anchor_demo/anchor_fix_init.jpg}
                \caption{}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/anchor_demo/anchor_fix_learned.jpg}
                \caption{}
        \end{subfigure}

        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/anchor_demo/anchor_proposal.jpg}
                \caption{}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/anchor_demo/gt.jpg}
                \caption{}
        \end{subfigure}
        \caption{Anchor settings of different methods. (a) The initial anchor settings of CLRNet. (b) The learned anchor settings of CLRNet trained on CULane. (c) The proposed anchors of our method. (d) The ground truth.}
        \label{anchor setting}
\end{figure}


\begin{figure}[t]
        \centering
        \def\subwidth{0.24\textwidth}
        \def\imgwidth{\linewidth}
        \def\imgheight{0.5625\linewidth}
        
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/nms_demo/less_gt.jpg}
                \caption{}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/nms_demo/less_pred.jpg}
                \caption{}
        \end{subfigure}

        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/nms_demo/redun_gt.jpg}
                \caption{}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/nms_demo/redun_pred.jpg}
                \caption{}
        \end{subfigure}

        \caption{Comparison of different anchor thresholds in sparse and dense scenarios. (a) Ground truth in a dense scenario, where some lane instances are close with each others. (b) Predictions with large NMS thresholds in a dense scenario, where a lane prediction is mistakenly suppressed. (c) Ground truth in a sparse scenario, where the lane instance are far apart. (d) Predictions with small NMS threshold in a sparse scenario, where redundant prediction results fail to be removed.}
        \label{NMS setting}
\end{figure}





Drawing inspiration from object detection methods such as Yolos \cite{yolov10} and Faster R-CNN \cite{fasterrcnn}, several anchor-based approaches have been introduced for lane detection, the representative works including LaneATT \cite{laneatt} and CLRNet \cite{clrnet}. These methods have demonstrated superior performance by leveraging anchor priors and enabling larger receptive fields for feature extraction. However, anchor-based methods encounter similar drawbacks as anchor-based general object detection method as follows:

(1) A large number of lane anchors are placed throughout the image, even in sparse scenarios. Sparse scenarios refer to situations where lanes are distributed sparsely and are located far from each other, as illustrated in Fig. \ref{anchor setting}(d).

(2) Non-maximum suppression (NMS) post-processing is required to remove redundant predictions but may struggle in dense scenarios. Dense scenarios involve situations where lanes are close to each other, such as forked lanes and double lanes, as depicted in Fig. \ref{NMS setting}(a).

Regrading the first issue, \cite{clrnet} introduced learned anchors, where the anchor parameters are optimized during training to adapt to the lane distributions (see Fig. \ref{anchor setting}(b)) in real dataset. Additionally, they employ cascade cross-layer anchor refinement to bring the anchors closer to the ground truth. However, the anchors are still numerous to cover the potential distributions of lanes. Moving further, \cite{adnet} proposes flexible anchors for each image by generating start points, rather than using a fixed set of anchors for all images. Nevertheless, the start points of lanes are subjective and lack clear visual evidence due to the global nature of lanes, which affects its performance. \cite{srlane} uses a local angle map to propose sketch anchors according to the direction of ground truth. This approach only considers the direction and neglects the accurate positioning of anchors, resulting in suboptimal performance without cascade anchor refinement. Overall, numerous anchors are unnecessary in sparse scenarios (where lane ground truths are sparse). The trend in newly proposed methods is to reduce the number of anchors and offer more flexible anchor configurations.

Regarding the second issue, nearly all anchor-based methods (including those mentioned above) require direct or indirect NMS post-processing to eliminate redundant predictions. Although it is necessary to eliminate redundant predictions, NMS remains a suboptimal solution. On the one hand, NMS is not deployment-friendly because it involves defining and calculating distances (e.g., Intersection over Union) between lane pairs. This is more challenging than bounding boxes in general object detection due to the complexity of lane geometry. On the other hand, NMS fails in some dense scenarios where the lane ground truths are closer together compared to sparse scenarios. A large distance threshold may result in false negatives, as some true positive predictions might be eliminated (as shown in Fig. \ref{NMS setting}(a)\&(b)) by mistake. Conversely, a small distance threshold may not eliminate redundant predictions effectively and can leave false positives (as shown in Fig. \ref{NMS setting}(c)\&(d)). Achieving an optimal trade-off in all scenarios by manually setting the distance threshold is challenging. The root cause of this problem is that the distance definition in NMS considers only geometric parameters while ignoring the semantic context in the image. Thus, when two predictions are “close” to each other, it is nearly impossible to determine whether one of them is redundant.

To address the two issues outlined above, we propose Polar R-CNN, a novel anchor-based method for lane detection. For the first issue, we introduce local and global heads based on the polar coordinate system to create anchors with more accurate locations and reduce the number of proposed anchors in sparse scenarios, as illustrated in Fig. \ref{anchor setting}(c). Compared to state-of-the-art previous work \cite{clrnet}\cite{clrernet} which uses 192 anchors, Polar R-CNN employs only 20 anchors to cover potential lane ground truths. For the second issue, we have revised Fast NMS to Graph-based Fast NMS and introduced a new heuristic graph neural network block (Polar GNN block) integrated into the NMS head. The Polar GNN block offers an interpretable structure, achieving nearly equivalent performance in sparse scenarios and superior performance in dense scenarios. We conducted experiments on five major benchmarks: TuSimple \cite{tusimple}, CULane \cite{scnn}, LLAMAS \cite{llamas}, CurveLanes \cite{curvelanes}, and DL-Rail \cite{dalnet}. Our proposed method demonstrates competitive performance compared to state-of-the-art methods.

Our main contributions are summarized as follows:

\begin{itemize}
\item We simplified the anchor parameters using local and global polar coordinate systems and applied them to two-stage lane detection frameworks. Compared to other anchor-based methods, the number of proposed anchors is greatly reduced while achieving better performance.
\item We introduced a novel heuristic Polar GNN block to implement a NMS-free paradigm. The GNN architecture is designed with reference to Graph-based Fast NMS, providing interpretability. Our model supports end-to-end training and testing, but traditional NMS post-processing can still be used as an option for a NMS version of our model.
\item Our method utilizes two-stage architectures and achieves competitive performance compared to state-of-the-art methods across five datasets. The high performance with fewer anchors and a NMS-free paradigm demonstrates the effectiveness of our approach. Additionally, our model is designed with a straightforward structure (without cascade refinement or attention strategies), which simplifies deployment.
\end{itemize}

\section{Related Works}
The lane detection aims to detect lane instances in an image. In this section, we only introduce deep-leanrning based methods for lane detection. The lane detection methods can be categorized by segmentation based, parameter-based methods and anchor-based methods.

\textbf{Segmentation-based Methods.} Segmentation-based methods focus on pixel-wise prediction. They predefined each pixel into different categories according to different lane instances and background\cite{lanenet} and predicted information pixel by pixel. However, these methods overly focus on low-level and local features, neglecting global semantic information and real-time detection. SCNN uses a larger receptive field to overcome this problem. Some methods such as UFLDv1 and v2\cite{ufld}\cite{ufldv2} and CondLaneNet\cite{CondLaneNet} utilize row-wise or column-wise classification instead of pixel classification to improve detection speed. Another issue with these methods is that the lane instance prior is learned by the model itself, leading to a lack of prior knowledge. Lanenet uses post-clustering to distinguish each lane instance. UFLD divides lane instances by angles and locations and can only detect a fixed number of lanes. CondLaneNet utilizes different conditional dynamic kernels to predict different lane instances. Some methods such as FOLOLane\cite{fololane} and GANet\cite{ganet} use bottom-up strategies to detect a few key points and model their global relations to form lane instances.

\textbf{Parameter-based Methods.} Instead of predicting a series of points locations or pixel classes, parameter-based methods directly generate the curve parameters of lane instances. PolyLanenet\cite{polylanenet} and LSTR\cite{lstr} consider the lane instance as a polynomial curve and output the polynomial coefficients directly. BézierLaneNet\cite{bezierlanenet} treats the lane instance as a Bézier curve and generates the locations of control points of the curve. BSLane uses B-Spline to describe the lane, and the curve parameters focus on the local shapes of lanes. Parameter-based methods are mostly end-to-end without post-processing, which grants them faster speed. However, since the final visual lane shapes are sensitive to the lane shape, the robustness and generalization of parameter-based methods may be less than ideal.


\textbf{Anchor-Based Methods.} Inspired by general object detection methods like YOLO \cite{yolov10} and Faster R-CNN \cite{fasterrcnn}, anchor-based approaches have been proposed for lane detection. Line-CNN is, to our knowledge, the earliest method that utilizes line anchors for detecting lanes. These lines are designed as rays emitted from the three edges (left, bottom, and right) of an image. However, the model’s receptive field is limited to the edges, which makes it suboptimal for capturing the entire lane. LaneATT \cite{laneatt} improves upon this by employing anchor-based feature pooling to aggregate features along the entire line anchor, achieving faster speeds and better performance. Nevertheless, its grid sampling strategy and label assignment pose limitations. CLRNet \cite{clrnet} enhances anchor-based performance with cross-layer refinement strategies, SimOTA label assignment \cite{yolox}, and Liou loss, surpassing many previous methods. A key advantage of anchor-based methods is their adaptability, allowing the integration of strategies from anchor-based general object detection, such as label assignment, bounding box refinement, and GIOU loss. However, existing anchor-based lane detection methods also have notable drawbacks. Lane anchors are often handcrafted and numerous, which can be cumbersome. Some approaches, such as ADNet \cite{adnet}, SRLane \cite{srlane}, and Sparse Laneformer \cite{sparse}, attempt to reduce the number of anchors and provide flexible proposals, but this can slightly impact performance. Additionally, methods such as \cite{clrernet} \cite{adnet} still rely on NMS post-processing, complicating NMS threshold settings and model deployment. Although one-to-one label assignment (during training) without NMS \cite{detr}\cite{o2o} (during evaluation) alleviates this issue, its performance remains less satisfactory compared to NMS-based models.

\begin{figure*}[ht]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/ovarall_architecture.png} % 替换为你的图片文件名
        \caption{The overall pipeline of Polar R-CNN. The architecture is simple and lightweight. The local polar head proposes sparse line anchors, and after pooling features along these anchors, the global polar head produces the final predictions. The global polar head includes a triplet head, which comprises a one-to-one classification head (O2O cls head), a one-to-many classification head (O2M cls head), and a one-to-many regression head (O2M reg head). The O2O classification head replaces NMS post-processing by selecting a single positive prediction sample for each ground truth from the redundant predictions generated by the O2M head.}
        \label{overall_architecture}
\end{figure*}

\textbf{NMS-Free Object Detections.}. Non-maximum suppression (NMS) is an important post-processing step in most general object detection methods. Detr \cite{detr} employs one-to-one label assignment to avoid redundant predictions without using NMS. Other NMS-free methods \cite{learnNMS} have also been proposed, addressing this issue from two aspects: model architecture and label assignment. Studies \cite{date} \cite{yolov10} suggest that one-to-one assignments are crucial for NMS-free predictions, but maintaining one-to-many assignments is still necessary to ensure effective feature learning of the model. Other works \cite{o3d} \cite{relationnet} consider the model’s expressive capacity to provide non-redundant predictions. However, few studies have analyzed the NMS-free paradigm for anchor-based lane detection methods as thoroughly as in general object detection. Most anchor-based lane detection methods still rely on NMS post-processing. In our work, besides label assignment, we extend the analysis to the detection head’s structure, focusing on achieving non-redundant (NMS-free) lane predictions.

In this work, we aim to address to two issues in anchor-based lane detection mentioned above, the sparse lane anchor setting and NMS-free predictions.

\section{Proposed method}
The overall architecture of Polar R-CNN is illustrated in Fig. \ref{overall_architecture}. Our model adheres to the Faster R-CNN \cite{fasterrcnn} framework, consisting of a backbone, Feature Pyramid Network (FPN), Region Proposal Network (RPN), and Region of Interest (RoI) pooling. To investigate the fundamental factors affecting model performance, such as anchor settings and NMS post-processing, and make the model easier to deploy, Polar R-CNN employs a simple and straightforward network structure. It relies on basic components including convolutional layers, Multi-Layer Perceptrons (MLPs), and pooling operations, deliberately excluding advanced elements like attention mechanisms, dynamic kernels, and cross-layer refinement used in pervious works \cite{clrnet}\cite{clrernet}.

% \begin{table}[h]
%         \centering
%         \caption{Notations of some important variable}
%         \begin{adjustbox}{width=\linewidth}
%         \begin{tabular}{lll}
%         \toprule
%         \textbf{Variable} & \textbf{Type} & \hspace{10em}\textbf{Defination} \\
%         \midrule
%         $\mathbf{P}_{i}$ & tensor& The $i_{th}$ output feature map from FPN\\
%         $H^{l}$& scalar& The height of the local polar map\\
%         $W^{l}$& scalar& The weight of the local polar map\\
%         $K_{a}$   & scalar& The number of anchors selected during evaluation\\
%         $\mathbf{c}^{g}$& tensor& The origin point of global polar coordinate\\
%         $\mathbf{c}^{l}$& tensor& The origin point of local polar coordinate\\
%         $r^{g}_{i}$& scalar& The $i_{th}$ anchor radius under global polar coordinate\\
%         $r^{l}_{i}$& scalar& The $i_{th}$ anchor radius under global polar coordinate\\
%         $\theta_{i}$& scalar& The $i_{th}$ anchor angle under global/local polar coordinate\\
%         \midrule
%         $\mathbf{X}^{pool}_{i}$& tensor& The pooling feature of the $i_{th}$ anchor\\
%         $N^{nbr}_{i}$& set& The adjacent node set of the $i_{th}$ of anchor node\\
%         $C_{o2m}$ & scalar& The positive threshold of one-to-many confidence\\
%         $C_{o2o}$ & scalar& The positive threshold of one-to-one confidence\\
%         $d_{dim}$ & scalar& Dimension of the distance tensor.\\
%         $w_{b}$ & scalar& Base width of the lane instance.\\
%         % \midrule
%         % & & \\
%         % & & \\
%         % & & \\
%         % & & \\
%         % & & \\
%         \bottomrule
%         \end{tabular}
%         \end{adjustbox}
% \end{table}



\subsection{Lane and Lane Anchor Representation}

Lanes are characterized by their thin and elongated curved shapes. A suitable lane prior aids the model in extracting features, predicting locations, and modeling the shapes of lane curves with greater accuracy. Consistent with previous studies \cite{linecnn}\cite{laneatt}, our lane priors (also referred to as lane anchors) consists of straight lines. We sample a sequence of 2D points along each lane anchor, denoted as $ P\doteq \left\{ \left( x_1, y_1 \right) , \left( x_2, y_2 \right) , ....,\left( x_n, y_n \right) \right\} $, where N is the number of sampled points. The y-coordinates of these points are uniformly sampled from the vertical axis of the image, specifically $y_i=\frac{H}{N-1}*i$, where H is the image height.  These y-coordinates are also sampled from the ground truth lane, and the model is tasked with regressing the x-coordinate offset from the lane anchor to the lane instance ground truth. The primary distinction between Polar R-CNN and previous approaches lies in the description of the lane anchors, which will be detailed in the following sections.
\begin{figure}[t]
        \centering
        \def\subwidth{0.24\textwidth}
        \def\imgwidth{\linewidth}
        \def\imgheight{0.4\linewidth}
        
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth]{thesis_figure/coord/ray.png}
                \caption{}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth]{thesis_figure/coord/polar.png}
                \caption{}
        \end{subfigure}
        \caption{Different descriptions for anchor parameters: (a) Ray: defined by its start point and orientation. (b) Polar: defined by its radius and angle.}
        \label{coord}
\end{figure}

\textbf{Polar Coordinate system.} Since lane anchors are typically represented as straight lines, they can be described using straight line parameters. Previous approaches have used rays to describe 2D lane anchors, with the parameters including the coordinates of the starting point and the orientation/angle, denoted as $\left\{\theta, P_{xy}\right\}$, as shown in Fig. \ref{coord}(a). \cite{linecnn}\cite{laneatt} define the start points as lying  on the three image boundaries. However, \cite{adnet} argue that this approach is problematic because the actual starting point of a lane could be located anywhere within the image. In our analysis, using a ray can lead to ambiguity in line representation because a line can have an infinite number of starting points, and the choice of the starting point for a lane is subjective. As illustrated in Fig. \ref{coord}(a), the yellow (the visual start point) and green (the point located on the image boundary) starting points with the same orientation $\theta$ describe the same line, and either could be used in different datasets \cite{scnn}\cite{vil100}. This ambiguity arises because a straight line has two degrees of freedom, whereas a ray has three (two for the start point and one for orientation). To resolve this issue , we propose using polar coordinates to describe a lane anchor with only two parameters: radius and angle, deoted as $\left\{\theta, r\right\}$, where $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and $r \in \left(-\infty, +\infty\right)$. This representation isillustrated in Fig. \ref{coord}(b).
 
\begin{figure}[t]
        \centering
        \includegraphics[width=0.45\textwidth]{thesis_figure/local_polar_head.png}
        \caption{The main architecture of LPH.}
        \label{lph}
\end{figure}
We define two types of polar coordinate systems: the global coordinate system and the local coordinate system, with the origin points denoted as the global origin $\boldsymbol{c}^{g}$ and the local origin $\boldsymbol{c}^{l}$, respectively. For convenience, the global origin is positioned near the static vanishing point of the entire lane image dataset, while the local origins are set at lattice points within the image. As illustrated in Fig. \ref{coord}(b), only the radius parameters are affected by the choice of the origin point, while the angle/orientation parameters remain consistent.
\subsection{Local Polar Head}
\textbf{Anchor formulation in local polar head.}. Inspired by the region proposal network in Faster R-CNN \cite{fasterrcnn}, the local polar head (LPH) aims to propose flexible, high-quality anchors aorund the lane ground truths within an image. As Fig. \ref{lph} and Fig. \ref{overall_architecture} demonstrate, the highest level $P_{3} \in \mathbb{R}^{C_{f} \times H_{f} \times W_{f}}$ of FPN feature maps is selected as the input for LPH. Following a downsampling operation, the feature map is then fed into two branches: the regression branch $\phi _{reg}^{lph}\left(\cdot \right)$ and the classification branch $\phi _{cls}^{lph}\left(\cdot \right)$:
\begin{equation}
        \begin{aligned}
                &F_d\gets DS\left( P_{3} \right), \,F_d\in \mathbb{R} ^{C_f\times H^{l}\times W^{l}},\\
                &F_{reg\,\,}\gets \phi _{reg}^{lph}\left( F_d \right), \,F_{reg\,\,}\in \mathbb{R} ^{2\times H^{l}\times W^{l}},\\
                &F_{cls}\gets \phi _{cls}^{lph}\left( F_d \right), \,F_{cls}\in \mathbb{R} ^{H^{l}\times W^{l}}.
        \end{aligned}
        \label{lph equ}
\end{equation}

 The regression branch aims to propose lane anchors by predicting two parameters $F_{reg\,\,} \equiv \left\{\theta_{j}, r^{l}_{j}\right\}_{j=1}^{H^{l}\times W^{l}}$, within the local polar coordinate system. These parameters represent the angles and the radius.The classification branch predicts the heat map $F_{cls\,\,}\equiv \left\{ c_j \right\} _{j=1}^{H^l\times W^l}$ of the local polar origin grid. By discarding local origin points with lower confidence, the module increases the likelihood of selecting potential positive foreground lane anchors while removing background lane anchors to the greatest extent. Keeping it simple, the regression branch $\phi _{reg}^{lph}\left(\cdot \right)$ consists of one $1\times1$ convolutional layer while the classification branch $\phi _{cls}^{lph}\left(\cdot \right)$ consists of two $1\times1$ convolutional layers.

\textbf{Loss Function.} During the training phase, as illustrated in Fig. \ref{lphlabel}, the ground truth labels for LPH are constructed as follows. The radius ground truth is defined as the shortest distance from a grid point (local origin point) to the ground truth lane curve. The angle ground truth is defined as the orientation of the vector from the grid point to the nearest point on the curve. A grid point is designated as a positive sample if its radius label is less than a threshold $\tau_{L}$ ; otherwise, it is considered a negative sample.

Once the regression and classification labels are established, the LPH can be trained using the smooth-L1 loss $d\left(\cdot \right)$ for regression and the binary cross-entropy loss $BCE\left( \cdot , \cdot \right)$ for classification. The LPH loss function is defined as follows:
\begin{equation}
        \begin{aligned}
                \mathcal{L} _{lph}^{cls}&=BCE\left( F_{cls},F_{gt} \right), \\
                \mathcal{L} _{lph}^{r\mathrm{e}g}&=\frac{1}{N_{lph}^{pos}}\sum_{j\in \left\{j|\hat{r}_i<\tau_{L} \right\}}{\left( d\left( \theta _j-\hat{\theta}_j \right) +d\left( r_j^L-\hat{r}_j^L \right) \right)}.\\
        \end{aligned}
        \label{loss_lph}
\end{equation}

\textbf{Top-$K_{a}$ Anchor Selectoin.}. During the training stage, all $H^{l}\times W^{l}$ anchors are considered as candidate anchors and fed into the R-CNN module. This approach helps the R-CNN module to learn from sufficient features of negative (background) anchor samples. In the evaluation stage, however, only the top-$K_{a}$ anchors with the highest confidence scores are selected and fed into the R-CNN module. This strategy is designed to filter out potential negative (background) anchors and reduce the computational complexity of the R-CNN module. By doing so, it maintains the adaptability and flexibility of anchor distribution while decreasing the total number of anchors. The following experiments will demonstrate the effectiveness of our top-$K_{a}$ anchor selection strategy.

\begin{figure}[t]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/coord/localpolar.png}
        \caption{Label construction for LPH.}
        \label{lphlabel}
\end{figure}

\subsection{Global Polar Head.}
Global polar head (GPH) is a crucial component in the second stage of Polar R-CNN. It takes lane anchor pooling features as input and predicts the precise lane location and confidence. Fig. \ref{gph} illustrates the structure and pipeline of GPH. GPH comprises RoI pooling modules and three subheads (triplet head module), which will be introduced in detail.

\textbf{RoI Pooling Module.} RoI pooling module is designed to transform features sampled from lane anchors into a standard feature tensor. Once the local polar parameters of a lane anchor are given, they can be converted to global polar coordinates using the following equation:
\begin{equation}
        \begin{aligned}
        r^{g}_{j}=r^{l}_{j}+\left( \textbf{c}^{l}_{j}-\textbf{c}^{g}_{j} \right) ^{T}\left[\cos\theta_{j}; \sin\theta_{j} \right].
        \end{aligned}
\end{equation}
where $\textbf{c}^{l}_{j} \in \mathbb{R}^{2}$ and $\textbf{c}^{g} \in \mathbb{R}^{2}$ represent the Cartesian coordinates of local and global origins correspondingly.

Next, feature points are sampled on the lane anchor. The y-coordinates of these points are uniformly sampled vertically from the image, as previously mentioned. The $x_{i}$ coordinates are computed using the global polar axis with the following equation:
\begin{equation}
        \begin{aligned}
        x_{i\,\,}=-y_i\tan \theta +\frac{r^{g}}{\cos \theta}.
        \end{aligned}
\end{equation}

\begin{figure}[t]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/detection_head.png} % 替换为你的图片文件名
        \caption{The main architecture of GPH.}
        \label{gph}
\end{figure}

Suppose the $P_{0}$, $P_{1}$ and $P_{2}$ denote the last three levels from FPN and $\boldsymbol{F}_{L}^{s}\in \mathbb{R} ^{N_p\times d_f}$ represent the $L_{th}$ sample point feature from $P_{L}$. The grid featuers from the three levels are extracted and fused together without cross layer cascade refinenment unlike CLRNet. To reduce the number of parameters, we employ a weight sum strategy to combine features from different layers (denoted as $L$), similar to \cite{detr}, but in a more compact form:
\begin{equation}
        \begin{aligned}
                \boldsymbol{F}^s=\sum_{L=0}^2{\boldsymbol{F}_{L}^{s}\times \frac{e^{\boldsymbol{w}_{L}^{s}}}{\sum_{L=0}^2{e^{\boldsymbol{w}_{L}^{s}}}}},
        \end{aligned}
\end{equation}
where $\boldsymbol{w}_{L}^{s}\in \mathbb{R} ^{N_p}$ represents the learnable aggregate weight, serving as a learned model weight. Instead of concatenating the three sampling features into $\boldsymbol{F}^s\in \mathbb{R} ^{N_p\times d_f\times 3}$ directly, the adaptive summation significantly reduces the feature dimensions to $\boldsymbol{F}^s\in \mathbb{R} ^{N_p\times d_f}$, which is one-third of the original dimension. The weighted sum tensors are then fed into fully connected layers to obtain the pooled RoI features of an anchor:
 \begin{equation}
        \begin{aligned}
                \boldsymbol{F}^{roi}\gets FC_{pooling}\left( \boldsymbol{F}^s \right), \boldsymbol{F}^{roi}\in \mathbb{R} ^{d_r},
        \end{aligned}
\end{equation} 

\textbf{Triplet Head.} The triplet head comprises three distinct heads: the one-to-one classification (O2O cls) head, the one-to-many classification (O2M cls) head, and the one-to-many regression (O2M reg) head. In various studies \cite{laneatt}\cite{clrnet}\cite{adnet}\cite{srlane}, the detection head predominantly follows the one-to-many paradigm. During the training phase, multiple positive samples are assigned to a single ground truth. Consequently, during the evaluation stage, redundant detection results are often predicted for each instance. These redundancies are typically addressed using NMS, which eliminates duplicate results and retains the highest confidence detection for each groung truth. However, NMS relies on the definition of distance between detection results, and this calculation can be complex for curved lanes and other irregular geometric shapes. To achieve non-redundant detection results with a NMS-free paradigm, the one-to-one paradigm becomes crucial during training, as highlighted in \cite{o2o}. Nevertheless, merely adopting the one-to-one paradigm is insufficient; the structure of the detection head also plays a pivotal role in achieving NMS-free detection. This aspect will be further analyzed in the following sections.

\textbf{NMS vs NMS-free.} Let $\boldsymbol{F}^{roi}_{i}$ denotes the ROI features extracted from $i_{th}$ anchors and the three subheads using $\boldsymbol{F}^{roi}_{i}$ as input. For now, let us focus on the O2M classification (O2M cls) head and the O2M regression (O2M reg) head, which follow the old paradigm used in previous work and can serve as a baseline for the new one-to-one paradigm. To maintain simplicity and rigor, both the O2M classification head and the O2M regression head consist of two layers with activation functions, featuring a plain structure without any complex mechanisms such as attention or deformable convolution. as previously mentioned, merely replacing the one-to-many label assignment with one-to-one label assignment is insufficient for eliminating NMS post-processing. This is because anchors often exhibit significant overlap or are positioned very close to each other, as shown in Fig. \ref{anchor setting}(b)\&(c). Let the $\boldsymbol{F}^{roi}_{i}$ and $\boldsymbol{F}^{roi}_{j}$ represent the features from two overlapping (or very close) anchors, implying that $\boldsymbol{F}^{roi}_{i}$ and $\boldsymbol{F}^{roi}_{j}$ will be almost identical. Let $f_{plain}^{cls}$ denotes the neural structure used in O2M classification head and suppose it's trained with one-to-one label assignment. If $\boldsymbol{F}^{roi}_{i}$ is a positive sample and the $\boldsymbol{F}^{roi}_{j}$ is a negative sample, the ideal output should be as follows:
\begin{equation}
        \begin{aligned}
                &\boldsymbol{F}_{i}^{roi}\approx \boldsymbol{F}_{j}^{roi},
                \\
                &f_{cls}^{plain}\left( \boldsymbol{F}_{i}^{roi} \right) \rightarrow 1,
                \\
                &f_{cls}^{plain}\left( \boldsymbol{F}_{i}^{roi} \right) \rightarrow 0.
        \end{aligned}
        \label{sharp fun}
\end{equation}

The Eq. (\ref{sharp fun}) suggests that the property of $f_{cls}^{plain}$ need to be ``sharp'' enough to differentiate between two similar features. That is to say, the output of $f_{cls}^{plain}$ changes rapidly over short periods or distances,  it implies that $f_{cls}^{plain}$ need to captures information with higher frequency. This issue is also discussed in \cite{o3d}. Capturing the high frequency with a plain structure is difficult because a naive MLP tends to capture information with lower frequency \cite{xu2022overview}. In the most extreme case, where $\boldsymbol{F}_{i}^{roi} = \boldsymbol{F}_{j}^{roi}$, it becomes impossible to distinguish the two anchors to positive and negative samples completely; in practice, both confidences converge to around 0.5. This problem arises from the limitations of the input format and the structure of the naive MLP, which restrict its expressive capability for information with higher frequency. Therefore, it is crucial to establish relationships between anchors and design a new model structure to effectively represent ``sharp'' information.

It is easy to see that the ``ideal'' one-to-one branch is equivalence to O2M cls branch with O2M regression and NMS post-processing. If the NMS could be replaced by some equivalent but learnable functions (e.g. a neural network with specific structure), the O2O head could be trained to handle the one-to-one assignment. However, the NMS involves sequential iteration and confidence sorting, which are challenging to reproduce with a neural network. Although previous works, such as RNN-based approaches \cite{stewart2016end}, utilize an iterative format, they are time-consuming and introduce additional complexity into the model training process due to their iterative nature. To eliminate the iteration process, we proposed a equivalent format of Fast NMS\cite{yolact}.


\begin{algorithm}[t]
        \caption{The Algorithm of the Graph-based Fast NMS}
        \begin{algorithmic}[1] %这个1 表示每一行都显示数字
        \REQUIRE ~~\\ %算法的输入参数：Input 
            The index of positive predictions, $1, 2, ..., i, ..., N_{pos}$;\\
            The positive corresponding anchors, $[\theta_i, r_{i}^{global}]$;\\
            The x axis of sampling points from positive anchors, $\boldsymbol{x}_{i}^{b}$;\\
            The positive confidence get from o2m classification head, $s_i$;\\
            The positive regressions get from o2m regression head, the horizontal offset $\varDelta \boldsymbol{x}_{i}^{roi}$ and end point location $\boldsymbol{e}_{i}$.\\
        \ENSURE ~~\\ %算法的输出：Output
            \STATE Calculate the confidential adjacent matrix $\boldsymbol{C} \in \mathbb{R} ^{N_{pos} \times N_{pos}} $, where the element $C_{ij}$ in $\boldsymbol{C}$ is caculate as follows: 
            \begin{equation}
                \begin{aligned}
                C_{ij}=\begin{cases}
                1,  s_i<s_j\,\,| \left( s_i=s_j \land i<j \right)\\
                        0,  others\\
                        \end{cases}
                \end{aligned}
                \label{al_1-1}
            \end{equation} 
            where the $\land$  denotes (element wise) logical ``AND'' operation between two Boolean values/tensors.
            \STATE Calculate the geometric prior adjacent matrix $\boldsymbol{M} \in \mathbb{R} ^{N_{pos} \times N_{pos}} $, where the element $M_{ij}$ in $\boldsymbol{M}$ is caculate as follows: 
            \begin{equation}
                \begin{aligned}
                M_{ij}=\begin{cases}
                        1,\left| \theta _i-\theta _j \right|<\theta _{\tau}\land \left| r_{i}^{global}-r_{j}^{global} \right|<r_{\tau}\\
                        0, others\\
                        \end{cases}
                \end{aligned}
                \label{al_1-2}
            \end{equation} 

            \STATE Calculate the inverse distance matrix $\boldsymbol{D}  \in \mathbb{R} ^{N_{pos} \times N_{pos}}$, where the element $D_{ij}$ in $\boldsymbol{D}$ is defined as follows: 
            \begin{equation}
                \begin{aligned}
            D_{ij} = 1-d\left( \boldsymbol{x}_{i}^{b} + \varDelta \boldsymbol{x}_{i}^{roi}, \boldsymbol{x}_{j}^{b} + \varDelta \boldsymbol{x}_{j}^{roi}, \boldsymbol{e}_{i}, \boldsymbol{e}_{j}\right),
                \end{aligned}
                \label{al_1-3}
            \end{equation} 
            where $d\left(\cdot, \cdot, \cdot, \cdot \right)$ is some predefined function to quantify the distance between two lane predictions.
            \STATE Define the adjacent matrix $\boldsymbol{T}=\,\,\boldsymbol{C}\land\boldsymbol{M}$ and the final confidence $\tilde{s}_i$ is calculate as following:
            \begin{equation}
                \tilde{s}_i = \begin{cases}
                    1, & \text{if } \underset{j \in \{ j \mid T_{ij} = 1 \}}{\max} D_{ij} < \delta_{\tau} \\
                    0, & \text{otherwise}
                \end{cases}
                \label{al_1-4}
            \end{equation}
                
                        
        \RETURN The final confidence $\tilde{s}_i$. % the return result of the algorithm
        \end{algorithmic}
        \label{Graph Fast NMS}
\end{algorithm}

\begin{figure}[t]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/gnn.png} % 替换为你的图片文件名
        \caption{The main architecture of O2O classification head.}
        \label{o2o_cls_head}
\end{figure}

The key rule of the NMS post-processing is as follows:
Given a series of positive detections with redundancy, a detection result A is suppressed by another detection result B if and only if:

(1) The confidence of A is lower than that of B. 

(2) The predefined distance (e.g. IoU distance and L1 distance) between A and B is smaller than a threshold.

(3) B is not suppressed by any other detection results.

For simplicity, Fast NMS only satisfies the condition (1) and (2), which may lead to an increase in false negative predictions but offers faster processing without sequential iteration. Leveraging the “iteration-free” property, we propose a further refinement called “sort-free” Fast NMS. This new approach, named Graph-based Fast NMS, is detailed in Algorithm \ref{Graph Fast NMS}.

It is straightforward to demonstrate that, when all elements in $\boldsymbol{M}$ are all set to 1 (regardless of geometric priors), Graph-based Fast NMS is equivalent to Fast NMS. Building upon our newly proposed Graph-based Fast NMS, we can design the structure of the one-to-one classification head in a manner that mirrors the principles of following Graph-based Fast NMS.

According to the analysis of the shortcomings of traditional NMS post-processing shown in Fig. \ref{NMS setting}, the fundamental issue arises from the definition of the distance between predictions. Traditional NMS relies on geometric properties to define distances between predictions, which often neglects the contextual semantics. For example, in some scenarios, two predicted lanes with a small geometric distance should not be suppressed, such as the case of double lines or fork lines. Although setting a threshold $d_{\tau}$ can mitigate this problem, it is challenging to strike a balance between precision and recall.

To address this, we replace the explicit definition of the inverse distance function with an implicit graph neural network. Additionally, the coordinates of anchors is also replace with the anchor features ${F}_{i}^{roi}$. According to information bottleneck theory \cite{alemi2016deep}, ${F}_{i}^{roi}$ , which contains the location and classification information, is sufficient for modelling the explicit geometric distance by neural network. Besides the geometric information, features ${F}_{i}^{roi}$ containes the implicit contextual information of an anchor, which provides additional clues for establishing implicit contextual distances between two anchors. The implicit contextual distance is calculated as follows:
 \begin{equation}
        \begin{aligned}
                \tilde{\boldsymbol{F}}_{i}^{roi}\gets& \mathrm{Re}LU\left( FC_{o2o}^{roi}\left( \boldsymbol{F}_{i}^{roi} \right) \right),
                \\
                \boldsymbol{F}_{ij}^{edge}\gets& FC_{in}\left( \tilde{\boldsymbol{F}}_{i}^{roi} \right) -FC_{out}\left( \tilde{\boldsymbol{F}}_{i}^{roi} \right)
                \\
                &+FC_{base}\left( \boldsymbol{x}_{i}^{b}-\boldsymbol{x}_{j}^{b} \right),
                \\
                \boldsymbol{D}_{ij}^{edge}\gets& MLP_{edge}\left( \boldsymbol{F}_{ij}^{graph} \right).
                \\
        \end{aligned}
        \label{edge_layer}
\end{equation}

Eq. (\ref{edge_layer}) represents the implicit expression of Eq. (\ref{al_1-3}), where the inverse distance $\boldsymbol{D}_{ij}^{edge}$ is no longer a scalar but a semantic tensor with dimension $d_{dis}$. $\boldsymbol{D}_{ij}^{edge}$ containes more complex information compared to traditional geometric distance. The confidence caculation is expressed as follows:
\begin{equation}
        \begin{aligned}
                &\boldsymbol{D}_{i}^{node}\gets \underset{j\in \left\{ j|T_{ij}=1 \right\}}{\max}\boldsymbol{D}_{ij}^{edge},
                \\
                &\boldsymbol{F}_{i}^{node}\gets MLP_{node}\left( \boldsymbol{D}_{i}^{node} \right),
                \\
                &\tilde{s}_i\gets \sigma \left( FC_{o2o}^{out}\left( \boldsymbol{F}_{i}^{node} \right) \right).
        \end{aligned}
        \label{node_layer}
\end{equation}

The Eq. (\ref{node_layer}) serves as the implicit replacement for Eq. (\ref{al_1-4}). In this approach, we use elementwise max pooling of tensors instead of scalar-based max operations. The pooled tensor is then fed into a neural network with a sigmoid activation function to directly obtain the confidence. By eliminating the need for a predefined distance threshold, all confidence calculation patterns are derived from the training data.

It should be noted that the O2O classification head depends on the predictons of O2M classification head as outlined in Eq. (\ref{al_1-1}). From a probablity percpective, the confidence output by O2M classification head, $s_{j}$, represents the probability that the $j_{th}$ detection is a positive sample. The confidence output by O2O classification head, $\tilde{s}_i$, denotes the conditional probablity that $i_{th}$ sample shouldn't be suppressed given the condition that the $i_{th}$ sample identified as a positive sample:
\begin{equation}
        \begin{aligned}
                &s_j|_{j=1}^{N_a}\equiv P\left( a_j\,\,is\,\,pos \right), \,\,  
                \\
                &\tilde{s}_i|_{i=1}^{N_{pos}}\equiv P\left( a_i\,\,is\,\,retained|a_i\,is\,\,pos \right),
        \end{aligned}
        \label{probablity}
\end{equation}
where $N_a$ equals $H^{l}\times W^{l}$ during the training stage and  $K_{a}$ during the testing stage. The overall architecture of O2O classification head is illustrated in Fig. \ref{o2o_cls_head}.

\textbf{Label assignment and Cost function.} We use the label assignment (SimOTA) similar to previous works \cite{clrnet}\cite{clrernet}. However, to make the function more compact and consistent with general object detection works \cite{iouloss}\cite{giouloss}, we have redefined the lane IoU. As illustrated in Fig. \ref{glaneiou}, the newly-defined lane IoU, which we refer to as GLaneIoU, is redefined as follows:
\begin{figure}[t]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/GLaneIoU.png} % 替换为你的图片文件名
        \caption{Illustrations of GLaneIoU redefined in our work.}
        \label{glaneiou}
\end{figure}
\begin{equation}
        \begin{aligned}
                &w_{i}^{k}=\frac{\sqrt{\left( \Delta x_{i}^{k} \right) ^2+\left( \Delta y_{i}^{k} \right) ^2}}{\Delta y_{i}^{k}}w_{b},
                \\
                &\hat{d}_{i}^{\mathcal{O}}=\min \left( x_{i}^{p}+w_{i}^{p}, x_{i}^{q}+w_{i}^{q} \right) -\max \left( x_{i}^{p}-w_{i}^{p}, x_{i}^{q}-w_{i}^{q} \right),
                \\
                &\hat{d}_{i}^{\xi}=\max \left( x_{i}^{p}-w_{i}^{p}, x_{i}^{q}-w_{i}^{q} \right) -\min \left( x_{i}^{p}+w_{i}^{p}, x_{i}^{q}+w_{i}^{q} \right), 
                \\
                &d_{i}^{\mathcal{U}}=\max \left( x_{i}^{p}+w_{i}^{p}, x_{i}^{q}+w_{i}^{q} \right) -\min \left( x_{i}^{p}-w_{i}^{p}, x_{i}^{q}-w_{i}^{q} \right),
                \\
                &d_{i}^{\mathcal{O}}=\max \left( \hat{d}_{i}^{\mathcal{O}},0 \right), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,   d_{i}^{\xi}=\max \left( \hat{d}_{i}^{\xi},0 \right),
        \end{aligned}
\end{equation}
where $w_{b}$ is the base semi-width of the lane instance. The definations of $d_{i}^{\mathcal{O}}$ and $d_{i}^{\mathcal{\xi}}$ is similar but slightly different from those in \cite{clrnet} and \cite{adnet}, with adjustments made to ensure the values are non-negative. This format is intended to maintain consistency with the IoU definitions used for bounding boxes. Therefore, the overall GLaneIoU is given as follows:
\begin{equation}
        \begin{aligned}
GLaneIoU\,\,=\,\,\frac{\sum\nolimits_{i=j}^k{d_{i}^{\mathcal{O}}}}{\sum\nolimits_{i=j}^k{d_{i}^{\mathcal{U}}}}-g\frac{\sum\nolimits_{i=j}^k{d_{i}^{\xi}}}{\sum\nolimits_{i=j}^k{d_{i}^{\mathcal{U}}}},
\end{aligned}
\end{equation}
where j and k are the indices of the valid points (the start point and the end point). It's straightforward to observed that when $g=0$, the GLaneIoU is correspond to GIoU\cite{giouloss} for bounding box, with a value range of $\left[0, 1 \right]$. When $g=1$, the GLaneIoU is correspond to GIoU for bounding box, with a value range of $\left(-1, 1 \right]$. In general, when $g>0$, the value range of GLaneIoU is $\left(-g, 1 \right]$. 
We then define the cost function between $i_{th}$ prediction and $j_{th}$ ground truth as follows like \cite{detr}:
\begin{equation}
        \begin{aligned}
                \mathcal{C} _{ij}=\left(s_i\right)^{\beta_c}\times \left( GLaneIoU_{ij, g=0} \right) ^{\beta_r}.
\end{aligned}
\end{equation}

This cost function is more compact than those in previous works\cite{clrnet}\cite{adnet} and takes both location and confidence into account. For label assignment, SimOTA (with k=4) \cite{yolox} is used for the two O2M heads with one-to-many assignment, while the Hungarian \cite{detr} algorithm is employed for the O2O classification head for one-to-one assignment.
\begin{figure}[t]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/auxloss.png} % 
        \caption{Auxiliary loss for segment parameter regression.}
        \label{auxloss}
\end{figure}

\textbf{Loss function.} We use focal loss \cite{focal} for O2O classification head and O2M classification head:
\begin{equation}
        \begin{aligned}
                \mathcal{L} _{o2m}^{cls}&=\sum_{i\in \varOmega _{pos}^{o2m}}{\alpha _{o2m}\left( 1-s_i \right) ^{\gamma}\log \left( s_i \right)}\\&+\sum_{i\in \varOmega _{neg}^{o2m}}{\left( 1-\alpha _{o2m} \right) \left( s_i \right) ^{\gamma}\log \left( 1-s_i \right)},
                \\
                \mathcal{L} _{o2o}^{cls}&=\sum_{i\in \varOmega _{pos}^{o2o}}{\alpha _{o2o}\left( 1-\tilde{s}_i \right) ^{\gamma}\log \left( \tilde{s}_i \right)}\\&+\sum_{i\in \varOmega _{neg}^{o2o}}{\left( 1-\alpha _{o2o} \right) \left( \tilde{s}_i \right) ^{\gamma}\log \left( 1-\tilde{s}_i \right)}.
                \\
        \end{aligned}
\end{equation}
where the set of the one-to-one sample, $\varOmega _{pos}^{o2o}$ and $\varOmega _{neg}^{o2o}$, is restricted to the positive sample set of O2M classification head:
\begin{equation}
        \begin{aligned}
                \varOmega _{pos}^{o2o}\cup \varOmega _{neg}^{o2o}=\left\{ i|s_i>C_{o2m} \right\}.
        \end{aligned}
\end{equation}

Only one sample with confidence larger than $C_{o2m}$ is chosed as the canditate sample of O2O classification head. According to \cite{pss},  to maintain feature quality during training stage, the gradient of O2O classification head are stopped from propagating back to the rest of the network (stop from the roi feature of the anchor $\boldsymbol{F}_{i}^{roi}$). Additionally, we use the rank loss to increase the gap between positive and negative confidences of O2O classification head:
\begin{equation}
        \begin{aligned}
                &\mathcal{L} _{\,\,rank}=\frac{1}{N_{rank}}\sum_{i\in \varOmega _{pos}^{o2o}}{\sum_{j\in \varOmega _{neg}^{o2o}}{\max \left( 0, \tau _{rank}-\tilde{s}_i+\tilde{s}_j \right)}},\\
                &N_{rank}=\left| \varOmega _{pos}^{o2o} \right|\left| \varOmega _{neg}^{o2o} \right|.
        \end{aligned}
\end{equation}

We directly use the GLaneIoU loss, $\mathcal{L}_{GLaneIoU}$, to regression the offset of xs (with g=1) and Smooth-L1 loss for the regression of end points (namely the y axis of the start point and the end point), denoted as $\mathcal{L} _{end}$. In order to make model learn the global features, we proposed the auxiliary loss illustrated in Fig. \ref{auxloss}:
\begin{align}
        \begin{aligned}
        \mathcal{L}_{aux} &= \frac{1}{\left| \varOmega_{pos}^{o2m} \right| N_{seg}} \sum_{i \in \varOmega_{pos}^{o2o}} \sum_{m=j}^k \Bigg[ l \left( \theta_i - \hat{\theta}_{i}^{seg,m} \right) \\
        &\quad + l \left( r_{i}^{global} - \hat{r}_{i}^{seg,m} \right) \Bigg].
        \end{aligned}
\end{align}

The anchors and ground truth are divided into several segments. Each anchor segment is regressed to the main components of the corresponding segment of the assigned ground truth. This trick assists the anchors in learning more about the global geometric shape.

\subsection{Loss function}

The overall loss function of Polar R-CNN is given as follows:
\begin{equation}
        \begin{aligned}
                \mathcal{L}_{overall} &=\mathcal{L} _{lph}^{cls}+w_{lph}^{reg}\mathcal{L} _{lph}^{reg}\\&+w_{o2m}^{cls}\mathcal{L} _{o2m}^{cls}+w_{o2o}^{cls}\mathcal{L} _{o2o}^{cls}+w_{rank}\mathcal{L} _{rank}\\&+w_{IoU}\mathcal{L} _{IoU}+w_{end}\mathcal{L} _{end}+w_{aux}\mathcal{L} _{aux}.
        \end{aligned}
\end{equation}
The first line in the loss function represents the loss for LPH, which includes both classification and regression components. The second line pertains to the losses associated with the two classification heads (O2M and O2O), while the third line represents the loss for the regression head within the triplet head. Each term in the equation is weighted by a factor to balance the contributions of each component to the gradient. The entire training process is end-to-end.

\begin{table*}[htbp]
        \centering
        \caption{Infos and hyperparameters for five datasets. For CULane, $*$ denotes the actual number of training samples used to train our model. Please note that labels for some validation/test sets are missing; therefore, we have selected different splits (test or validation set) for different datasets.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{l|l|ccccc}
        \toprule
        \multicolumn{2}{c|}{\textbf{Dataset}} & CULane & TUSimple & LLAMAS & DL-Rail & CurveLanes \\
        \midrule
        \multirow{7}*{Dataset Description}
        & Train      &88,880/$55,698^{*}$&3,268 &58,269&5,435&100,000\\
        & Validation &9,675 &358   &20,844&-    &20,000 \\
        & Test       &34,680&2,782 &20,929&1,569&-      \\
        & Resolution &$1640\times590$&$1280\times720$&$1276\times717$&$1920\times1080$&$2560\times1440$, etc\\
        & Lane &$\leqslant4$&$\leqslant5$&$\leqslant4$&$=2$&$\leqslant10$\\
        & Environment &urban and highway & highway&highway&railay&urban and highway\\
        & Distribution &sparse&sparse&sparse&sparse&sparse and dense\\
        \midrule
        \multirow{2}*{Dataset Split}
        & Evaluation &Test&Test&Test&Test&Val\\
        & Visualization &Test&Test&Val&Test&Val\\
        \midrule
        \multirow{1}*{Data Preprocess}
        & Crop Height &270&160&300&560&640, etc\\
        \midrule
        \multirow{6}*{Training Hyperparameter}
        & Epoch Number &32&70&20&90&32\\
        & Batch Size   &40&24&32&40&40\\
        & Warm up iterations &800&200&800&400&800\\
        & $w_{aux}$  &0.2&0  &0.2&0.2&0.2\\
        & $w_{rank}$ &0.7&0.7&0.1&0.7&0  \\
        \midrule
        \multirow{4}*{Evaluation Hyperparameter}
        & $H^{l}\times W^{l}$ &$4\times10$&$4\times10$&$4\times10$&$4\times10$&$6\times13$\\
        & $K_{a}$ &20&20&20&12&50\\
        & $C_{o2m}$ &0.48&0.40&0.40&0.40&0.45\\
        & $C_{o2o}$ &0.46&0.46&0.46&0.46&0.44\\
        \bottomrule
        \end{tabular}
        \end{adjustbox}
        \label{dataset_info}
    \end{table*}

\section{Experiment}
\subsection{Dataset and Evaluation Metric}
We conducted experiments on four widely used lane detection benchmarks and one rail detection dataset: CULane\cite{scnn}, TuSimple\cite{tusimple}, LLAMAS\cite{llamas}, CurveLanes\cite{curvelanes}, and DL-Rail\cite{dalnet}. Among these datasets, CULane and CurveLanes are particularly challenging. The CULane dataset consists various scenarios but has sparse lane distributions, whereas CurveLanes includes a large number of curved and dense lane types, such as forked and double lanes. The DL-Rail dataset, focused on rail detection across different scenarios, is chosen to evaluate our model’s performance beyond traditional lane detection. The details for five dataset are shown in Table. \ref{dataset_info}

We use the F1-score to evaluate our model on the CULane, LLAMAS, DL-Rail, and Curvelanes datasets, maintaining consistency with previous works. The F1-score is defined as follows:
\begin{equation}
        \begin{aligned}
        F1=\frac{2\times Precision\times Recall}{Precision\,\,+\,\,Recall},
        \\
        Precision\,\,=\,\,\frac{TP}{TP+FP},
        \\
        Recall\,\,=\,\,\frac{TP}{TP+FN}.
        \end{aligned}
\end{equation}
In our experiment, we use different IoU thresholds to calculate the F1-score for different datasets: F1@50 and F1@75 for CULane \cite{clrnet}, F1@50 for LLAMAS \cite{clrnet} and Curvelanes \cite{CondLaneNet}, and F1@50, F1@75, and mF1 for DL-Rail \cite{dalnet}. The mF1 is defined as:
\begin{equation}
        \begin{aligned}
                mF1=\left( F1@50+F1@55+...+F1@95 \right) /10.
        \end{aligned}
\end{equation}

For Tusimple, the evaluation is formulated as follows:
\begin{equation}
        \begin{aligned}
                Accuracy=\frac{\sum{C_{clip}}}{\sum{S_{clip}}}.
        \end{aligned}
\end{equation}
where $C_{clip}$ and $S_{clip}$ represent the number of correct points (predicted points within 20 pixels of the ground truth) and the ground truth points, respectively. If the accuracy exceeds 85\%, the prediction is considered correct. TuSimples also report the False Positive Rate (FPR=1-Precision) and False Negative Rate (FNR=1-Recall) formular. 

\subsection{Implement Detail}
All input images are cropped and resized to $800\times320$. Similar to \cite{clrnet}, we apply random affine transformations and random horizontal flips. For the optimization process, we use the AdamW \cite{adam} optimizer with a learning rate warm-up and a cosine decay strategy. The initial learning rate is set to 0.006. The number of sampled points and regression points for each lane anchor are set to 36 and 72, respectively. The power coefficients of cost function, $\beta_{c}$ and $\beta_{r}$, are set to 1 and 6 respectively. We set different base semi-widths, denoted as $w_{b}^{assign}$, $w_{b}^{cost}$ and $w_{b}^{loss}$ for label assignment, cost function and loss function, respectively, as demonstrated in previous work\cite{clrernet}. Other parameters, such as batch size and loss weights for each dataset, are detailed in Table \ref{dataset_info}. Since some test/validation sets for the five datasets are not accessible, the test/validation sets used are also listed in Table \ref{dataset_info}. All the expoeriments are conducted on a single NVIDIA A100-40G GPU. To make our model simple, we only use CNN-based backbone, namely ResNet\cite{resnet} and DLA34\cite{dla}. 


\begin{table*}[htbp]
        \centering
        \caption{Comparision results on CULane test set with other methods.}
        \normalsize
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{lrlllllllllll}
        \toprule
        \textbf{Method}& \textbf{Backbone}&\textbf{F1@50}$\uparrow$& \textbf{F1@75}$\uparrow$& \textbf{Normal}$\uparrow$&\textbf{Crowded}$\uparrow$&\textbf{Dazzle}$\uparrow$&\textbf{Shadow}$\uparrow$&\textbf{No line}$\uparrow$& \textbf{Arrow}$\uparrow$& \textbf{Curve}$\uparrow$& \textbf{Cross}$\downarrow$ & \textbf{Night}$\uparrow$  \\
        \hline
        \textbf{Seg \& Grid} \\
        \cline{1-1}
        SCNN\cite{scnn}        &VGG-16   &71.60&39.84&90.60&69.70&58.50&66.90&43.40&84.10&64.40&1900&66.10\\
        RESA\cite{resa}        &ResNet50 &75.30&53.39&92.10&73.10&69.20&72.80&47.70&83.30&70.30&1503&69.90\\
        LaneAF\cite{laneaf}     &DLA34    &77.41&-    &91.80&75.61&71.78&79.12&51.38&86.88&72.70&1360&73.03\\
        UFLDv2\cite{ufldv2}      &ResNet34 &76.0 &-    &92.5 &74.8 &65.5 &75.5 &49.2 &88.8 &70.1 &1910&70.8 \\
        CondLaneNet\cite{CondLaneNet} &ResNet101&79.48&61.23&93.47&77.44&70.93&80.91&54.13&90.16&75.21&1201&74.80\\
        \cline{1-1}
        \textbf{Parameter} \\
        \cline{1-1}
        BézierLaneNet\cite{bezierlanenet} &ResNet18&73.67&-&90.22&71.55&62.49&70.91&45.30&84.09&58.98&\textbf{996} &68.70\\
        BSNet\cite{bsnet}         &DLA34   &80.28&-&93.87&78.92&75.02&82.52&54.84&90.73&74.71&1485&75.59\\
        Eigenlanes\cite{eigenlanes}    &ResNet50&77.20&-&91.7 &76.0 &69.8 &74.1 &52.2 &87.7 &62.9 &1509&71.8 \\
        \cline{1-1}
        \textbf{Keypoint} \\
        \cline{1-1}
        CurveLanes-NAS-L\cite{curvelanes} &-  &74.80&-&90.70&72.30&67.70&70.10&49.40&85.80&68.40&1746&68.90\\
        FOLOLane\cite{fololane}         &ResNet18 &78.80&-&92.70&77.80&75.20&79.30&52.10&89.00&69.40&1569&74.50\\
        GANet-L\cite{ganet}          &ResNet101&79.63&-&93.67&78.66&71.82&78.32&53.38&89.86&77.37&1352&73.85\\
        \cline{1-1}
        \textbf{Dense Anchor} \\
        \cline{1-1}
        LaneATT\cite{laneatt}  &ResNet18 &75.13&51.29&91.17&72.71&65.82&68.03&49.13&87.82&63.75&1020&68.58\\
        LaneATT\cite{laneatt}  &ResNet122&77.02&57.50&91.74&76.16&69.47&76.31&50.46&86.29&64.05&1264&70.81\\
        CLRNet\cite{laneatt}   &Resnet18 &79.58&62.21&93.30&78.33&73.71&79.66&53.14&90.25&71.56&1321&75.11\\
        CLRNet\cite{laneatt}   &DLA34    &80.47&62.78&93.73&79.59&75.30&82.51&54.58&90.62&74.13&1155&75.37\\
        CLRerNet\cite{clrernet} &DLA34    &81.12&64.07&94.02&80.20&74.41&\textbf{83.71}&56.27&90.39&74.67&1161&\textbf{76.53}\\
        \cline{1-1}
        \textbf{Sparse Anchor} \\
        \cline{1-1}
        ADNet \cite{adnet}            &ResNet34&78.94&-&92.90&77.45&71.71&79.11&52.89&89.90&70.64&1499&74.78\\
        SRLane \cite{srlane}         &ResNet18&79.73&-&93.52&78.58&74.13&81.90&55.65&89.50&75.27&1412&74.58\\
        Sparse Laneformer\cite{sparse} &Resnet50&77.83&-&-    &-    &-    &-    &-    &-    &-    &-   &-    \\
        \hline
        \textbf{Proposed Method} \\
        \cline{1-1}
        Polar R-CNN-NMS   &ResNet18&80.81&63.97&94.12&79.57&76.53&83.33&55.10&90.70&79.50&1088&75.25\\
        Polar R-CNN       &ResNet18&80.81&63.96&94.12&79.57&76.53&83.33&55.06&90.62&79.50&1088&75.25\\
        Polar R-CNN       &ResNet34&80.92&63.97&94.24&79.76&76.70&81.93&55.40&\textbf{91.12}&79.85&1158&75.71\\
        Polar R-CNN       &ResNet50&81.34&64.77&94.45&\textbf{80.42}&75.82&83.61&56.62&91.10&80.05&1356&75.94\\
        Polar R-CNN-NMS   &DLA34   &\textbf{81.49}&64.96&\textbf{94.44}&80.36&\textbf{76.79}&83.68&56.52&90.85&\textbf{80.09}&1133&76.32\\
        Polar R-CNN       &DLA34   &\textbf{81.49}&\textbf{64.97}&\textbf{94.44}&80.36&\textbf{76.79}&83.68&\textbf{56.55}&90.81&\textbf{79.80}&1133&76.33\\
        \bottomrule
        \end{tabular}  
      \end{adjustbox}
        \label{culane result}
\end{table*}





\begin{table}[h]
        \centering
        \caption{Comparision results on TuSimple test set with other methods.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{lrcccc}
        \toprule
        \textbf{Method}& \textbf{Backbone}& \textbf{Acc(\%)}&\textbf{F1(\%)}&\textbf{FP(\%)}&\textbf{FN(\%)} \\
        \midrule
        SCNN\cite{scnn}      &VGG16         &96.53&95.97&6.17&\textbf{1.80}\\
        PolyLanenet\cite{polylanenet}&EfficientNetB0&93.36&90.62&9.42&9.33\\
        UFLDv2\cite{ufld}     &ResNet34      &88.08&95.73&18.84&3.70\\
        LaneATT\cite{laneatt}    &ResNet34      &95.63&96.77&3.53&2.92\\
        FOLOLane\cite{laneatt}   &ERFNet        &\textbf{96.92}&96.59&4.47&2.28\\
        CondLaneNet\cite{CondLaneNet}&ResNet101     &96.54&97.24&2.01&3.50\\
        CLRNet\cite{clrnet}     &ResNet18      &96.84&97.89&2.28&1.92\\
        \midrule
        Polar R-CNN-NMS &ResNet18&96.21&\textbf{97.98}&2.17&1.86\\
        Polar R-CNN       &ResNet18&96.20&97.94&2.25&1.87\\ 
        \bottomrule
        \end{tabular}
        \end{adjustbox}
        \label{tusimple result}
\end{table}


\begin{table}[h]
        \centering
        \caption{Comparision results on LLAMAS test set with other methods.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{lrcccc}
        \toprule
        \textbf{Method}& \textbf{Backbone}&\textbf{F1@50(\%)}&\textbf{Precision(\%)}&\textbf{Recall(\%)} \\
        \midrule
        SCNN\cite{scnn}          &ResNet34&94.25&94.11&94.39\\
        BézierLaneNet\cite{bezierlanenet} &ResNet34&95.17&95.89&94.46\\
        LaneATT\cite{laneatt}       &ResNet34&93.74&96.79&90.88\\
        LaneAF\cite{laneaf}        &DLA34   &96.07&\textbf{96.91}&95.26\\
        DALNet\cite{dalnet}        &ResNet18&96.12&96.83&95.42\\
        CLRNet\cite{clrnet}       &DLA34   &96.12&-    &-    \\
        \midrule
        
        Polar R-CNN-NMS          &ResNet18&96.05&96.80&95.32\\
        Polar R-CNN                &ResNet18&96.06&96.81&95.32\\
        Polar R-CNN-NMS          &DLA34&96.13&96.80&\textbf{95.47}\\
        Polar R-CNN                &DLA34&\textbf{96.14}&96.82&\textbf{95.47}\\

        \bottomrule
        \end{tabular}
        \end{adjustbox}
        \label{llamas result}
\end{table}

    \begin{table}[h]
        \centering
        \caption{Comparision results on DL-Rail test set with other methods.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{lrccc}
        \toprule
        \textbf{Method}& \textbf{Backbone}&\textbf{mF1(\%)}&\textbf{F1@50(\%)}&\textbf{F1@75(\%)} \\
        \midrule
        BézierLaneNet\cite{bezierlanenet} &ResNet18&42.81&85.13&38.62\\
        GANet-S\cite{ganet}       &Resnet18&57.64&95.68&62.01\\
        CondLaneNet\cite{CondLaneNet}   &Resnet18&52.37&95.10&53.10\\
        UFLDv1\cite{ufld}       &ResNet34&53.76&94.78&57.15\\
        LaneATT(with RPN)\cite{dalnet} &ResNet18&55.57&93.82&58.97\\
        DALNet\cite{dalnet}      &ResNet18&59.79&96.43&65.48\\
        \midrule
        Polar R-CNN-NMS &ResNet18&\textbf{61.53}&\textbf{97.01}&\textbf{67.86}\\
        Polar R-CNN       &ResNet18&61.52&96.99&67.85\\
        \bottomrule
        \end{tabular}
        \end{adjustbox}
        \label{dlrail result}
    \end{table}


\begin{table}[h]
        \centering
        \caption{Comparision results on CurveLanes validation set with other methods.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{lrcccc}
        \toprule
        \textbf{Method}& \textbf{Backbone}&\textbf{F1@50 (\%)}&\textbf{Precision (\%)}&\textbf{Recall (\%)} \\
        \midrule
        SCNN\cite{scnn}          &VGG16    &65.02&76.13&56.74\\
        Enet-SAD\cite{enetsad}      &-        &50.31&63.60&41.60\\
        PointLanenet\cite{pointlanenet}  &ResNet101&78.47&86.33&72.91\\
        CurveLane-S\cite{curvelanes}   &-        &81.12&93.58&71.59\\
        CurveLane-M\cite{curvelanes}   &-        &81.80&93.49&72.71\\
        CurveLane-L\cite{curvelanes}   &-        &82.29&91.11&75.03\\
        UFLDv2\cite{ufldv2}        &ResNet34 &81.34&81.93&80.76\\
        CondLaneNet-M\cite{CondLaneNet} &ResNet34 &85.92&88.29&83.68\\
        CondLaneNet-L\cite{CondLaneNet} &ResNet101&86.10&88.98&83.41\\
        CLRNet\cite{clrnet}        &DLA34    &86.10&91.40&81.39\\
        CLRerNet\cite{clrernet}      &DLA34    &86.47&91.66&81.83\\
        \hline
        Polar R-CNN &DLA34&\textbf{87.29}&90.50&\textbf{84.31}\\
        \hline
        \end{tabular}
        \end{adjustbox}
        \label{curvelanes result}
\end{table}

\subsection{Comparison with the state-of-the-art method}
The comparison results of our proposed model with other methods are shown in Tables \ref{culane result}, \ref{tusimple result}, \ref{llamas result}, \ref{dlrail result}, and \ref{curvelanes result}. We present results for two versions of our model: the NMS-based version, denoted as Polar R-CNN-NMS, and the NMS-free version, denoted as Polar R-CNN. The NMS-based version utilizes predictions obtained from the O2M head followed by NMS post-processing, while the NMS-free version derives predictions directly from the O2O classification head without NMS.

To ensure a fair comparison, we also include results for CLRerNet \cite{clrernet} on the CULane and CurveLanes datasets, as we use a similar training strategy and data split. As illustrated in the comparison results, our model demonstrates competitive performance across five datasets. Specifically, on the CULane, TuSimple, LLAMAS, and DL-Rail datasets (sparse scenarios), our model outperforms other anchor-based methods. Additionally, the performance of the NMS-free version is nearly identical to that of the NMS-based version, highlighting the effectiveness of the O2O head in eliminating redundant predictions. On the CurveLanes dataset, the NMS-free version achieves superior F1-measure and Recall compared to both NMS-based and segment\&grid-based methods.

We also compare the number of anchors and processing speed with other methods. Fig. \ref{anchor_num_method} illustrates the number of anchors used by several anchor-based methods on CULane. Our proposed model utilizes the fewest proposal anchors (20 anchors) while achieving the highest F1-score on CULane. It remains competitive with state-of-the-art methods like CLRerNet, which uses 192 anchors and a cross-layer refinement strategy. Conversely, the sparse Laneformer, which also uses 20 anchors, does not achieve optimal performance. It is important to note that our model is designed with a simpler structure without additional refinement, indicating that the design of flexible anchors is crucial for performance in sparse scenarios. Furthermore, due to its simple structure and fewer anchors, our model exhibits lower latency compared to most methods, as shown in Fig. \ref{speed_method}. The combination of fast processing speed and a straightforward architecture makes our model highly deployable.

\subsection{Ablation Study and Visualization}
To validate and analyze the effectiveness and influence of different component of Polar R-CNN, we conduct serveral ablation expeoriments on CULane and CurveLanes dataset to show the performance.

\textbf{Ablation study on polar coordinate system and anchor number.} To assess the importance of local polar coordinates of anchors, we examine the contribution of each component (i.e., angle and radius) to model performance. As shown in Table \ref{aba_lph}, both angle and radius contribute to performance to varying degrees. Additionally, we conduct experiments with auxiliary loss using fixed anchors and Polar R-CNN. Fixed anchors refer to using anchor settings trained by CLRNet, as illustrated in Fig. \ref{anchor setting}(b). Model performance improves by 0.48% and 0.3% under the fixed anchor paradigm and proposal anchor paradigm, respectively.

We also explore the effect of different local polar map sizes on our model, as illustrated in Fig. \ref{anchor_num_testing}. The overall F1 measure improves with increasing the local polar map size and tends to stabilize when the size is sufficiently large. Specifically, precision improves, while recall decreases. A larger polar map size includes more background anchors in the second stage (since we choose k=4 for SimOTA, with no more than four positive samples for each ground truth). Consequently, the model learns more negative samples, enhancing precision but reducing recall. Regarding the number of anchors chosen during the evaluation stage, recall and F1 measure show a significant increase in the early stages of anchor number expansion but stabilize in later stages. This suggests that eliminating some anchors does not significantly affect performance. Fig. \ref{cam} displays the heat map and top-$K_{a}$ selected anchors’ distribution in sparse scenarios. Brighter colors indicate a higher likelihood of anchors being foreground anchors. It is evident that most of the proposed anchors are clustered around the lane ground truth.

\begin{figure}[t]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/anchor_num_method.png}
        \caption{Anchor numbers vs F1@50 of different methods on CULane lane detection benchmark.}
        \label{anchor_num_method}
\end{figure}


\begin{figure}[t]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/speed_method.png}
        \caption{Latency vs F1@50 of different methods on CULane lane detection benchmark.}
        \label{speed_method}
\end{figure}

\begin{table}[h]
        \centering
        \caption{Ablation study of anchor proposal strategies}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{c|ccc|cc}
        \toprule
        \textbf{Anchor strategy}&\textbf{Local R}& \textbf{Local Angle}&\textbf{Auxloss}&\textbf{F1@50 (\%)}&\textbf{F1@75 (\%)}\\
        \midrule
        \multirow{2}*{Fixed}
                &-         &-         &          &79.90         &60.98\\
                &-         &-         &\checkmark&80.38         &62.35\\
        \midrule
        \multirow{5}*{Porposal}
                &          &          &          &75.85         &58.97\\
                &\checkmark&          &          &78.46         &60.32\\
                &          &\checkmark&          &80.31         &62.13\\
                &\checkmark&\checkmark&          &80.51         &63.38\\
                &\checkmark&\checkmark&\checkmark&\textbf{80.81}&\textbf{63.97}\\
        \bottomrule
        \end{tabular}
        \end{adjustbox}
        \label{aba_lph}
\end{table}


\begin{figure*}[t]
        \centering
        \def\subwidth{0.325\textwidth}
        \def\imgwidth{\linewidth}
        
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth]{thesis_figure/anchor_num/anchor_num_testing_p.png}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth]{thesis_figure/anchor_num/anchor_num_testing_r.png}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth]{thesis_figure/anchor_num/anchor_num_testing.png}
        \end{subfigure}
        \caption{F1@50 preformance of different polar map sizes and different top-$K_{a}$ anchor selections on CULane test set.}
        \label{anchor_num_testing}
\end{figure*}



\begin{figure}[t]
        \centering
        \def\subwidth{0.24\textwidth}
        \def\imgwidth{\linewidth}
        \def\imgheight{0.4\linewidth}
        
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/heatmap/cam1.jpg}
                \caption{}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/heatmap/anchor1.jpg}
                \caption{}
        \end{subfigure}

        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/heatmap/cam2.jpg}
                \caption{}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/heatmap/anchor2.jpg}
                \caption{}
        \end{subfigure}
        \caption{(a)\&(c): The heap map of the local polar map; (b)\&(d): The final anchor selection during the evaluation stage.}
        \label{cam}
\end{figure}

\textbf{Ablation study on NMS-free block in sparse scenarios.} We conduct several experiments on the CULane dataset to evaluate the performance of the NMS-free head in sparse scenarios. As shown in Table \ref{aba_NMSfree_block}, without using the GNN to establish relationships between anchors, Polar R-CNN fails to achieve a NMS-free paradigm, even with one-to-one assignment. Furthermore, the classification matrix (cls matrix) proves crucial, indicating that conditional probability is effective. Other components, such as the neighbor matrix (provided as a geometric prior) and rank loss, also contribute to the performance of the NMS-free block.

To compare the NMS-free paradigm with the traditional NMS paradigm, we perform experiments with the NMS-free block under both proposal and fixed anchor strategies. Table \ref{NMS vs NMS-free} presents the results of these experiments. Here, O2M-B refers to the O2M classification head, O2O-B refers to the O2O classification head with a plain structure, and O2O-G refers to the O2O classification head with Polar GNN block. To assess the ability to eliminate redundant predictions, NMS post-processing is applied to each head. The results show that NMS is necessary for the traditional O2M classification head. In the fixed anchor paradigm, although the O2O classification head with a plain structure effectively eliminates redundant predictions, it is less effective than the proposed Polar GNN block. In the proposal anchor paradigm, the O2O classification head with a plain structure fails to eliminate redundant predictions due to high anchor overlap and similar RoI features. Thus, the GNN structure is essential for Polar R-CNN in the NMS-free paradigm. Both in the fixed and proposal anchor paradigms, the O2O classification head with the GNN structure successfully eliminates redundant predictions, indicating that our GNN-based O2O classification head can replace NMS post-processing in sparse scenarios without a decrease in performance. This confirms our earlier theory that both structure and label assignment are crucial for a NMS-free paradigm.

We also explore the stop-gradient strategy for the O2O classification head. As shown in Table \ref{stop}, the gradient of the O2O classification head negatively impacts both the O2M classification head (with NMS post-processing) and the O2O classification head. This suggests that one-to-one assignment introduces critical bias into feature learning.

\begin{table}[h]
        \centering
        \caption{Ablation study on Polar GNN block.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{cccc|ccc}
        \toprule
        \textbf{GNN}&\textbf{cls Mat}& \textbf{Nbr Mat}&\textbf{Rank Loss}&\textbf{F1@50 (\%)}&\textbf{Precision (\%)} & \textbf{Recall (\%)} \\
        \midrule
                  &          &          &          &16.19&69.05&9.17\\
        \checkmark&\checkmark&          &          &79.42&88.46&72.06\\
        \checkmark&          &\checkmark&          &71.97&73.13&70.84\\
        \checkmark&\checkmark&\checkmark&          &80.74&88.49&74.23\\
        \checkmark&\checkmark&\checkmark&\checkmark&\textbf{80.78}&\textbf{88.49}&\textbf{74.30}\\
        \bottomrule
        \end{tabular}\
        \end{adjustbox}
        \label{aba_NMSfree_block}
\end{table}






\begin{table}[h]
        \centering
        \caption{The ablation study for NMS and NMS-free on CULane test set.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{c|l|lll}
        \toprule
        \multicolumn{2}{c|}{\textbf{Anchor strategy~/~assign}} & \textbf{F1@50 (\%)} & \textbf{Precision (\%)} & \textbf{Recall (\%)} \\
        \midrule
        \multirow{6}*{Fixed}     
                                &O2M-B w/~ NMS &80.38&87.44&74.38\\
                                &O2M-B w/o NMS &44.03\textcolor{darkgreen}{~(36.35$\downarrow$)}&31.12\textcolor{darkgreen}{~(56.32$\downarrow$)}&75.23\textcolor{red}{~(0.85$\uparrow$)}\\
                                \cline{2-5}
                                &O2O-B w/~ NMS &78.72&87.58&71.50\\
                                &O2O-B w/o NMS &78.23\textcolor{darkgreen}{~(0.49$\downarrow$)}&86.26\textcolor{darkgreen}{~(1.32$\downarrow$)}&71.57\textcolor{red}{~(0.07$\uparrow$)}\\
                                \cline{2-5}
                                &O2O-G w/~ NMS &80.37&87.44&74.37\\
                                &O2O-G w/o NMS &80.27\textcolor{darkgreen}{~(0.10$\downarrow$)}&87.14\textcolor{darkgreen}{~(0.30$\downarrow$)}&74.40\textcolor{red}{~(0.03$\uparrow$)}\\
        \midrule
        \multirow{6}*{Proposal}     
                                &O2M-B w/~ NMS &80.81&88.53&74.33\\
                                &O2M-B w/o NMS &36.46\textcolor{darkgreen}{~(44.35$\downarrow$)}&24.09\textcolor{darkgreen}{~(64.44$\downarrow$)}&74.93\textcolor{red}{~(0.6$\uparrow$)}\\
                                \cline{2-5}
                                &O2O-B w/~ NMS &77.27&92.64&66.28\\
                                &O2O-B w/o NMS &47.11\textcolor{darkgreen}{~(30.16$\downarrow$)}&36.48\textcolor{darkgreen}{~(56.16$\downarrow$)}&66.48\textcolor{red}{~(0.20$\uparrow$)}\\
                                \cline{2-5}
                                &O2O-G w/~ NMS &80.81&88.53&74.32\\
                                &O2O-G w/o NMS &80.81\textcolor{red}{~(0.00$\uparrow$)}&88.52\textcolor{darkgreen}{~(0.01$\downarrow$)}&74.33\textcolor{red}{~(0.01$\uparrow$)}\\
        \bottomrule
        \end{tabular}
        \end{adjustbox}
        \label{NMS vs NMS-free}
\end{table}



    

\begin{table}[h]
        \centering
        \caption{The ablation study for the stop grad strategy on CULane test set.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{c|c|lll}
        \toprule
        \multicolumn{2}{c|}{\textbf{Paradigm}} & \textbf{F1 (\%)} & \textbf{Precision (\%)} & \textbf{Recall (\%)} \\
        \midrule
        \multirow{2}*{Baseline}  
                                 &O2M-B w/~ NMS &78.83&88.99&70.75\\
                                 &O2O-G w/o NMS &71.68\textcolor{darkgreen}{~(7.15$\downarrow$)}&72.56\textcolor{darkgreen}{~(16.43$\downarrow$)}&70.81\textcolor{red}{~(0.06$\uparrow$)}\\
        \midrule
        \multirow{2}*{Stop Grad} 
                                 &O2M-B w/~ NMS &80.81&88.53&74.33\\
                                 &O2O-G w/o NMS &80.81\textcolor{red}{~(0.00$\uparrow$)}&88.52\textcolor{darkgreen}{~(0.01$\downarrow$)}&74.33\textcolor{red}{~(0.00$\uparrow$)} \\
        \bottomrule
        \end{tabular}
        \end{adjustbox}
        \label{stop}
\end{table}




\textbf{Ablation study on NMS-free block in dense scenarios.} Despite demonstrating the feasibility of replacing NMS with the O2O classification head in sparse scenarios, the shortcomings of NMS in dense scenarios remain. To investigate the performance of the NMS-free block in dense scenarios, we conduct experiments on the CurveLanes dataset, as detailed in Table \ref{aba_NMS_dense}.

In the traditional NMS post-processing \cite{clrernet}, the default IoU threshold is set to 50 pixels. However, this default setting may not always be optimal, especially in dense scenarios where some lane predictions might be erroneously eliminated. Lowering the IoU threshold increases recall but decreases precision. To find the most effective IoU threshold, we experimented with various values and found that a threshold of 15 pixels achieves the best trade-off, resulting in an F1-score of 86.81\%. In contrast, the NMS-free paradigm with the GNN-based O2O classification head achieves an overall F1-score of 87.29\%, which is 0.48\% higher than the optimal threshold setting in the NMS paradigm. Additionally, both precision and recall are improved under the NMS-free approach. This indicates that the GNN-based O2O classification head is capable of learning both explicit geometric distance and implicit semantic distances between anchors in addition to geometric distances, thus providing a more effective solution for dense scenarios compared to the traditional NMS post-processing.

\begin{table}[h]
        \centering
        \caption{NMS vs NMS-free on CurveLanes validation set.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{l|l|ccc}
        \toprule
        \textbf{Paradigm} & \textbf{NMS thres(pixel)} & \textbf{F1@50(\%)} & \textbf{Precision(\%)} & \textbf{Recall(\%)} \\
        \midrule
        \multirow{7}*{Polar R-CNN-NMS} 
                                & 50 (default) &85.38&\textbf{91.01}&80.40\\
                                & 40           &85.97&90.72&81.68\\
                                & 30           &86.26&90.44&82.45\\
                                & 25           &86.38&90.27&82.83\\
                                & 20           &86.57&90.05&83.37\\
                                & 15 (optimal) &86.81&89.64&84.16\\
                                & 10           &86.58&88.62&\textbf{84.64}\\
        \midrule
        Polar R-CNN & - &\textbf{87.29}&90.50&84.31\\
        \bottomrule
        \end{tabular}
        \end{adjustbox}
        \label{aba_NMS_dense}
    \end{table}

\textbf{Visualization.}  We present the Polar R-CNN predictions for both sparse and dense scenarios. Fig. \ref{vis_sparse} displays the predictions for sparse scenarios across four datasets. LPH effectively proposes anchors that are clustered around the ground truth, providing a robust prior for the RoI stage to achieve the final lane predictions. Moreover, the number of anchors has significantly decreased compared to previous works, making our method faster than other anchor-based methods in theory. Fig. \ref{vis_dense} shows the predictions for dense scenarios. We observe that NMS@50 mistakenly removes some predictions, leading to false negatives, while NMS@15 fails to eliminate redundant predictions, resulting in false positives. This highlights the trade-off between using a large IoU threshold and a small IoU threshold. The visualization clearly demonstrates that geometric distance becomes less effective in dense scenarios. Only the O2O classification head, driven by data, can address this issue by capturing semantic distance beyond geometric distance. As shown in Fig. \ref{vis_dense}, the O2O classification head successfully eliminates redundant true predictions while retaining dense predictions with small geometric distances.




    \begin{figure*}[htbp]
        \centering
        \def\pagewidth{0.49\textwidth}
        \def\subwidth{0.47\linewidth}
        \def\imgwidth{\linewidth}
        \def\imgheight{0.5625\linewidth}
        \def\dashheight{0.8\linewidth}

        \begin{subfigure}{\pagewidth}
            \rotatebox{90}{\small{GT}}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/culane/1_gt.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/culane/2_gt.jpg}
            \end{minipage}
        \end{subfigure}
        \begin{subfigure}{\pagewidth}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/tusimple/1_gt.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/tusimple/2_gt.jpg}
            \end{minipage}
        \end{subfigure}
        \vspace{0.5em}

        
        \begin{subfigure}{\pagewidth}
            \raisebox{-1.5em}{\rotatebox{90}{\small{Anchors}}}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/culane/1_anchor.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/culane/2_anchor.jpg}
            \end{minipage}
        \end{subfigure}
        \begin{subfigure}{\pagewidth}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/tusimple/1_anchor.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/tusimple/2_anchor.jpg}
            \end{minipage}
        \end{subfigure}
        \vspace{0.5em}
        
        \begin{subfigure}{\pagewidth}
            \raisebox{-2em}{\rotatebox{90}{\small{Predictions}}}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/culane/1_pred.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/culane/2_pred.jpg}
            \end{minipage}
            \caption{CULane}
        \end{subfigure}
        \begin{subfigure}{\pagewidth}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/tusimple/1_pred.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/tusimple/2_pred.jpg}
            \end{minipage}
            \caption{TuSimple}
        \end{subfigure}
        \vspace{0.5em}

        % \begin{tikzpicture}
        %         \draw[dashed, pattern=on 8pt off 2pt, color=gray, line width=1pt] (-\textwidth/2,0) -- (\textwidth/2.,0);
        % \end{tikzpicture}
        % \vspace{0.05em}
        
        \begin{subfigure}{\pagewidth}
            \rotatebox{90}{\small{GT}}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/llamas/1_gt.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/llamas/2_gt.jpg}
            \end{minipage}
        \end{subfigure}
        \begin{subfigure}{\pagewidth}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/dlrail/1_gt.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/dlrail/2_gt.jpg}
            \end{minipage}
        \end{subfigure}
        \vspace{0.5em}
        
        \begin{subfigure}{\pagewidth}
            \raisebox{-1.5em}{\rotatebox{90}{\small{Anchors}}}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/llamas/1_anchor.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/llamas/2_anchor.jpg}
            \end{minipage}
        \end{subfigure}
        \begin{subfigure}{\pagewidth}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/dlrail/1_anchor.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/dlrail/2_anchor.jpg}
            \end{minipage}
        \end{subfigure}
        \vspace{0.5em}

        \begin{subfigure}{\pagewidth}
            \raisebox{-2em}{\rotatebox{90}{\small{Predictions}}}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/llamas/1_pred.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/llamas/2_pred.jpg}
            \end{minipage}
            \caption{LLAMAS}
        \end{subfigure}
        \begin{subfigure}{\pagewidth}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/dlrail/1_pred.jpg}
            \end{minipage}
            \begin{minipage}{\subwidth}
                    \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_dataset/dlrail/2_pred.jpg}
            \end{minipage}
            \caption{DL-Rail}
        \end{subfigure}
        \vspace{0.5em}
        
        \caption{The visualization of the detection results of sparse scenarios.}
        \label{vis_sparse}
\end{figure*}



\begin{figure*}[htbp!]
        \centering
        \def\subwidth{0.24\textwidth}
        \def\imgwidth{\linewidth}
        \def\imgheight{0.5625\linewidth}

        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/redun_gt.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/redun_pred50.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/redun_pred15.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/redun_NMSfree.jpg}
        \end{subfigure}
        \vspace{0.5em}

        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/redun2_gt.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/redun2_pred50.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/redun2_pred15.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/redun2_NMSfree.jpg}
        \end{subfigure}
        \vspace{0.5em}


        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/less_gt.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/less_pred50.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/less_pred15.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/less_NMSfree.jpg}
        \end{subfigure}
        \vspace{0.5em}

        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/less2_gt.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/less2_pred50.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/less2_pred15.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/less2_NMSfree.jpg}
        \end{subfigure}
        \vspace{0.5em}

        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/all_gt.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/all_pred50.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/all_pred15.jpg}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/all_NMSfree.jpg}
        \end{subfigure}
        \vspace{0.5em}

        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/all2_gt.jpg}
                \caption{GT}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/all2_pred50.jpg}
                \caption{NMS@50}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/all2_pred15.jpg}
                \caption{NMS@15}
        \end{subfigure}
        \begin{subfigure}{\subwidth}
                \includegraphics[width=\imgwidth, height=\imgheight]{thesis_figure/view_nms/all2_NMSfree.jpg}
                \caption{NMSFree}
        \end{subfigure}
        \vspace{0.5em}

        \caption{The visualization of the detection results of sparse scenarios.}
        \label{vis_dense}
\end{figure*}


\section{Conclusion and Future Work}
In this paper, we propose Polar R-CNN to address two key issues in anchor-based lane detection methods. By incorporating a local and global polar coordinate system, our Polar R-CNN achieves improved performance with fewer anchors. Additionally, the introduction of the O2O classification head with Polar GNN block allows us to replace the traditional NMS post-processing, and the NMS-free paradigm demonstrates superior performance in dense scenarios. Our model is highly flexible and the number of anchors can be adjusted based on the specific scenario. Users have the option to use either the O2M classification head with NMS post-processing or the O2O classification head for a NMS-free approach. Polar R-CNN is also deployment-friendly due to its simple structure, making it a potential new baseline for lane detection. Future work could explore incorporating new structures, such as large kernels or attention mechanisms, and experimenting with new label assignment, training, and anchor sampling strategies. We also plan to extend Polar R-CNN to video instance lane detection and 3D lane detection, utilizing advanced geometric modeling for these new tasks.
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\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{thesis_figure/wsq.jpg}}]{Shengqi Wang}
        received the Master degree from Xi'an Jiaotong University, Xi'an, China, in 2022. He is now pursuing for the Ph.D. degree in statistics at Xi'an Jiaotong University. His research interests include low-level computer vision, deep learning, and so on. 
\end{IEEEbiography}

\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{thesis_figure/ljm.pdf}}]{Junmin Liu}
        was born in 1982. He received the Ph.D. degree in Mathematics from Xi'an Jiaotong University, Xi'an, China, in 2013. From 2011 to 2012, he served as a Research Assistant with the Department of Geography and Resource Management at the Chinese University of Hong Kong, Hong Kong, China. From 2014 to 2017, he worked as a Visiting Scholar at the University of Maryland, College Park, USA. He is currently a full Professor at the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China. His research interests are mainly focused on the theory and application of machine learning and image processing. He has published over 60+ research papers in international conferences and journals. 
\end{IEEEbiography}

\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{thesis_figure/xiangyongcao.jpg}}]{Xiangyong Cao (Member, IEEE)}
        received the B.Sc. and Ph.D. degrees from Xi’an Jiaotong University, Xi’an, China, in 2012 and 2018, respectively. From 2016 to 2017, he was a Visiting Scholar with Columbia University, New York, NY, USA. He is an Associate Professor with the School of Computer Science and Technology, Xi’an Jiaotong University. His research interests include statistical modeling
        and image processing.
\end{IEEEbiography}
\vfill
\end{document}


