\documentclass[lettersize,journal]{IEEEtran}
\usepackage{amsmath,amsfonts}
\usepackage{algorithmic}
\usepackage{algorithm}
\usepackage{array}
% \usepackage[caption=false,font=normalsize,labelfont=sf,textfont=sf]{subfig}
\usepackage{textcomp}
\usepackage{stfloats}
\usepackage{url}
\usepackage{verbatim}
\usepackage{graphicx}
\usepackage{cite}
\usepackage{subcaption}
\usepackage{multirow}
\usepackage[T1]{fontenc}
\usepackage{adjustbox}
\usepackage{amssymb}
\usepackage{booktabs}
\usepackage{tikz}
\usepackage{tabularx}
\usepackage{mathrsfs}
\usepackage[colorlinks,bookmarksopen,bookmarksnumbered, linkcolor=red]{hyperref}
% \usepackage[table,xcdraw]{xcolor}

\definecolor{darkgreen}{RGB}{17,159,27} % 
\aboverulesep=0pt
\belowrulesep=0pt
\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.}}

\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}

\maketitle

\begin{abstract}
    Lane detection is a critical and challenging task in autonomous driving, particularly in real-world scenarios where traffic lanes can be slender, lengthy, and often obscured by other vehicles, complicating detection efforts. Existing anchor-based methods typically rely on prior lane anchors to extract features and  subsequently refine the location and shape of lanes. While these methods achieve high performance, manually setting prior anchors is cumbersome, and ensuring sufficient  coverage across diverse datasets often requires a large amount of dense anchors. Furthermore,
    the use of \textit{Non-Maximum Suppression} (NMS) to eliminate redundant predictions complicates real-world deployment and may underperform in complex scenarios. In this paper, we propose \textit{Polar R-CNN}, a NMS-free anchor-based method for lane detection. By incorporating both local and global polar coordinate systems, Polar R-CNN facilitates flexible anchor proposals and significantly reduces the number of anchors required without compromising performance. Additionally, we introduce a triplet head with heuristic structure that supports nms-free paradigm, enhancing deployment efficiency and performance in scenarios with dense lanes. Our method achieves competitive results on five popular lane detection benchmarks—\textit{Tusimple}, \textit{CULane}, \textit{LLAMAS}, \textit{CurveLanes}, and \textit{DL-Rail}—while maintaining a lightweight design and straightforward structure. Our source code is 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 critical task in computer vision and autonomous driving, aimed at identifying and tracking lane markings on the road \cite{adas}. While extensive research has been conducted in ideal environments, it is still challenging in adverse scenarios such as night driving, glare, crowd, and rainy conditions, where lanes may be occluded or damaged \cite{scnn}. Moreover, the slender shapes and complex topologies of lanes further complicate detection efforts \cite{polylanenet}. %Therefore, 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 performances in a real-time application. along with their global properties,
\par
In the past few decades, a lot of methods primarily focus on handcrafted local feature extraction and lane shape modeling. Techniques such as the \textit{Canny edge detector}\cite{cannyedge},\textit{ Hough transform}\cite{houghtransform}, and \textit{deformable templates}\cite{kluge1995deformable} have been widely employed for lane fitting. However, these approaches often face limitations in real-world scenarios, especially when low-level and local features lack clarity and distinctiveness.
\par
In recent years, advancements in deep learning and the availability of large datasets have led to significant progress in lane detection, especially deep models such as \textit{Convolutional Neural Networks} (CNNs)\cite{scnn} and \textit{transformer-based} architectures \cite{lstr}. Based on this, earlier approaches typically framed lane detection as a \textit{segmentation task} \cite{lanenet}, which, despite its straightforward, required time-consuming computations. There are still some methods that rely on \textit{parameter-based} models, which directly output lane curve parameters rather than pixel locations \cite{polylanenet}\cite{lstr}\cite{bezierlanenet}. Although these segmentation-based and parameter-based methods provide end-to-end solutions, their sensitivity to lane shape compromises their 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 (\textit{i.e.}, the yellow lines) settings of different methods and the ground truth lanes. (a) The initial anchor settings of CLRNet. (b) The learned anchor settings of CLRNet trained on CULane. (c) The flexible proposal 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/redun_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_pred.jpg}
                \caption{}
        \end{subfigure}

        \caption{Comparison of NMS thresholds in \textit{sparse} and \textit{dense} scenarios. (a) and (b) Ground truths in a dense and sparse scenarios, respectively. (c) Predictions with large NMS thresholds in a dense scenario, resulting in a lane prediction being mistakenly suppressed. (d) Predictions with small NMS thresholds in a sparse scenario, where redundant prediction results are not effectively removed.}
        \label{NMS setting}
\end{figure}
%, where some lane instances are close with each others; , where the lane instance are far apart
\par
Drawing inspiration from object detection methods such as \textit{YOLO} \cite{yolov10} and \textit{Faster R-CNN} \cite{fasterrcnn}, several anchor-based approaches have been introduced for lane detection, with representative works including \textit{LaneATT} \cite{laneatt} and \textit{CLRNet} \cite{clrnet}. These methods have shown superior performance by leveraging anchor \textit{priors} (as shown in Fig. \ref{anchor setting}) and enabling larger receptive fields for feature extraction. However, anchor-based methods encounter similar drawbacks to those in general object detection, including the following:
\begin{itemize}
\item As shown in Fig. \ref{anchor setting}(a), a large amount of lane anchors are predefined in the image, even in \textbf{\textit{sparse scenarios}}---the situations where lanes are distributed widely and located far apart from each other, as illustrated in the Fig. \ref{anchor setting}(d).
\item A \textit{Non-Maximum Suppression} (NMS) \cite{nms} post-processing step is required to eliminate redundant predictions but may struggle in \textbf{\textit{dense scenarios}} where lanes are close to each other, such as forked lanes and double lanes, as illustrated in the Fig. \ref{NMS setting}(a).
\end{itemize}
\par
Regrading the first issue, \cite{clrnet} introduced learned anchors that optimize the anchor parameters during training to better adapt to lane distributions, as shown in Fig. \ref{anchor setting}(b). However, the number of anchors remains excessive to adequately cover the diverse potential distributions of lanes. Furthermore, \cite{adnet} proposes flexible anchors for each image by generating start points with directions, rather than using a fixed set of anchors. Nevertheless, these start points of lanes are subjective and lack clear visual evidence due to the global nature of lanes. In contrast, \cite{srlane} uses a local angle map to propose sketch anchors according to the direction of ground truth. While this approach considers directional alignment, it neglects precise anchor positioning, resulting in suboptimal performance. Overall, the abundance of anchors is unnecessary in sparse scenarios.% where lane ground truths are sparse. The trend in new methodologies is to reduce the number of anchors while offering more flexible anchor configurations.%, which negatively impacts its performance. They also employ cascade cross-layer anchor refinement to bring the anchors closer to the ground truth.  in the absence of cascade anchor refinement
\par
Regarding the second issue, nearly all anchor-based methods \cite{laneatt}\cite{clrnet}\cite{adnet}\cite{srlane} rely on direct or indirect NMS post-processing to eliminate redundant predictions. Although it is necessary to eliminate redundant predictions, NMS remains a suboptimal solution. On one hand, NMS is not deployment-friendly because it requires defining and calculating distances between lane pairs using metrics such as \textit{Intersection over Union} (IoU). This task is more challenging than in general object detection due to the intricate geometry of lanes. On the other hand, NMS can struggle in dense scenarios. Typically, a large distance threshold may lead to false negatives, as some true positive predictions could be mistakenly eliminated, as illustrated in Fig. \ref{NMS setting}(a)(c). Conversely, a small distance threshold may fail to eliminate redundant predictions effectively, resulting in false positives, as shown in Fig. \ref{NMS setting}(b)(d). Therefore, achieving an optimal trade-off across all scenarios by manually setting the distance threshold is challenging. %The root of this problem lies in the fact that the distance definition in NMS considers only geometric parameters while ignoring the semantic context in the image. As a result, when two predictions are ``close'' to each other, it is nearly impossible to determine whether one of them is redundant.% where lane ground truths are closer together than in sparse scenarios;including those mentioned above,
\par
To address the above two issues, we propose Polar R-CNN, a novel anchor-based method for lane detection. For the first issue, we introduce \textit{Local Polar Module} based on the polar coordinate system to create anchors with more accurate locations, thereby reducing the number of proposed anchors in sparse scenarios, as illustrated in Fig. \ref{anchor setting}(c). In contrast to \textit{State-Of-The-Art} (SOTA) methods \cite{clrnet}\cite{clrernet}, which utilize 192 anchors, Polar R-CNN employs only 20 anchors to effectively cover potential lane ground truths. For the second issue, we have incorporated a triplet head with a new heuristic \textit{Graph Neural Network} (GNN) \cite{gnn} block. The 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: \textit{TuSimple} \cite{tusimple}, \textit{CULane} \cite{scnn}, \textit{LLAMAS} \cite{llamas}, \textit{CurveLanes} \cite{curvelanes}, and \textit{DL-Rail} \cite{dalnet}. Our proposed method demonstrates competitive performance compared to SOTA approaches. Our main contributions are summarized as follows:
\begin{itemize}
\item We design a strategy to simplify the anchor parameters by using local and global polar coordinate systems and applied these to the two-stage lane detection framework. Compared to other anchor-based methods, this strategy significantly reduces the number of proposed anchors while achieving better performance.
\item We propose a novel triplet detection head with GNN block to implement a NMS-free paradigm. The block is inspired by Fast NMS, providing enhanced interpretability. Our model supports end-to-end training and testing while still allowing for traditional NMS post-processing as an option for a NMS version of our model.
\item By integrating the polar coordinate systems and NMS-free paradigm, we present a Polar R-CNN model for fast and efficient lane detection. And we conduct extensive experiments on five benchmark datasets to demonstrate the effectiveness of our model in high performance with fewer anchors and a NMS-free paradigm. %Additionally, our model features a straightforward structure—lacking cascade refinement or attention strategies—making it simpler to deploy. 
\end{itemize}
%
\begin{figure*}[ht]
	\centering
	\includegraphics[width=0.99\linewidth]{thesis_figure/ovarall_architecture.png} 
	\caption{An illustration of the Polar R-CNN architecture. It has a similar pipeline with the Faster R-CNN for the task of object detection, and consists of a backbone, a \textit{Feature Pyramid Network} with three levels of feature maps, respectively denote by $P_1, P_2, P_3$, followed by a \textit{Local Polar Module}, and a \textit{Global Polar Module} for lane detection. Based on the designed lane representation and lane anchor representation in polar coordinate system, the local polar module can propose sparse line anchors and the global polar module can produce the final accurate lane predictions. The global polar module includes a triplet head, which comprises a \textit{one-to-one (O2O)} classification head, a \textit{one-to-many} (O2M) classification head, and a \textit{one-to-many} (O2M) regression head.}
	\label{overall_architecture}
\end{figure*}
\section{Related Works}
%As mentioned above, our model is based on deep learning.
Generally, deep learning-based lane detection methods can be categorized into three groups: segmentation-based, parameter-based, and anchor-based methods. Additionally, NMS-free is an important technique for anchor-based methods, and it will also be described in this section.
\par
\textbf{Segmentation-based Methods.} These 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, they often overly emphasize low-level and local features, neglecting global semantic information and real-time detection. To address this issue, \textit{SCNN} \cite{scnn} uses a larger receptive field. There are some methods such as \textit{UFLDv1-v2} \cite{ufld}\cite{ufldv2} and \textit{CondLaneNet}\cite{CondLaneNet} by utilizing 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. For example, \textit{LaneNet}\cite{lanenet} uses post-clustering to distinguish each lane instance, while \textit{UFLDv1-v2}
categorizes lane instances by angles and locations, allowing it to detect only a fixed number of lanes. In contrast, \textit{CondLaneNet} employs different conditional dynamic kernels to predict different lane instances. Additionally, some methods such as \textit{FOLOLane}\cite{fololane} and \textit{GANet}\cite{ganet} adopt bottom-up strategies to detect a few key points and model their global relations to form lane instances.
\par
\textbf{Parameter-based Methods.} Instead of predicting a series of points locations or pixel classifications, the parameter-based methods directly generate the curve parameters of lane instances. For example, \textit{PolyLanenet}\cite{polylanenet} and \textit{LSTR}\cite{lstr} consider the lane instance as a polynomial curve, outputting the polynomial coefficients directly. \textit{BézierLaneNet}\cite{bezierlanenet} treats the lane instance as a Bézier curve, generating the locations of their control points, while \textit{BSLane}\cite{bsnet} uses B-Spline to describe the lane, with curve parameters that emphasize local lane shapes. These parameter-based methods are mostly end-to-end and do not require post-processing, resulting in faster inference speed. However, since the final visual lane shapes are sensitive to their shapes, the robustness and generalization of these methods may not be optimal.
\par
\textbf{Anchor-Based Methods.} These methods are inspired by general object detection models, such as YOLO \cite{yolov10} and Faster R-CNN \cite{fasterrcnn}, for lane detection. The earliest work is Line-CNN, which utilizes line anchors 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, rendering it suboptimal for capturing the entirety of the 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 still pose limitations. A key advantage of the anchor-based methods is their flexibility, allowing the integration of strategies from anchor-based object detection. For example, \textit{CLRNet} \cite{clrnet} enhances the performance with \textit{cross-layer refinement strategies}, \textit{SimOTA label assignment} \cite{yolox}, and \textit{LIOU loss}, outperforming many previous methods. They also have some essential drawbacks, \textit{e.g.}, lane anchors are often handcrafted and numerous. Some approaches, such as \textit{ADNet} \cite{adnet}, \textit{SRLane} \cite{srlane}, and \textit{Sparse Laneformer} \cite{sparse}, attempt to reduce the number of anchors and provide more flexible proposals; however, this can slightly impact performance. Additionally, methods such as \cite{adnet}\cite{clrernet} 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 is still less satisfactory compared to NMS-based models.
\par
\textbf{NMS-free Methods.} Due to the threshold sensitivity and computational overhead of NMS, many studies attempt to NMF-free methods or models that do not use NMS during the detection process. For example, \textit{DETR} \cite{detr} employs one-to-one label assignment to avoid redundant predictions without using NMS. Other NMS-free methods \cite{yolov10}\cite{learnNMS}\cite{date} have also been proposed to addressing this issue from two aspects: \textit{model architecture} and \textit{label assignment}. For example, studies in \cite{yolov10}\cite{date} 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. While some works in \cite{o3d} \cite{relationnet} consider the model’s expressive capacity to provide non-redundant predictions. However, compared to the extensive studies conducted in general object detection, there has been limited research analyzing the NMS-free paradigm. 
\par
In this work, we aim to address the above two issues in the framework of anchor-based lane detection to achieve NMF-free and non-redundant lane predictions.
%
%
\section{Polar R-CNN}
\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 (\textit{e.g.} the green point $\left( x_{1}^{s},y_{1}^{s} \right)$ or the yellow point $\left( x_{2}^{s},y_{2}^{s} \right) $) and direction $\theta^{s}$. (b) Polar: defined by its radius $r$ and angle $\theta$.} %rectangular coordinates
	\label{coord}
\end{figure}
%
The overall architecture of our Polar R-CNN is illustrated in Fig. \ref{overall_architecture}. As shown in this figure, our Polar R-CNN for lane detection has a parallel  pipeline with Faster R-CNN \cite{fasterrcnn}, which consists of a backbone\cite{resnet}, a \textit{Feature Pyramid Network} (FPN) \cite{fpn}, a \textit{Local Polar Module} (LPM) as the \textit{Region Proposal Network} (RPN) \cite{fasterrcnn}, and a \textit{Global Polar Module} (GPM) as the \textit{Region of Interest} (RoI) \cite{fasterrcnn} pooling module. In the following, we first introduce the polar coordinate representation of lane anchors, and then present the designed LPM and GPM in our Polar R-CNN.

%
\subsection{Representation of Lane and Lane Anchor}
%
Lanes are characterized by their thin, elongated, and curved shapes. A well-defined lane prior aids the model in feature extraction and location prediction. 
\par
\textbf{Lane and Anchor Representation as Ray.} Given an input image with dimensions of width $W$ and height $H$, a lane is represented by a set of 2D points $X=\{(x_1,y_1),(x_2,y_2),\cdots,(x_N,y_N)\}$ with equally spaced y-coordinates, i.e., $y_i=i\times\frac{H}{N}$, where $N$ is the number of data points. Since the y-coordinate is fixed, a lane can be uniquely defined by its x-coordinates. Previous studies \cite{linecnn}\cite{laneatt} have introduced \textit{lane priors}, also known as \textit{lane anchors}, which are represented as straight lines in the image plane and served as references. From a geometric perspective, a lane anchor can be viewed as a ray defined by a start point $(x^{s},y^{s})$ located at the edge of an image (left/bottom/right boundaries), along with a direction $\theta^s$. The primary task of a lane detection model is to estimate the x-coordinate offset from the lane anchor to the ground truth of the lane instance. 
\par
However, the representation of lane anchors as rays presents certain limitations. Notably, a lane anchor can have an infinite number of potential start points, which makes the definition of its start point ambiguous and subjective. As illustrated in Fig. \ref{coord}(a), the studies in \cite{dalnet}\cite{laneatt}\cite{linecnn} define the start points as being located at the boundaries of an image, such as the green point in Fig. \ref{coord}(a). In contrast, the research presented in \cite{adnet} defines the start points, exemplified by the purple point in Fig. \ref{coord}(a), based on their actual visual locations within the image. Moreover, occlusion and damage to the lane significantly affect the detection of these start points, highlighting the need for the model to have a large receptive field \cite{adnet}. Essentially, a straight lane has two degrees of freedom: the slope and the intercept, under a Cartesian coordinate system, implying that the lane anchor could be described using just two parameters instead of the three redundant parameters (\textit{i.e.}, two for the start point and one for the direction) employed in ray representation.
%
\begin{figure}[t]
	\centering
	\includegraphics[width=0.87\linewidth]{thesis_figure/coord/localpolar.png}
	\caption{The local polar coordinate system. The ground truth of the radius $\hat{r}_{i}^{l}$ of the $i$-th local pole is defines as the minimum distance from the pole to the lane curve instance. A positive pole has a radius $\hat{r}_{i}^{l}$ that is below a threshold $\tau^{l}$, and vice versa. Additionally, the ground truth angle $\hat{\theta}_i$ is determined by the angle formed between the radius vector (connecting the pole to the closest point on the lanes) and the polar axis.}
	\label{lpmlabel}
\end{figure}
\par
\textbf{Representation in Polar Coordinate.} As stated above, lane anchors represented by rays have some drawbacks. To address these issues, we introduce a polar coordinate representation of lane anchors. In mathematics, the polar coordinate is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point (\textit{i.e.}, pole) and an angle $\theta$ from a reference direction (\textit{i.e.}, polar axis). As shown in Fig. \ref{coord}(b), given a polar corresponding to the yellow point, a lane anchor for a straight line can be uniquely defined by two parameters: the radial distance from the pole (\textit{i.e.}, radius), $r$, and the counterclockwise angle from the polar axis to the perpendicular line of the lane anchor, $\theta$, with $r \in \mathbb{R}$ and $\theta\in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. 
\par
To better leverage the local inductive bias properties of CNNs, we define two types of polar coordinate systems: the local and global coordinate systems. The local polar coordinate system is to generate lane anchors, while the global coordinate system expresses these anchors in a form within the entire image and regresses them to the ground truth lane instances. Given the distinct roles of the local and global systems, we adopt a two-stage framework for our Polar R-CNN, similar to Faster R-CNN\cite{fasterrcnn}.  
\par
The local polar system is designed to predict lane anchors adaptable to both sparse and dense scenarios. In this system, there are many poles with each as the lattice point of the feature map, referred to as local poles. As illustrated on the left side of Fig. \ref{lpmlabel}, there are two types of local poles: positive and negative. Positive local poles (\textit{e.g.}, the blue points) have a radius $r_{i}^{l}$ below a threshold $\lambda^l$, otherwise, they are classified as negative local poles (\textit{e.g.}, the red points). Each local pole is responsible for predicting a single lane anchor. While a lane ground truth may generate multiple lane anchors, as shown in Fig. \ref{lpmlabel}, there are three positive poles around the lane instance (green lane), which are expected to generate three lane anchors. 

%This one-to-many approach is essential for ensuring comprehensive anchor proposals, especially since some local features around certain poles may be lost due to damage or occlusion of the lane curve. 
\par
In the local polar coordinate system, the parameters of each lane anchor are determined based on the location of its corresponding local pole. However, in practical terms, once a lane anchor is generated, its definitive position becomes immutable and independent of its original local pole. To simplify the representation of lane anchors in the second stage of Polar-RCNN, a global polar system has been designed, featuring a singular and unified pole that serves as a reference point for the entire image. The location of this global pole is manually set, and in this case, it is positioned near the static vanishing point observed across the entire lane image dataset. This approach ensures a consistent and unified polar coordinate for expressing lane anchors within the global context of the image, facilitating accurate regression to the ground truth lane instances.

\begin{figure}[t]
        \centering
        \includegraphics[width=0.45\textwidth]{thesis_figure/local_polar_head.png}
        \caption{An illustration of the structure of LPM.}
        \label{lpm}
\end{figure}
\subsection{Local Polar Module}
As shown in Fig. \ref{overall_architecture}, three levels of feature maps, denoted as $\boldsymbol{P}_1, \boldsymbol{P}_2, \boldsymbol{P}_3$, are extracted using a \textit{Feature Pyramid Network} (FPN). To generate high-quality anchors around the lane ground truths within an image, we introduce the \textit{Local Polar Module} (LPM), which takes the highest feature map $\boldsymbol{P}_3\in\mathbb{R}^{C_{f} \times H_{f} \times W_{f}}$  as input and outputs a set of lane anchors along with their confidence scores. As demonstrated in Fig. \ref{lpm}, it undergoes a \textit{downsampling} operation $DS(\cdot)$ to produce a lower-dimensional feature map of a size $H^l\times W^l$:
\begin{equation}
		\boldsymbol{F}_d\gets DS\left( \boldsymbol{P}_{3} \right)\ \text{and}\ \boldsymbol{F}_d\in \mathbb{R} ^{C_f\times H^{l}\times W^{l}}.
\end{equation}
The downsampled feature map $\boldsymbol{F}_d$ is then fed into two branches: a \textit{regression} branch $\phi _{reg}^{l}\left(\cdot \right)$ and a \textit{classification} branch $\phi _{cls}^{l}\left(\cdot \right)$, \textit{i.e.},
\begin{align}
\boldsymbol{F}_{reg}\gets \phi _{reg}^{l}\left( \boldsymbol{F}_d \right)\ &\text{and}\ \boldsymbol{F}_{reg\,\,}\in \mathbb{R} ^{2\times H^{l}\times W^{l}},\\
\boldsymbol{F}_{cls}\gets \phi _{cls}^{l}\left( \boldsymbol{F}_d \right)\ &\text{and}\ \boldsymbol{F}_{cls}\in \mathbb{R} ^{H^{l}\times W^{l}}.	\label{lpm equ}
\end{align}
The regression branch consists of a single $1\times1$ convolutional layer and with the goal of generating lane anchors by outputting their angles $\theta_j$ and the radius $r^{l}_{j}$, \textit{i.e.}, $\boldsymbol{F}_{reg\,\,} \equiv \left\{\theta_{j}, r^{l}_{j}\right\}_{j=1}^{H^{l}\times W^{l}}$, in the defined local polar coordinate system previously introduced. Similarly, the classification branch $\phi _{cls}^{l}\left(\cdot \right)$ only consists of two $1\times1$ convolutional layers for simplicity. This branch is to predict the confidence heat map $\boldsymbol{F}_{cls\,\,}\equiv \left\{ s_j^l \right\} _{j=1}^{H^l\times W^l}$ for local poles, each associated with a feature point. By discarding local poles with lower confidence, the module increases the likelihood of selecting potential positive foreground lane anchors while effectively removing background lane anchors.
\par
\textbf{Loss Function for LPM.} To train the LPM, we define the ground truth labels for each local pole as follows: the ground truth radius, $\hat{r}^l_i$, is set to be the minimum distance from a local pole to the corresponding lane curve, while the ground truth angle, $\hat{\theta}_i$, is set to be the orientation of the vector extending from the local pole to the nearest point on the curve. Consequently, we have a label set of local poles $\hat{\boldsymbol{F}}_{cls}=\{\hat{s}_j^l\}_{j=1}^{H^l\times W^l}$, where $\hat{s}_j^l=1$ if the $j$-th local pole is positive and $\hat{s}_j^l=0$ if it is negative. Once the regression and classification labels are established, as shown in Fig. \ref{lpmlabel}, LPM can be trained using the $Smooth_{L1}$ loss $S_{L1}\left(\cdot \right)$ for regression branch and the \textit{binary cross-entropy} loss $BCE\left( \cdot , \cdot \right)$ for classification branch. The loss functions for LPM are given as follows:
\begin{align}
\mathcal{L} ^{l}_{cls}&=BCE\left( \boldsymbol{F}_{cls},\hat{\boldsymbol{F}}_{cls} \right)\\
\mathcal{L} _{reg}^{l}&=\frac{1}{N_{pos}^{l}}\sum_{j\in \left\{ j|\hat{r}_{j}^{l}<\lambda^l \right\}}{\left( S_{L1}\left( \theta _{j}^{l}-\hat{\theta}_{j}^{l} \right) +S_{L1}\left( r_{j}^{l}-\hat{r}_{j}^{l} \right) \right)}
\label{loss_lph}
\end{align}
where $N^{l}_{pos}=\left|\{j|\hat{r}_j^l<\tau^{l}\}\right|$ is the number of positive local poles in LPM.
\par
\textbf{Top-$K$ Anchor Selection.} As discussed above, all $H^{l}\times W^{l}$ anchors, each associated with a local pole in the feature map, are all considered as candidates during the training stage. However, some of these anchors serve as background anchors. We select $K$ anchors with the top-$K$ highest confidence scores as the foreground candidates to feed into the second stage (\textit{i.e.}, global polar module). During training, all anchors are chosen as candidates, where $K=H^{l}\times W^{l}$  assists it assists \textit{Global Polar Module} (the second stage) in learning from a diverse range of features, including various negative background anchor samples. Conversely, during the evaluation stage, some anchors with lower confidence can be excluded, where  $K\leqslant H^{l}\times W^{l}$. This strategy effectively filters out potential negative anchors and reduces the computational complexity of the second stage. By doing so, it maintains the adaptability and flexibility of anchor distribution while decreasing the total number of anchors especially in the sparse scenarios. The following experiments will demonstrate the effectiveness of different top-$K$ anchor selection strategies.

\begin{figure}[t]
	\centering
	\includegraphics[width=\linewidth]{thesis_figure/detection_head.png} 
	\caption{The primary pipeline of GPM integrates the RoI Pooling Layer with the triplet head. The triplet head comprises three components: the O2O classification head, the O2M classification head, and the O2M regression head. The O2O classification head serves as a replacement for NMS; the dashed path with ``$\times$'' indicates that NMS is no longer necessary. Both sets of $\left\{s_i^g\right\}$ and $\left\{\tilde{s}_i^g\right\}$ participate in the process of selecting the ultimate non-redundant outcomes, a procedure referred to as dual confidence selection. During the backward training phase, the gradients from the O2O classification head (the blue dashed route with  ``$\times$'') are stopped.}
        \label{gpm}
\end{figure}

\subsection{Global Polar Module}
 We introduce a novel \textit{Global Polar Module} (GPM) as the second stage to achieve final lane prediction. As illustrated in Fig. \ref{overall_architecture}, GPM takes features samples from anchors proposed by LPM and provides the precise location and confidence scores of final lane detection results. The overall architecture of GPM is illustrated in the Fig. \ref{gpm}.
\par
\textbf{RoI Pooling Layer.} It is designed to extract sampled features from lane anchors. For ease of the sampling operation, we first transform the radius of the positive lane anchors in a local polar coordinate, $r_j^l$, into the equivalent in a global polar coordinate system, $r_j^g$, by the following equation:
\begin{align}
	r_{j}^{g}&=r_{j}^{l}+\left[ \cos \theta _j; \sin \theta _j \right] ^T\left( \mathbf{c}_{j}^{l}-\mathbf{c}^g \right), \label{l2g}\\
	j &= 1, 2, \cdots, K, \notag
\end{align}
where $\boldsymbol{c}^{g} \in \mathbb{R}^{2}$ and $\boldsymbol{c}^{l}_{j} \in \mathbb{R}^{2}$ represent the Cartesian coordinates of the global pole and the $j$-th local pole, respectively. It is noteworthy that the angle $\theta_j$ remains unaltered, as the local and global polar coordinate systems share the same polar axis. And next, the feature points are sampled on each lane anchor as follows:
\begin{align}
        x_{i,j}^{s}&=-y_{i,j}^{s}\tan \theta _j+\frac{r_{j}^{g}+\left[ \cos \theta _j;\sin \theta _j \right] ^T\boldsymbol{c}^g}{\cos \theta _j},\label{positions}\\
        i&=1,2,\cdots,N,\notag
\end{align}
where the y-coordinates $\boldsymbol{y}_{j}^{s}\equiv \{y_{1,j}^s,y_{2,j}^s,\cdots ,y_{N,j}^s\}$ of the $j$-th lane anchor are uniformly sampled vertically from the image, as previously mentioned. The x-coordinates  $\boldsymbol{x}_{j}^{s}\equiv \{x_{1,j}^s,x_{2,j}^s,\cdots ,x_{N,j}^s\}$ are then calculated by Eq. (\ref{positions}). The derivation of Eq. (\ref{l2g})-(\ref{positions}) can be found in Appendix \ref{appendix_coord}.
\par
Given the feature maps $\boldsymbol{P}_1, \boldsymbol{P}_2, \boldsymbol{P}_3$ from FPN, we can extract feature vectors corresponding to the positions of feature points $\{(x_{1,j}^s,y_{1,j}^s),(x_{2,j}^s,y_{2,j}^s),\cdots,(x_{N,j}^s,y_{N,j}^s)\}_{j=1}^{K}$, respectively denoted as $\boldsymbol{F}_{1,j}, \boldsymbol{F}_{2,j}, \boldsymbol{F}_{3,j}\in \mathbb{R} ^{N\times C_f}$. To enhance representation, similar to \cite{srlane}, we employ a weighted sum strategy to combine features from different levels as:
\begin{equation}
\boldsymbol{F}^s_j=\sum_{k=1}^3{\frac{e^{\boldsymbol{w}_{k}}}{\sum_{k=1}^3{e^{\boldsymbol{w}_{k}}}}\circ \boldsymbol{F}_{k,j} },
\end{equation}
where $\boldsymbol{w}_{k}\in \mathbb{R}^{N}$ represents trainable aggregate weight ascribed to $N$ sampled points. The symbol ``$\circ$'' represents element-wise multiplication (\textit{i.e.}, Hadamard product). Instead of concatenating the three sampling features into $\boldsymbol{F}^s_j\in \mathbb{R} ^{N\times 3C_f}$ directly, the adaptive summation significantly reduces the feature dimensions to $\boldsymbol{F}^s_j\in \mathbb{R} ^{N\times C_f}$, which is one-third of the initial dimension. The weighted sum of the tensors is flattened into a vector $\bar{\boldsymbol{F}}^s_j\in \mathbb{R} ^{NC_f}$, and then subjected to a linear transformation:
\begin{align}
        \boldsymbol{F}_{j}^{roi}&\gets \boldsymbol{W}_{pool}\bar{\boldsymbol{F}}_{j}^{s},\\
        j&=1,2,\cdots,K,\notag.
\end{align}
Here, $\boldsymbol{W}_{pool}\in \mathbb{R} ^{d_r\times NC_f}$ is employed to reduce the dimension of $\bar{\boldsymbol{F}}_{j}^{s}$, thereby yielding the final RoI feature $\boldsymbol{F}_{j}^{roi}\in \mathbb{R} ^{d_r}$, where $d_r\ll NC_f$.

\textbf{Triplet Head.} With the $\left\{ \boldsymbol{F}_{i}^{roi} \right\} _{i=1}^{K}$ as input of the Triplet Head, it encompasses three distinct components: the one-to-one (O2O) classification head, the one-to-many (O2M) classification head, and the one-to-many (O2M) regression head, as depicted in Fig. \ref{gpm}. To attain optimal non-redundant detection outcomes within a NMS-free paradigm (\textit{i.e.}, end-to-end detection), both the one-to-one and one-to-many label assignments become essential during the training stage, as underscored in \cite{o2o}. Drawing inspiration from \cite{o3d}\cite{pss} but with subtle variations, we architect the triplet head to achieve a NMS-free paradigm.
%In numerous studies \cite{laneatt}\cite{clrnet}\cite{adnet}\cite{srlane}, the detection head predominantly adheres to the one-to-many paradigm. During the training phase, multiple positive samples are assigned to a single ground truth. Consequently, during the evaluation phase, redundant detection outcomes are frequently predicted for each instance. These redundancies are conventionally mitigated using Non-Maximum Suppression (NMS), which eradicates duplicate results. Nevertheless, NMS relies on the definition of the geometric distance between detection results, rendering this calculation intricate for curvilinear lanes. Moreover, NMS post-processing introduces challenges in balancing recall and precision, a concern highlighted in our previous analysis.
%As illustrated in Fig. \ref{gpm}, it is important to note that the detection process of the O2O classification head is not independent; rather, the confidence $\left\{ \tilde{s}_i^g \right\}$ output by the O2O classificatoin head relies upon the confidence $\left\{ s_i^g \right\} $ output by the O2M classification head.
\begin{figure}[t]
        \centering
        \includegraphics[width=0.9\linewidth]{thesis_figure/gnn.png} % 替换为你的图片文件名
        \caption{An example of the graph construction in O2O classification head. In the illustration, the elements $A_{12}$, $A_{32}$ and $A_{54}$ are equal to $1$ in the adjacent matrix $\boldsymbol{A}$, thereby indicating the presence of directed edges between the respective node pairs (\textit{i.e.}, $1\rightarrow2$, $3\rightarrow2$ and $5\rightarrow4$). This implies that the detection result $2$ may be potentially suppressed by $1$ and $3$, whereas detection result $4$ may be potentially suppressed by $5$.}
        \label{graph}
\end{figure}

To ensure both simplicity and efficiency in our model, the O2M regression head and the O2M classification head are architected with a straightforward design with \textcolor{red}{two-layer Multi-Layer Perceptrons (MLPs)}. To facilitate the model’s transition to a NMS-free paradigm, we have developed an extended O2O classification head. In this section, we focus on elaborating the structure of the O2O classification head. The comprehensive details of the structure design can be located in Appendix \ref{NMS_appendix}; 

Disregarding the intricate details, the fundamental prerequisites of Fast NMS is as follows. The detection result A is suppressed by another detection result B if:
\begin{itemize}
\item (1) The confidence score of B exceeds that of A;
\item (2) The distance between A and B is less than a predefined threshold.
\end{itemize}
According to the two above conditions, we can construct a relation graph between anchors, as illustrated in the Fig. \ref{graph}.

In a graph, the essential components consist of nodes and edges. We have constructed a directed graph as follows. Each anchor is conceptualized as a node, with the ROI features $\boldsymbol{F}_{i}^{roi}$ serving as the input features (\textit{i.e.}, initial signals) of these nodes. Directed edges between nodes are represented by the adjacent matrix $\boldsymbol{A}\in\mathbb{R}^{K\times K}$.
Specifically, if one element $A_{ij}$ in $\boldsymbol{A}$ equals $1$, a directed edge exists from the $i$-th node and $j$-th node, which implies that the $j$-th prediction may be suppressed by the $i$-th prediction. The existence of an edge is determined by two matrices corresponding to the above two conditions in Fast NMS.

The first matrix is the confidence comparison matrix $\boldsymbol{A}^{C}\in\mathbb{R}^{K\times K}$, which is defined as follows:
\begin{align}
        A_{ij}^{C}=\begin{cases}
                1,\,\,s_i^g>s_j^g\,\,or\,\,\left( s_i^g=s_j^g\,\,and\,\,i>j \right)\\
                0,\,\,others.
        \end{cases}
        \label{confidential matrix}
\end{align}
This matrix facilitates the comparison of scores for each pair of anchors. The edge from the $i$-th and $j$-th nodes exists \textit{only if} the two anchors satisfy the new condition (accounting the situation involving two equal confidence scores) derived from condition (1).

The second matrix is the geometric prior matrix, denoted as $\boldsymbol{A}^{G}\in\mathbb{R}^{K\times K}$:
\begin{align}
        A_{ij}^{G}=\begin{cases}
                1,\,\,\left| \theta _i-\theta _j \right|<\tau^{\theta}\,\,and\,\,\left| r_{i}^{g}-r_{j}^{g} \right|<\tau^r\\
                0,\,\,others.
        \end{cases}
        \label{geometric prior matrix}
\end{align}
This matrix indicates that an edge is considered to exist between two nodes \textit{only if} the two corresponding anchors are sufficiently close to each other. The distance between anchors is characterized by their global polar parameters. This criterion, which takes into account the distance between anchors, introduces a slight variation of condition (2), which accounts for the distance of detection outcomes.

With the aforementioned two matrices, the overall adjacency matrix is formulated as $\boldsymbol{A} = \boldsymbol{A}^{C} \odot \boldsymbol{A}^{G}$; where ``$\odot$'' signifies the element-wise multiplication. Though we have constructed the suppressing  relation graph of each pair of anchors, the distance still remains undefined. In fast NMS, the distance is delineated by geometric properties of the detection results, constraining the model's performance in dense scenarios as we analyzed before. Some forked lanes or dashed lanes have a small geometric distance, which may cause a difficulty in the trade-off of predictions distance. So we replace the geometric distance with the high-dimension semantic distance. The semantic distance is formulated by the graph neural network, which is data-driven. Consequently, the semantic distance between the $i$-th anchor and the $j$-th anchor can be modeled as follows:
\begin{align}
	\tilde{\boldsymbol{F}}_{i}^{roi}&\gets \mathrm{ReLU}\left( \boldsymbol{W}_{roi}\boldsymbol{F}_{i}^{roi}+\boldsymbol{b}_{roi} \right),\label{edge_layer_1}\\
	\boldsymbol{F}_{ij}^{edge}&\gets \boldsymbol{W}_{in}\tilde{\boldsymbol{F}}_{j}^{roi}-\boldsymbol{W}_{out}\tilde{\boldsymbol{F}}_{i}^{roi},\label{edge_layer_2}\\
	\tilde{\boldsymbol{F}}_{ij}^{edge}&\gets \boldsymbol{F}_{ij}^{edge}+\boldsymbol{W}_s\left( \boldsymbol{x}_{j}^{s}-\boldsymbol{x}_{i}^{s} \right) +\boldsymbol{b}_s,\label{edge_layer_3}\\
	\boldsymbol{D}_{ij}^{edge}&\gets \mathrm{MLP}_{edge}\left( \tilde{\boldsymbol{F}}_{ij}^{edge} \right).\label{edge_layer_4}
\end{align}
Eq. (\ref{edge_layer_1})-(\ref{edge_layer_4}) calculate the semantic distance $\boldsymbol{D}_{ij}^{edge}\in \mathbb{R}^{d_n}$ from the $i$-th node and the $j$-th node corresponding to the edge $E_{i\rightarrow j}$ with a directional characteristic. With the directed semantic distances provided for linked node pairs, we employ an element-wise max pooling layer to aggregate all the \textit{incoming edges} of a node to refine its node features to $\boldsymbol{D}_{i}^{node}\in \mathbb{R}^{d_n}$:
\begin{align}
        D_{i,m}^{node}&\gets {\max}\,D_{ki,m}^{edge}, \\
        m&=1,2,\cdots,d_n,\notag
\end{align}
where $D_{i,m}^{node}$ and $D_{ki,m}^{edge}$ are the $m$-th elements of $\boldsymbol{D}_{i}^{node}$ and $\boldsymbol{D}_{ki}^{edge}$, respectively. And Additionally, $k$ is an element of set $\left\{ k|A_{ki}=1 \right\}$. In this context, drawing inspiration from by \cite{o3d}\cite{pointnet}, the max pooling aims to extract the most distinctive features alone the column of the adjacent matrix (\textit{i.e.}, the set of the incoming nodes that may potentially suppress the refined node). With the refined node features, the ultimate confidence scores $\tilde{s}_{i}^{g}$ are generated by the subsequent layers:
\begin{align}
        \boldsymbol{F}_{i}^{node}&\gets \mathrm{MLP}_{node}\left( \boldsymbol{D}_{i}^{node} \right) ,
        \\
        \tilde{s}_{i}^{g}&\gets \sigma \left( \boldsymbol{W}_{node}\boldsymbol{F}_{i}^{node} + \boldsymbol{b}_{node} \right) ,
\label{node_layer}
\end{align}
Equations (\ref{edge_layer_1})-(\ref{node_layer}) are referred to as the newly proposed \textit{graph neural network} (GNN) in our study, which serves as the structural foundation of the O2O classification head, replacing the traditional NMS post-processing.

\textbf{Dual Confidence Selection.} Within the conventional NMS framework, the predictions emanating from the O2M classification heads with confidences $\left\{ s_{i}^{g} \right\} $ surpassing $\lambda_{o2m}^s$ are designated as positive candidates. They are subsequently fed into the NMS post-processing stage to remove redundant predictions. In the NMS-free paradigm of our work, the final non-redundant predictions are selected through the following certerion:
\begin{align}
        \varOmega _{o2o}^{pos}\equiv \left\{ i|\tilde{s}_{i}^{g}>\lambda _{o2o}^{s} \right\} \cap \left\{ i|s_{i}^{g}>\lambda _{o2m}^{s} \right\}.
\end{align}
We employ dual confidence thresholds, denoted as $\lambda_{o2m}^s$ and $\lambda_{o2o}^s$, to select the final non-redundant positives predictions. $\varOmega _{o2o}^{pos}$ signifies the ultimate collection of non-redundant predictions, wherein both confidences satisfy the aforementioned conditions in conjunction with the dual confidence thresholds. This methodology of selecting non-redundant predictions is termed \textit{dual confidence selection}.

% \textbf{Label Assignment and Cost Function for GPM.} As the previous work \cite{o3d}\cite{pss}, we use the dual assignment strategy for label assignment of triplet head. The cost function for the $i$-th prediction and $j$-th ground truth is given as follows:
% \begin{align}
%         \mathcal{C} _{ij}^{o2m}&=s_i^g\times \left( GIoU_{lane, \,ij} \right) ^{\beta},\\
%         \mathcal{C} _{ij}^{o2o}&=\tilde{s}_i^g\times \left( GIoU_{lane, \,ij} \right) ^{\beta},
% \end{align}
% where $\mathcal{C} _{ij}^{o2m}$ is the cost function for the O2M classification and regression head while $\mathcal{C} _{ij}^{o2o}$ for O2O classification head, with $\beta$ serving as the trade-off hyperparameter for location and confidence. This cost function is more compact than that in previous works\cite{clrnet}\cite{adnet}, considering both location and confidence into account. We have redefined IoU function between lane instances: $GIOU_{lane}$, which differs slightly from previous work \cite{clrernet}. More details about $GIOU_{lane}$ can be found in the Appendix \ref{giou_appendix}.



\textbf{Loss function for GPM.}
We use SimOTA \cite{yolox} (one-to-many assignment) for the O2M classification head and the O2M regression head while Hungarian \cite{detr} algorithm (one-to-one assignment) for the O2O classification head. More details about the label assignment can be found in Appendix \ref{giou_appendix}. Focal loss \cite{focal} is utilized for both O2O classification head and the O2M classification head, dentoed as $\mathcal{L}^{o2m}_{cls}$ and $\mathcal{L}^{o2o}_{cls}$, respectively. The set of candidate samples involved in the computation of $\mathcal{L}^{o2o}_{cls}$, denoted as $\varOmega_{o2o}$, is confined to the positive sample set of the O2M classification head:
\begin{align}
        \varOmega _{o2o}=\left\{ i\mid s_i^g>\lambda_{o2m}^s \right\}.
\end{align}
In essence, certain samples with lower $\left\{ s_{i}^{g} \right\} $ are excluded from the computation of $\mathcal{L}^{o2o}_{cls}$. Furthermore, we harness the rank loss $\mathcal{L} _{rank}$ as referenced in \cite{pss} to amplify the disparity between the positive and negative confidences of the O2O classification head. Given the disparity between the label assignments of the O2O classification head and the O2M classification head, to preserve the quality of RoI feature learning, the gradient is stopped from the O2O classification head during the training process. This technique is also utilized in \cite{pss}.
\begin{figure}[t]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/auxloss.png} % 
        \caption{Auxiliary loss for segment parameter regression. The ground truth of a lane curve is partitioned into several segments, with the parameters of each segment denoted as $\left( \hat{\theta}_{i,\cdot}^{seg},\hat{r}_{i,\cdot}^{seg} \right)$. The model output the parameter offsets $\left( \varDelta \theta _{j,\cdot},\varDelta r_{j,\cdot}^{g} \right)$ to regress from the original anchor to each target line segments.}
        \label{auxloss}
\end{figure}

We directly apply the redefined GIoU loss (refer to Appendix \ref{giou_appendix}), $\mathcal{L}_{GIoU}$, to regress the offset of x-axis coordinates of sampled points and $Smooth_{L1}$ loss for the regression of end points of lanes, denoted as $\mathcal{L}_{end}$. 

To facilitate the learning of global features, we propose the auxiliary loss $\mathcal{L}_{aux}$ depicted in Fig. \ref{auxloss}. The anchors and ground truth are segmented into several divisions. Each anchor segment is regressed to the primary components of the corresponding segment of the designated ground truth. This approach aids the detection head in acquiring a deeper comprehension of the global geometric form. 

The final loss functions for GPM are given as follows:
\begin{align}
        \mathcal{L} _{cls}^{g}&=w^{o2m}_{cls}\mathcal{L}^{o2m}_{cls}+w^{o2o}_{cls}\mathcal{L}^{o2o}_{cls}+w_{rank}\mathcal{L}_{rank},
        \\
        \mathcal{L} _{reg}^{g}&=w_{GIoU}\mathcal{L}_{GIoU}+w_{end}\mathcal{L}_{end}+w_{aux}\mathcal{L} _{aux}.
\end{align}
% \begin{align}
%         \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}^{g} - \hat{r}_{i}^{seg,m} \right) \Bigg].
% \end{align}
\subsection{The Overall Loss Function.} The entire training process is orchestrated in an end-to-end manner, wherein both LPM and GPM are trained concurrently. The overall loss function is delineated as follows:
\begin{align}
\mathcal{L} =\mathcal{L} _{cls}^{l}+\mathcal{L} _{reg}^{l}+\mathcal{L} _{cls}^{g}+\mathcal{L} _{reg}^{g}.
\end{align}

\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. 

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{align}
Pre\,\,&=\,\,\frac{TP}{TP+FP},
\\
Rec\,\,&=\,\,\frac{TP}{TP+FN}.
\\
F1&=\frac{2\times Pre\times Rec}{Pre\,\,+\,\,Rec},
\end{align}
where $TP$, $FP$ and $FN$ represent the true positives, false positives, and false negatives of the entire dataset, respectively. 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{align}
                mF1=\left( F1@50+F1@55+...+F1@95 \right) /10.
        \end{align}

For Tusimple, the evaluation is formulated as follows:
        \begin{align}
                Accuracy=\frac{\sum{C_{clip}}}{\sum{S_{clip}}}.
        \end{align}
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 ($\mathrm{FPR}=1-\mathrm{Precision}$) and False Negative Rate ($\mathrm{FNR}=1-\mathrm{Recall}$) 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 coefficient of cost function $\beta$ is set to 6. The training processing of the whole model (including LPM and GPM) is end-to-end just like \cite{adnet}\cite{srlane}.  All the experiments 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}. Other details for datasets and training process can be seen in Appendix \ref{vis_appendix}.


\begin{table*}[htbp]
        \centering
        \caption{Comparison 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{Comparison 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{FPR(\%)}&\textbf{FNR(\%)} \\
        \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{Comparison 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{Comparison 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{Comparison 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 $\left\{s_i^g\right\}$ obtained from the O2M head followed by NMS post-processing, while the NMS-free version derives predictions via dual confidence selection.

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 dataset splits. As illustrated in the comparison results, our model demonstrates competitive performance across five datasets. Specifically, on the CULane, TuSimple, LLAMAS, and DL-Rail datasets of 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 classification head in eliminating redundant predictions in the sparse scenarios. On the CurveLanes dataset, the NMS-free version achieves superior F1-measure and Recall compared to other 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 dataset. 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. 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 complicated components such as cross-layer refinement, indicating the pivotal role of  flexible anchors under polar coordinates in enhaning 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}.
\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}

\subsection{Ablation Study}
To validate and analyze the effectiveness and influence of different component of Polar R-CNN, we conduct serveral ablation studies on CULane and CurveLanes datasets.

\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 parameters 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 dynamic $k=4$ for SimOTA, with no more than 4 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-$K$ selected anchors’ distribution in sparse scenarios. Brighter colors indicate a higher likelihood of anchors being foreground. 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/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-$K$ 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) and (c) The heap map of the local polar map; (b) and (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 paradigm 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 confidence comparison matrix $\boldsymbol{A}^{C}$ proves crucial, indicating that conditional probability is effective. Other components, such as the geometric prior matrix $\boldsymbol{A}^{G}$ 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 (employing a fixed set of anchors as illustrated in Fig. \ref{anchor setting}(b)). Table \ref{NMS vs NMS-free} presents the results of these experiments. In the table, ``O2M'' and ``O2O'' refer to the NMS (the gray dashed route in Fig. \ref{o2o_cls_head}) and NMS-free paradigms (the green route in Fig. \ref{o2o_cls_head}) respectively. The suffix ``-B'' signifies that the head consists solely of MLPs, whereas ``-G'' indicates that the head is equipped with the GNN architecture. In the fixed anchor paradigm, although the O2O classification head without GNN effectively eliminates redundant predictions, the performance still improved by incorporating GNN structure. In the proposal anchor paradigm, the O2O classification head without GNN fails to eliminate redundant predictions due to high anchor overlaps. Thus, the GNN structure is essential for Polar R-CNN in the NMS-free paradigm. In both the fixed and proposed anchor paradigms, the O2O classification head with the GNN structure successfully eliminates redundant predictions, indicating that our GNN-based O2O classification head can supplant the NMS post-processing in sparse scenarios without decline in performance.

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 observation indicates that the one-to-one assignment induces significant bias into feature learning, thereby underscoring the necessity of the stop-gradient strategy to preserve optimal performance.

\begin{table}[h]
        \centering
        \caption{Ablation study on GNN block.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{cccc|ccc}
        \toprule
        \textbf{GNN}&$\boldsymbol{A}^{C}$&$\boldsymbol{A}^{G}$&\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 gradient strategy on CULane test set.}
        \begin{adjustbox}{width=\linewidth}
        \begin{tabular}{c|c|lll}
        \toprule
        \multicolumn{2}{c|}{\textbf{Paradigm}} & \textbf{F1@50 (\%)} & \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 the O2O classification head with proposed GNN structure is capable of learning both explicit geometric distance and implicit semantic distances between anchors, 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}




\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 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.
%
%
%
\bibliographystyle{IEEEtran}
\bibliography{reference} 
%\newpage
%
\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


\newpage
% 附录有多个section时
\appendices
\setcounter{table}{0}   %从0开始编号，显示出来表会A1开始编号
\setcounter{figure}{0}
\setcounter{section}{0}
\setcounter{equation}{0}
\renewcommand{\thetable}{A\arabic{table}}
\renewcommand{\thefigure}{A\arabic{figure}}
\renewcommand{\thesection}{A\arabic{section}}
\renewcommand{\theequation}{A\arabic{equation}}
\section{Details about the Coordinate Systems}
In this section, we introduce the details about the coordinate systems employed in our model and coordinate transformations between them.
For convenience, we adopted Cartesian coordinate system instead of the image coordinate system, wherein the y-axis is oriented from bottom to top and the x-axis from left to right. The coordinates of of the local poles $\left\{\boldsymbol{c}^l_i\right\}$, the global pole $\boldsymbol{c}^g$, and the sampled points $\{(x_{1,j}^s,y_{1,j}^s),(x_{2,j}^s,y_{2,j}^s),\cdots,(x_{N,j}^s,y_{N,j}^s)\}_{j=1}^{K}$ of anchors are all within this coordinate by default. 

We now furnish the derivation of the Eq. (\ref{l2g}) and Eq. (\ref{positions}), with the crucial symbols elucidated in Fig. \ref{elu_proof}. These geometric transformations can be demonstrated with Analytic geometry theory in Euclidean space. The derivation of Eq. (\ref{l2g}) is presented as follows:
\begin{align}
        r_{j}^{g}&=\left\| \overrightarrow{c^gh_{j}^{g}} \right\| =\left\| \overrightarrow{h_{j}^{a}h_{j}^{l}} \right\| =\left\| \overrightarrow{h_{j}^{a}h_{j}^{l}} \right\| \notag\\
	&=\left\| \overrightarrow{c_{j}^{l}h_{j}^{l}}-\overrightarrow{h_{j}^{a}c_{j}^{l}} \right\| =\left\| \overrightarrow{c_{j}^{l}h_{j}^{l}} \right\| -\left\| \overrightarrow{c_{j}^{l}h_{j}^{a}} \right\| \notag\\
	&=\left\| \overrightarrow{c_{j}^{l}h_{j}^{l}} \right\| - \frac{\overrightarrow{c_{j}^{l}h_{j}^{a}}}{\left\| \overrightarrow{c_{j}^{l}h_{j}^{a}} \right\|}\cdot \overrightarrow{c_{j}^{l}h_{j}^{a}} =\left\| \overrightarrow{c_{j}^{l}h_{j}^{l}} \right\| +\frac{\overrightarrow{c_{j}^{l}h_{j}^{a}}}{\left\| \overrightarrow{c_{j}^{l}h_{j}^{a}} \right\|}\cdot \overrightarrow{c^gc_{j}^{l}} \notag\\
	&=r_{j}^{l}+\left[ \cos \theta _j;\sin \theta _j \right] ^T\left( \boldsymbol{c}_{j}^{l}-\boldsymbol{c}^g \right),
        \label{proof_l2g}
\end{align}
where $h_j^l$, $h_j^g$ and $h_j^a$ represent the foots of their respective perpendiculars in Fig. \ref{elu_proof}.
Analogously, the derivation of Eq. (\ref{positions}) is provided as follows:
\begin{align}
        &\overrightarrow{c^gp_{i,j}^{s}}\cdot \overrightarrow{c^gh_{j}^{g}}=\overrightarrow{c^gh_{j}^{g}}\cdot \overrightarrow{c^gh_{j}^{g}} \notag\\
        \Rightarrow &\overrightarrow{c^gp_{i,j}^{s}}\cdot \overrightarrow{c^gh_{j}^{g}}=\left\| \overrightarrow{c^gh_{j}^{g}} \right\| \left\| \overrightarrow{c^gh_{j}^{g}} \right\| \notag\\
        \Rightarrow &\frac{\overrightarrow{c^gh_{j}^{g}}}{\left\| \overrightarrow{c^gh_{j}^{g}} \right\|}\cdot \overrightarrow{c^gp_{i,j}^{s}}=\left\| \overrightarrow{c^gh_{j}^{g}} \right\| \notag\\
        \Rightarrow &\left[ \cos \theta _j;\sin \theta _j \right] ^T\left( \boldsymbol{p}_{i,j}^{s}-\boldsymbol{c}^g \right) =r_{j}^{g}\notag\\
        \Rightarrow &x_{i,j}^{s}\cos \theta _j+y_{i,j}^{s}\sin \theta _j=r_{j}^{g}+\left[ \cos \theta _j;\sin \theta _j \right] ^T\boldsymbol{c}^g \notag\\
        \Rightarrow &x_{i,j}^{s}=-y_{i,j}^{s}\tan \theta _j+\frac{r_{j}^{g}+\left[ \cos \theta _j;\sin \theta _j \right] ^T\boldsymbol{c}^g}{\cos \theta _j},
        \label{proof_sample}
\end{align}
where $p_{i,j}^{s}$ represents the $i$-th sampled point of the $j$-th lane anchor, whose coordinate is $\boldsymbol{p}_{i,j}^{s}\equiv(x_{i,j}^s, y_{i,j}^s)$.


\label{appendix_coord}





\section{The Design Principles of the One-to-one classification Head}
Two fundamental prerequisites of the NMS-free framework lie in the label assignment strategies and the head structures.

As for the label assignment strategy, previous work use one-to-many label assignments such as SimOTA\cite{yolox}. One-to-many label assignment make the detection head make redundant preidictions for one ground truth, resulting in the need of NMS post-processing. Thus, some works \cite{detr}\cite{learnNMS} proposed one-to-one label assignment such as Hungarian algorithm. This force the model to predict one positive samples for one ground truth.

However, directly using one-to-one label assignment damage the learning of the model, and the plain structure such as MLPs and CNNs struggle to assimilate the ``one-to-one'' characteristics, resulting in the decreasing of performance compared to one-to-many label assignments with NMS post-processing\cite{yolov10}\cite{o2o}. Consider a trival example: Let $\boldsymbol{F}^{roi}_{i}$ denotes the ROI features extracted from the $i$-th anchor, and the model is trained with one-to-one label assignment. Assuming that the $i$-th anchor and the $j$-th anchor are both close to the ground truth and overlap with each other, we can express as follows:
\begin{align}
        \boldsymbol{F}_{i}^{roi}\approx \boldsymbol{F}_{j}^{roi}.
\end{align}
This indicates that the RoI pooling features of the two anchors are similar. Suppose that $\boldsymbol{F}^{roi}_{i}$ is designated as a positive sample while $\boldsymbol{F}^{roi}_{j}$ as a negative sample,  the ideal outcome should manifest as:
\begin{align}
        \boldsymbol{F}_{cls}^{plain}\left( \boldsymbol{F}_{i}^{roi} \right) &\rightarrow 1,
        \\
        \boldsymbol{F}_{cls}^{plain}\left( \boldsymbol{F}_{j}^{roi} \right) &\rightarrow 0,
\label{sharp fun}
\end{align}
where $\boldsymbol{F}_{cls}^{plain}$ represents a classification head characterized by a plain architecture. The Eq. (\ref{sharp fun}) implies that the property of $\boldsymbol{F}_{cls}^{plain}$ need to be ``sharp'' enough to differentiate between two similar features. In other words, the output of $\boldsymbol{F}_{cls}^{plain}$ changes rapidly over short periods or distances. This ``sharp'' pattern is hard to train for MLPs or CNNs \cite{o3d} solely. Consequently, additional new heuristic structures like \cite{o3d}\cite{relationnet} need to be developed.

We draw inspiration from Fast NMS \cite{yolact} for the design of the O2O classification head. Fast NMS serves as an iteration-free post-processing algorithm based on traditional NMS. Furthermore, we have incorporated a sort-free strategy along with geometric priors into Fast NMS, with the specifics delineated in Algorithm \ref{Graph Fast NMS}.


\begin{figure}[t]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/elu_proof.png}
        \caption{The symbols employed in the derivation of coordinate transformations across different coordinate systems.}
        \label{elu_proof}
\end{figure}


\begin{algorithm}[t]
        \caption{Fast NMS with Geometric Prior.}
        \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 all anchors, $1, 2, ..., i, ..., K$;\\
            The positive corresponding anchors, $\left\{ \theta _i,r_{i}^{g} \right\} |_{i=1}^{K}$;\\
            The confidence emanating from the O2M classification head, $s_i^g$;\\
            The regressions emanating from the O2M regression head, denoted as $\left\{ Lane_i \right\} |_{i=1}^{K}$\\
            The predetermined thresholds $\tau^\theta$, $\tau^r$, $\tau_d$ and $\lambda _{o2m}^{s}$.
        \ENSURE ~~\\ %算法的输出：Output
            \STATE Caculate the confidence comparison matrix $\boldsymbol{A}^{C}\in\mathbb{R}^{K\times K}$, defined as follows:
            \begin{align}
                A_{ij}^{C}=\begin{cases}
                        1, s_i>s_j\,\,or\,\,\left( s_i^g=s_j^g\,\,and\,\,i>j \right)\\
                        0, others.\\
                \end{cases}
                \label{confidential matrix}
                \end{align}
            \STATE Calculate the geometric prior matrix $\boldsymbol{A}^{G}\in\mathbb{R}^{K\times K}$, which is defined as follows:
            \begin{align}
                A_{ij}^{G}=\begin{cases}
                        1, \left| \theta _i-\theta _j \right|<\tau^{\theta}\,\,and\,\,\left| r_{i}^{g}-r_{j}^{g} \right|<\tau^r\\
                        0, others.\\
                \end{cases}
                \label{geometric prior matrix}
        \end{align}
            \STATE Calculate the inverse distance matrix $\boldsymbol{D}  \in \mathbb{R} ^{K \times K}$ The element $D_{ij}$ in $\boldsymbol{D}$ is defined as follows: 
                \begin{align}
                        D_{ij}=d^{-1}\left( Lane_i,Lane_j \right) ,
                \label{al_1-3}
                \end{align}
            where $d\left(\cdot, \cdot \right)$ is some predefined function to quantify the distance between two lane predictions such as IoU.
            \STATE Define the adjacent matrix $\boldsymbol{A} = \boldsymbol{A}^{C} \odot \boldsymbol{A}^{G}$ and the final confidence $\tilde{s}_i^g$ is calculate as following:
            \begin{align}
                \tilde{s}_{i}^{g}=\begin{cases}
                        1, \mathrm{if}\underset{D_{ki}\in \{D_{ki}\mid A_{ki}=1\}}{\max}D_{ki}<\left( \tau ^d \right) ^{-1},\\
                        0, \mathrm{otherwise}\\
                \end{cases}
                \label{al_1-4}
            \end{align}
            \STATE Get the final selection set:
            \begin{align}
            \varOmega_{nms}^{pos}=\left\{ i|s_{i}^{g}>\lambda _{o2m}^{s}\,\,and\,\,\tilde{s}_{i}^{g}=1 \right\}
            \label{al_1-5}
            \end{align}
        
        \RETURN The final selection result $\varOmega_{nms}^{pos}$.
        \end{algorithmic}
        \label{Graph Fast NMS}
\end{algorithm}

The new algorithm has a distinct format from the original one\cite{yolact}. The geometric prior $\boldsymbol{A}_{G}$ indicated that predictions associated with adequately proximate anchors were likely to suppress one another. It is straightforward to demonstrate that, when all elements within $\boldsymbol{A}_{G}$ are all set to 1 (disregarding geometric priors), Algorithm \ref{Graph Fast NMS} is equivalent to Fast NMS. Building upon our newly proposed sort-free Fast NMS with geometric prior, we can design the structure of the one-to-one classification head.

The principal limitations of the NMS lie in the definitions of distance derived from geometry (i.e., Eq. (\ref{al_1-3})) and the threshold $\lambda^{g}$ employed to eliminate redundant predictions (i.e., Eq. (\ref{al_1-4})). For instance, in the scenario of double lines, despite the minimal geometric distance between the two lane instances, their semantic divergence is strikingly distinct. Consequently, we replace the above two steps with trainable neural networks, allowing them to learn the semantic distance in a data-driven fashion. The neural network blocks to replace Eq. (\ref{al_1-3}) are expressed as:
% \begin{align}
% 	\tilde{\boldsymbol{F}}_{i}^{roi}&\gets \mathrm{ReLU}\left( \boldsymbol{W}_{roi}\boldsymbol{F}_{i}^{roi}+\boldsymbol{b}_{roi} \right) ,\label{edge_layer_1_appendix}\\
% 	\boldsymbol{F}_{ij}^{edge}&\gets \boldsymbol{W}_{in}\tilde{\boldsymbol{F}}_{j}^{roi}-\boldsymbol{W}_{out}\tilde{\boldsymbol{F}}_{i}^{roi},\label{edge_layer_2_appendix}\\
% 	\tilde{\boldsymbol{F}}_{ij}^{edge}&\gets \boldsymbol{F}_{ij}^{edge}+\boldsymbol{W}_s\left( \boldsymbol{x}_{j}^{s}-\boldsymbol{x}_{i}^{s} \right) +\boldsymbol{b}_s,\label{edge_layer_3_appendix}\\
% 	\boldsymbol{D}_{ij}^{edge}&\gets \mathrm{MLP}_{edge}\left( \tilde{\boldsymbol{F}}_{ij}^{edge} \right) .\label{edge_layer_4_appendix}
% \end{align}
where the inverse distance $\boldsymbol{D}_{ij}^{edge}\in\mathbb{R}^{d_n}$ is no longer a scalar but a tensor. We use element-wise max pooling for tensor to repalce the max operation for scalear. So the $\left( \tau ^d \right) ^{-1}$ can vbe no longer employed as the threshold of the distance. Furthermore, the predetermined $\left( \tau ^d \right) ^{-1}$ can be no longer employed as the threshold of the distance. We defined a neural work as a implicit decision plane to formulate the final score $\tilde{s}_{i}^{g}$. The replacement of Eq. (\ref{al_1-4}) is constructed as follows:So  We also use a The replacement of Eq. (\ref{al_1-4}) is constructed as follows:
% \begin{align}
%         \boldsymbol{D}_{i}^{node}&\gets \underset{\boldsymbol{D}_{ki}^{edge}\in \left\{ \boldsymbol{D}_{ki}^{edge}|A_{ki}=1 \right\}}{\max}\boldsymbol{D}_{ki}^{edge}.
%         \\
%         \boldsymbol{F}_{i}^{node}&\gets \mathrm{MLP}_{node}\left( \boldsymbol{D}_{i}^{node} \right) ,
%         \\
%         \tilde{s}_{i}^{g}&\gets \sigma \left( \boldsymbol{W}_{node}\boldsymbol{F}_{i}^{node} + \boldsymbol{b}_{node} \right).
% \label{node_layer_appendix}
% \end{align}
In this expression, the score $\tilde{s}_{i}^{g}$ transitions from a binary score to a continuous soft score ranging from 0 to 1. We introduce a new threshold $\lambda^s_{o2o}$ within the replacement criteria of Eq. (\ref{al_1-5}):
\begin{align}
        \varOmega_{nms}^{pos}=\left\{ i|s_{i}^{g}>\lambda _{o2m}^{s}\,\,and\,\,\tilde{s}_{i}^{g}>\lambda^s_{o2o}\right\}.
\end{align}
This criteria is also referred to as the \textit{dual confidence selection} in the main text.
\label{NMS_appendix}

\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. Labels for some validation/test sets are missing; therefore, selected different splits (test or validation set) are selected 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}
        & $H^{l}\times W^{l}$ &$4\times10$&$4\times10$&$4\times10$&$4\times10$&$6\times13$\\
        & $K$ &20&20&20&12&50\\
        & $d$ &5&8&10&5&5\\
        & $C_{o2m}$ &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*}



\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}

\section{The Details of Intersection Over Union between Lane Instances}
To make the IoU between lane instances consistent with that of general object detection methods \cite{iouloss}\cite{giouloss}, we have redefined the lane IoU. As illustrated in Fig. \ref{glaneiou}, the newly-defined IoU of lanes, which we term as GLaneIoU, is articulated as follows:

\begin{align}
        \Delta x_{i,p}^{d}&=x_{i+1,p}^{d}-x_{i-1,p}^{d},\,\, \Delta y_{i,p}^{d}=y_{i+1,p}^{d}-y_{i-1,p}^{d}, \\
        w_{i,p}&=\frac{\sqrt{\left( \Delta x_{i,p}^{d} \right) ^2+\left( \Delta y_{i,p}^{d} \right) ^2}}{\Delta y_{i,p}^{d}}w^b,\\
        b_{i,p}^{l}&=x_{i,p}^{d}-w_{i,p},\,\, b_{i,p}^{r}=x_{i,p}^{d}+w_{i,p}\,\,   \\
\end{align}
where $w^{b}$ is the base semi-width parameter and $w_{i,p}&$ is the actual semi-width of $p$-th lane instance. $\left\{ b_{i,p}^{l} \right\} _{i=1}^{N}$ and $\left\{ b_{i,p}^{r} \right\} _{i=1}^{N}$ denotes the left boundaries and the right boundaries if the $p$-th lane instance. Then we defined inter and union between lane instances:
\begin{align}
        d_{i,pq}^{\mathcal{O}}&=\max \left( \min \left( b_{i,p}^{r}, b_{i,q}^{r} \right) -\max \left( b_{i,p}^{l}, b_{i,q}^{l} \right) , 0 \right),\\
        d_{i,pq}^{\xi}&=\max \left( \max \left( b_{i,p}^{l}, b_{i,q}^{l} \right) -\min \left( b_{i,p}^{l}, b_{i,q}^{l} \right) , 0 \right),\\
        d_{i,pq}^{\mathcal{U}}&=\max \left( b_{i,p}^{r}, b_{i,q}^{r} \right) -\min \left( b_{i,p}^{l}, b_{i,q}^{l} \right),
\end{align}
 The definations of $\left\{d_{i,pq}^{\mathcal{O}}\right\}_{i=1}^{N}$, $\left\{d_{i,pq}^{\xi}\right\}_{i=1}^{N}$ and $\left\{d_{i,pq}^{\mathcal{U}}\right\}_{i=1}^{N}$ denote the over distance, gap distance, and union distance, respectively. These definatons are 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 between the $p$-th and $q$-th lane instances is given as follows:
\begin{align}
        GIoU_{lane}\left( p,q \right)=\frac{\sum\nolimits_{i=j}^k{d_{i,pq}^{\mathcal{O}}}}{\sum\nolimits_{i=j}^k{d_{i,pq}^{\mathcal{U}}}}-g\frac{\sum\nolimits_{i=j}^k{d_{i,pq}^{\xi}}}{\sum\nolimits_{i=j}^k{d_{i,pq}^{\mathcal{U}}}},
\end{align}
where j and k are the indices of the start point and the end point, respectively. It's straightforward to observed that when $g=0$, the $GIoU_{lane}$ is correspond to IoU for bounding box, with a value range of $\left[0, 1 \right]$. When $g=1$, the $GIoU_{lane}$ is correspond to GIoU\cite{giouloss} for bounding box, with a value range of $\left(-1, 1 \right]$. 

% In general, when $g>0$, the value range of $GIoU_{lane}$ is $\left(-g, 1 \right]$. We set $g=0$ for cost function and IoU matrix in SimOTA, while $g=1$ for the loss function.

\label{giou_appendix}

\section{Details about The Label assignment and Loss function.}

\begin{figure}[t]
        \centering
        \includegraphics[width=\linewidth]{thesis_figure/detection_head_assign.png}
        \caption{Label assignment and loss function for the triplet head.}
        \label{head_assign}
\end{figure}
We furnish the cost function and label assignments for the triplet head. We use dual label assignment strategy \cite{date} to assign label for triplet head, as illustrated in Fig. \ref{head_assign}. Specifically, we use one-to-many label assignments for both O2O classification head and O2M regression head. This part is almost the same as previous work \cite{clrernet}. In order to equip our model with NMS-free paradigm, we additionally add a O2O classification head and employ one-to-one label assignment to it. 

The cost metric for one-to-one and one-to-many label assignments, are given as follows:

\begin{align}
        \mathcal{C} _{p,q}^{o2o}=\tilde{s}_{p}^{g}\times \left( GIoU_{lane}\left( p,q \right) \right) ^{\beta} \label{o2o_cost},\\
        \mathcal{C} _{p,q}^{o2m}=s_{p}^{g}\times \left( GIoU_{lane}\left( p,q \right) \right) ^{\beta}, \label{o2m_cost}
\end{align}
where $\mathcal{C} _{pq}^{o2o}$ and $\mathcal{C} _{pq}^{o2m}$ are the cost metric between $p$-th prediction and $q$-th ground truth and $g$ in $GIoU_{lane}$ are set to $0$ to keep it non-negative. These metrics imply that both confidence score and geometric distance contributes to the cost metrics.

Suppose that there are $K$ predictions and $G$ ground truth. Let $\pi$ denotes some one-to-one label assignment strategy and $\pi(q)$ represents that $\pi(q)$-th predictions are assign to the $q$-th anchor. Additionally, $\mathscr{S}_{K, G}$ denotes the set of all possible one-to-one assignment strategies for K predictions and Q ground truth. It's easy to demonstrate that the total number of one-to-one assignment strategies $\left| \mathscr{S} _{K,G} \right|$ is $\frac{K!}{\left( K-G \right)!}$. The final assignment $\hat{\pi}$ are determined as follows:
\begin{align}
        \hat{\pi}=\underset{\pi \in \mathscr{S}_{K,G}}{arg\max}\sum_{q=1}^G{\mathcal{C} _{\pi \left( q \right) ,q}^{o2o}}。
\end{align}
This assignment problem can be solved by Hungarian algorithm \cite{detr}. Finally, $G$ predictions are assigned as positive samples and $K-G$ predictons are assigned as negative samples.

In the one-to-many label assignment, we simply use SimOTA \cite{yolox}, which is the same as previous works \cite{clrernet}. Neglecting the detailed process of SimOTA, we only introduce the inputs of SimOTA, the cost matrix $\boldsymbol{M}^C\in \mathbb{R}^{G\times K}$ and the IoU matrix $\boldsymbol{M}^{IoU}\in \mathbb{R}^{G\times K}$. The elements in the two matrices are defined as $M^C_{qp}=\mathcal{C} _{p,q}^{o2m}$ and $M^{IoU}_{qp}= GIoU_{lane}\left( p,q \right)$ (with $g=0$), respectively. The number of assigned predictions for each ground truth is unfixed but no more than an upper bound $k_{dynamic}$, which is set to $4$ in our experiment. Finally, there are $K_{pos}$ positive samples and $K-K_{pos}$ negative samples, where $K_{pos}$ ranges from $0$ to $Gk_{dynamic}$.

Given the ground truth label generated by the label assignment strategy for each prediction, we can conduct the loss function during phase. As illustrated in Fig. \ref{head_assign}, $\mathcal{L}_{cls}^{o2o}$ and $\mathcal{L}_{rank}$ are for the O2O classification head, $\mathcal{L}_{cls}^{o2m}$ is for the O2M classification head whereas $\mathcal{L}_{GIOU}$ (with $g=1$), $\mathcal{L}_{end}$ and $\mathcal{L}_{aux}$ for the O2M head. The training phase of the O2M classification and regression heads are almost the same as previous works \cite{clrnet}. 





\begin{figure*}[t]
        \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*}[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/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 and dense scenarios on CurveLanes dataset.}
        \label{vis_dense}
\end{figure*}
\section{The Supplement of Implement Detail and The Visualization Results.}
Some important implement details for each dataset is shown in Table \ref{dataset_info}. 

Fig. \ref{vis_sparse} shows the visualization results 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 while maintaining accurate location around the ground truth compared to previous works, making our method faster than other anchor-based methods in theory. 

Fig. \ref{vis_dense} shows the visualization results 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 that the trade-off struggles  between 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 the last column of Fig. \ref{vis_dense}, the O2O classification head successfully eliminates redundant predictions while retaining dense predictions with small geometric distances. 
\label{vis_appendix}

\end{document}