An Autotuning Hybrid Method with Bayesian Optimization for Road Edge Extraction in Highway Systems from Point Clouds

Abstract

In transportation infrastructure systems, feature images and spatial characteristics are generally utilized as complementary elements derived from point clouds for road edge extraction, but the involvement of one or more hyperparameters in each makes the extraction complicated. This study proposes an autotuning hybrid method with Bayesian optimization for road edge extraction in highway systems. The hybrid method combines the strengths of 2D feature images and 3D spatial characteristics while also automatically tuning the hyperparameter combination using Bayesian optimization. The hyperparameters encompass high and low pixel gradient thresholds, neighborhood radius, and normal vector threshold. Later, the point cloud dataset of national highways in Henan Province, China, is taken as the case study to evaluate the performance of the proposed method against three benchmark methods in two typical road scenarios: straight and curved edges. Experimental results show that the proposed method outperforms the benchmarks in detection quality and accuracy. It can serve as a decision-making tool to complement traditional manual road surveying, enabling efficient and automated road edge extraction in highway systems.

1. Introduction

In the transportation domain, roads serve as fundamental components in transportation infrastructure systems, making the acquisition of road information essential for infrastructure planning and maintenance decisions. Road information is also a prerequisite for high-precision map navigation and intelligent transportation management. Accurate road edge extraction, as a vital element of road information, is conducive to enhancing the performance of vehicle tracking by narrowing down the search space within autonomous driving environments, particularly in highway systems [1]. In engineering practices, traditional manual road surveying by using total station or real-time kinematic technology is extremely time-consuming and inefficient. To surmount such deficiency, automatic extraction technologies have increasingly gained attention in both the industry and research community, primarily relying on two main data sources: imagery data and remote sensing data [2,3].

With the rapid advancement of technology, data sources from remote sensing have become increasingly prevalent, with light detection and ranging (LiDAR) being one of the most frequently employed. Equipped with navigation sensors (e.g., Global Navigation Satellite System (GNSS)), mobile laser scanning (MLS) is a representative technology within the LiDAR field, which enables the collection of high-density geo-referenced 3D point clouds in a cost-effective and time-efficient manner [4,5]. A key advantage of MLS technology is its robustness to varying illumination conditions, as it does not require direct contact with road objects while possessing high-precision distance measurements. This technology has been extensively used for the extraction of various on-road and roadside objects in highway systems [6]. Over the past two decades, numerous road extraction methods based on MLS point clouds have been developed by researchers, which subsume two main categories: feature image-based and spatial characteristics-based [7]. The former generates feature images from point clouds and employs image processing algorithms to extract road elements, while the latter directly utilizes spatial geometric and mathematical statistical features from 3D point clouds [8,9].

A promising alternative approach in road extraction methods is to fuse the advantages of both 2D feature-based and 3D spatial characteristics-based techniques, as the features of these two categories are considered complementary. The 2D methods, with their image-based processing algorithms, provide robust feature detection in the planar space, while 3D methods offer precise spatial geometric information. By combining these strengths, a more comprehensive and accurate extraction process can be achieved. However, each of these methods typically requires the tuning of one or more hyperparameters, such as thresholds and radius values, which significantly influence both the efficiency and accuracy of the extraction process. When integrating elements from both 2D and 3D methods, the challenge of determining the optimal hyperparameter combination becomes increasingly complex due to the added dimension and interaction effects.

In recent years, Bayesian optimization (BO) has emerged as a powerful and practical tool for automatic hyperparameter tuning in engineering and machine learning problems [10,11,12]. Unlike traditional grid or random search methods, BO is a non-parametric technique that leverages the principles of the Bayesian theorem to intelligently explore the hyperparameter space, improving search efficiency and enabling the discovery of optimal parameter values in fewer iterations [13]. BO constructs a probabilistic model to predict the performance of hyperparameter configuration and iteratively refines its model based on prior evaluations, leading to automatic and effective optimization. Given the complexity of tuning hyperparameters in road edge extraction methods that integrate 2D and 3D features, this study aims to propose an autotuning hybrid method integrating BO to automatically adjust and optimize hyperparameters. This method is designed to enhance the accuracy and performance of road edge extraction in highway systems by leveraging point cloud data.

1.1. Literature Review

The automated extraction of roads using remote sensing data has garnered considerable attention due to its practical importance in road information inventories derived from transportation infrastructure systems [3,14]. Instead of covering the extensive body of research accumulated in this field, we focus our literature review on studies relevant to the vehicle-based MLS technique. Based upon MPL point clouds, road extraction methods used in the existing research can be broadly classified into two main branches: (i) feature image-based and (ii) spatial characteristics-based [7,15].

In the first branch, 3D MPL point clouds are converted into 2D feature images, which significantly reduces the computational complexity associated with processing large-scale point cloud data. This simplification allows for more efficient data handling without compromising the essential spatial details needed for road extraction. In general, there are two common patterns to form 2D feature imagery [3]. The first pattern involves classifying points into ground and non-ground categories by analyzing elevation levels and elevation differentials. This classification is crucial for separating road surfaces from surrounding objects such as trees and buildings, which could otherwise interfere with the accuracy of road detection algorithms. The second pattern projects the point clouds onto the XOY plane, producing top-down 2D images that provide a bird’s-eye view of the road and its surrounding features. This projection simplifies the spatial representation of the road network, facilitating easier identification of road areas and edges. Based upon 2D feature images, the extraction of road areas or road edges depending on semantic knowledge can be efficiently achieved through different image processing algorithms, and a considerable body of scientific literature has accumulated on this topic in recent years, which is reviewed in several survey papers [16,17,18,19].

In the second branch, road extraction is performed by using characteristics of road properties or mathematical statistical attributes derived from MLS point clouds. Given that roads typically exhibit relatively steady geometric shape and appearance attributes, these road properties have wide usages in road extraction. For instance, Smadja et al. devised a two-step road extraction method considering the local road shape, in which the first step roughly estimates roadsides by the Random Sample Consensus (RANSAC) algorithm and the second step extracts road boundaries based on road width and curvature [20]. Similarly, Zhang identified candidate road regions by applying an elevation-based Gaussian differential filter, followed by the detection of road edge points using a fixed smoothness threshold [21]. Qiu et al. employed the RANSAC algorithm to perform an initial extraction of the planar ground and subsequently identified road edge points by considering road width and continuity [22]. Yadav et al. refined the process by first filtering out ground points and then extracting road area points based on a set of filtering criteria with fixed thresholds, including topology and smoothness [23].

Further, many beneficial mathematical statistics attributes (e.g., point density and intensity) can be derived from 3D raw point clouds to aid in road extraction. For instance, Wang et al. leveraged trajectory and scan lines to iteratively detect road areas and boundaries based on a predefined height threshold and the direction of scan line segments between adjacent trajectory points [24]. Kumar et al. utilized active contour models to identify road edges, using a grid of adjacent squares that covered the entire 3D point scene [25]. Yang et al. combined multiple feature aggregation levels and contextual features, first segmenting ground points and then applying two rules from trajectory data and point cloud normal vectors to classify filtering points into detailed facilities, including road areas and road edges [26]. Hu et al. employed the cloth simulation method to extract ground points and then applied a point density threshold to identify road area points [27]. Additionally, in urban environments, road edges are often delineated by curbs, prompting extensive research in this area [28,29,30,31,32,33].

In essence, the road extraction methods from the two research branches make use of MPL point clouds from different perspectives to extract road areas and edges. The two sources of information—2D feature images and 3D spatial characteristics—are considered complementary and combining the strengths of both method categories holds the potential to further enhance road extraction performance [3,4]. However, a significant challenge arises from the difficulty in selecting optimal parameters, as the integration of methods increases the number of hyperparameters. In most existing studies, the determination of these hyperparameter thresholds is typically based on either rule-of-thumb practices or trial-and-error approaches. In order to improve the robustness and effectiveness of the integrated method, it is essential to automate the process of optimizing the hyperparameter combination.

Currently, several automatic hyperparameter tuning approaches exist, including gradient optimization, Monte Carlo sampling, and Bayesian optimization (BO) [34,35]. Both gradient optimization and Monte Carlo sampling are parametric methods, meaning they depend on the existence of explicit functions that link hyperparameters to the objective function. In these approaches, the performance of the model is enhanced by adjusting hyperparameters according to predetermined relationships. However, they require a well-defined mathematical relationship between the hyperparameters and the model’s performance metrics, which may not always be valid for complex systems. In contrast, BO is a non-parametric method that does not require explicit knowledge of the analytical form of the objective function. Instead, it utilizes Bayesian probability theory to model the autotuning process, allowing it to derive acquisition functions that guide its search for the optimal hyperparameter set [36,37]. This makes BO particularly effective in handling complex functions with unknown or intractable analytical forms. BO’s ability to optimize complex, high-dimensional, and noisy objective functions has been demonstrated across a range of engineering applications, including tunnel disease detection [38], structural damage detection [39], and microstrip filter design [40]. Nevertheless, thus far, none of the existing studies have developed an integrated method to extract road edges that fuses the advantages of both 2D and 3D characteristics while also fine-tuning the augmented hyperparameters using BO. To fill the gap, this study endeavors to design an autotuning hybrid method incorporating Bayesian hyperparameter optimization for the purpose of road edge extraction from point clouds.

1.2. Objective and Contribution

The primary objective and contribution of this study is to propose a novel autotuning hybrid method with Bayesian optimization for the extraction of road edges in highway systems from MPL point clouds. This method seamlessly integrates the benefits of 2D feature images and 3D spatial characteristics derived from point clouds while employing Bayesian optimization to automatically tune a range of hyperparameters, including high and low pixel gradient thresholds, neighborhood radius, and normal vector threshold. Later, a real-world dataset from national highways in Henan Province is utilized as the case study to evaluate the efficacy of the proposed method against three benchmark methods in two typical road scenarios: straight and curved edges. The experimental results show that the proposed method outperforms the benchmarks in detection quality and accuracy, which underscore its potential as an automated decision-making tool, offering a highly efficient complement to traditional manual road surveying techniques and significantly enhancing the precision and efficiency of road edge extraction.

The outline for the paper is organized as follows: Section 2 describes the hybrid road edge extraction method incorporating Bayesian optimization. Section 3 presents the experimental results of the proposed method on various road environments. Finally, conclusions are drawn in Section 4.

2. Method

Road edge extraction methods that rely solely on either 2D feature images or 3D spatial characteristics are vulnerable to distortions or noise from individual features, potentially leading to missed edge point selection or the extraction of irrelevant points. Combining both characteristics in road edge extraction provides the potential to overcome these limitations by mitigating the risks of single-feature misjudgment. However, integrating additional characteristics inevitably introduces more hyperparameters from two different dimensions into the method. Bayesian optimization (BO) has shown strong efficacy in refining such augmented hyperparameter combinations, particularly when optimizing complex functions with no explicitly known analytical form. Following data preprocessing, the proposed autotuning hybrid method advances with the construction of the hybrid characteristics model and the optimization of hyperparameters using BO, as outlined in the methodological flowchart in Figure 1.

2.1. Data Preprocessing

In the process of data acquisition, it is not uncommon to encounter noise points within the collected point cloud data. These anomalies can arise from a range of external factors, including the precision limitations of the scanning equipment, the inherent properties of the objects being scanned, and various environmental influences such as lighting, weather conditions, or surface reflectivity. The presence of such noises may degrade the quality of the raw data, complicating the subsequent analysis and extraction of meaningful features. As such, data preprocessing becomes a crucial step in the overall workflow. This stage is essential for the early identification and potential removal of noise points, thereby improving the integrity of the dataset. By effectively mitigating these irregularities during preprocessing, the complexity of downstream data processing and feature extraction is greatly reduced, simplifying subsequent data extraction.

In this study, a data preprocessing pipeline widely utilized in the literature is adopted, which includes elevation filtering, RANSAC fitting, and Density-Based Spatial Clustering of Applications with Noise (DBSCAN) [3]. Initially, a significant portion of irrelevant data, such as tall buildings and trees with elevation characteristics not aligned with road edges, is filtered out. Following this, RANSAC fitting is applied based on the correlation of ground point clouds, effectively excluding most irrelevant points. Finally, DBSCAN clustering is employed to perform fine-scale denoising by judging the density distribution of point clouds, ensuring the removal of residual noises.

2.2. Hybrid Road Edge Extraction Model Construction

In the construction of the hybrid road edge extraction model, the feature image-based module and the characteristics-based module are combined to fuse the advantages of both 2D feature images and 3D spatial characteristics derived from point clouds. Specifically, in the feature image-based module, 2D feature images are generated by projecting point clouds onto the XOY plane. Once the pixel information is obtained, the Canny algorithm is applied as the image processing technique. In the spatial characteristics-based module, spatial geometric and mathematical statistical features (e.g., normal vectors) are extracted from the 3D point clouds. These two modules are merged by intersecting their respective extraction results, forming a hybrid road edge extraction model. By leveraging the complementary strengths of both modules, the hybrid model has the potential to enhance performance further. However, the integration of these two dimensions using MPL point clouds leads to an increased number of hyperparameters, adding complexity to the hybrid model.

The proper selection of hyperparameters is critical in determining the performance of the extraction model and forms the basis for its optimization. In this study, we adopt the most commonly-used hyperparameters from previous research in each module [7]. These hyperparameters include high and low pixel gradient thresholds, neighborhood radius, and normal vector threshold, the details of which are described as follows.

(1)

Pixel gradient thresholds

The gradient of a pixel quantifies the variation in brightness between adjacent pixels within an image, serving as a measure of intensity change. In typical images, edges are characterized by sharp transitions in pixel brightness, making pixel gradients an essential tool for edge detection algorithms. Specifically, areas with substantial changes in pixel intensity often correspond to edges, where the most critical information about object boundaries is found. As a result, pixel gradients are widely used to identify these regions. The pixel gradient thresholds are vital in edge extraction from 2D feature images, which contain two components: a high threshold and a low threshold. These thresholds dictate the classification of pixels based on their gradient values. When the 3D point cloud is projected onto a 2D plane to generate feature images, the gradient for each pixel is computed and then compared against the established pixel gradient threshold to assess whether it should be identified as an edge point.

Pixels with gradient values that exceed the high threshold are immediately classified as edge points, as their significant intensity shifts indicate the presence of a boundary. Conversely, pixels with gradient values below the low threshold are discarded, as their minor intensity changes suggest they are unlikely to belong to an edge. For pixels with gradient values that fall between the high and low thresholds, a more nuanced approach is applied. These pixels are accepted as edge points only if they are directly connected to a neighboring pixel with a gradient value above the high threshold. This ensures that even subtle edges are captured without falsely identifying noise or non-edge regions as boundaries. The careful balance between these two thresholds plays a crucial role in achieving reliable road edge detection in highway systems from 2D projected feature images.

(2)

Neighborhood radius

The selection of an appropriate neighborhood radius is paramount in determining the spatial structure required for extracting 3D spatial information from point clouds. This parameter directly influences the calculation of the normal vector, which is essential for defining the local geometric properties of the point cloud. Specifically, calculating the normal vector requires first determining the local plane that best fits a point’s immediate surroundings, which in turn necessitates support from neighboring points. The neighborhood size, typically defined by the radius or the number of neighboring points, governs the extent of the area considered for this calculation. The determination of the neighborhood radius is a delicate balancing act. Striking the right balance ensures that the local geometry is captured with enough precision to preserve key spatial features while minimizing the influence of noises.

The influence of the neighborhood radius on the final normal vector acquisition is profound. When the radius is set too large, the resulting normal vector becomes overly generalized, as the spatial structure encompasses a wide area. This leads to the loss of fine-grained details in the 3D geometry, which may obscure subtle but important features such as small curves, edges, or irregularities in the surface. In such cases, the calculated normal vectors may not accurately reflect the true local variations in the point cloud, resulting in a rough approximation of the surface geometry. Conversely, if the neighborhood radius is set too small, the calculation becomes susceptible to noises, as it relies on a limited number of points to define the local plane. This can cause the normal vectors to fluctuate significantly, particularly in regions where the point density is irregular or where the data includes noisy measurements. The lack of sufficient spatial context in these scenarios prevents the model from accurately capturing the true geometry of the surface, leading to potentially erroneous normal vector calculations.

(3)

Normal vector threshold

The identification of edge points in a 3D point cloud is predicated on the significant deviation between the normal vectors of edge points and those of their neighboring points. To effectively capture this deviation, a carefully chosen threshold for the normal angle between adjacent points can be used to distinguish edge points from non-edge points. The normal angle is calculated as the dot product of the normal vectors of neighboring points, which quantifies the angular difference between them. By comparing this dot product against a specified normal vector threshold, the 3D point cloud can be segmented into edge points and other points based on their geometric characteristics. Balancing this threshold is crucial to achieving accurate and reliable edge detection in point cloud processing.

The choice of the normal vector threshold is a critical factor in this process and should be tailored to the specific application, as well as the noise level inherent in the point cloud data. Opting for a lower threshold enhances the model’s sensitivity to subtle changes in the surface geometry, allowing for the detection of finer, more intricate edge features. However, this increased sensitivity comes with a trade-off: a greater susceptibility to noises, which may introduce false positives and degrade the overall accuracy of the edge detection process. Conversely, setting a higher normal vector threshold reduces the likelihood of noise contamination by imposing stricter criteria for edge identification. While this approach can effectively minimize the influence of noises, it may also result in the exclusion of valid edge points, particularly in regions where the surface geometry is less pronounced or where edge transitions are more gradual. Consequently, the model may fail to capture certain critical features, reducing the precision of the edge detection process.

2.3. Bayesian Hyperparameter Optimization

In the majority of existing research, hyperparameters derived from either 2D image features or 3D spatial characteristics are determined based on either rule-of-thumb practices or trial-and-error approaches. However, when dealing with road edge extraction methods that integrate multi-dimensional features, the optimization of an expanded set of hyperparameters through traditional methods may be unattainable. Bayesian optimization (BO) provides a promising alternative, particularly suited for managing the intricacies of multi-dimensional feature spaces. BO operates by exploiting the relationship between the objective function and the parameters, as inferred from observed data. Its primary objective is to iteratively optimize the hyperparameter configuration, continually refining the parameter space to achieve optimal model performance. By leveraging probabilistic models, BO enables a more systematic and efficient approach to hyperparameter optimization, surpassing the limitations of traditional methods in handling the complexities of multi-dimensional data.

The first key element of BO is the selection of a surrogate model, which serves to approximate the relationship between the parameters being optimized and the objective function. BO does not rely on an explicit functional relationship between the parameter configuration of the road edge extraction model and the accuracy of the extraction. Instead, it utilizes a parameter-free framework to model this interaction. In this study, a Gaussian process model is adopted as the surrogate model to capture the underlying complexities of the parameter space. The second element is the acquisition function, which strategically directs the optimization process. It determines the next point for evaluation based on the computational history, leveraging prior iterations to efficiently explore the parameter space and iteratively improve the model’s performance.

(1)

Gaussian process

A Gaussian process fits a function $f\left(x\right)$ to a finite number of observation points $\left\{x_{1:n},f_{1:n}\right\}$ and predicts its output based on new input points. Gaussian process regression assumes that the vector consisting of the function values of random variables at each observation point obeys an n-dimensional normal distribution $f\left(x_{1:n}\right)\sim \mathcal{N}({\mu }_{1:n},\sum (x_{1:n},x_{1:n}\left)\right)$. The core principle of the Gaussian process involves calculating the mean vector and covariance matrix of a normal distribution based on the given sample values.

The mean vector and covariance matrix are determined by the choice of the mean function and covariance function (also referred to as the kernel function), as shown in Equation (1). The mean function captures the overall trend of the objective function, while the covariance function represents the correlation between random variables. In most practical applications, the mean function is typically set to zero. The squared exponential kernel function, as represented in Equation (2), is one of the most widely used and validated kernel functions. The corresponding covariance matrix can be expressed as Equation (3):

$$f\left(x\right)\sim GP\left(m\right(x),K)$$

(1)

$$k(x_i,x_j)=exp(-1/2\parallel x_i-x_j{\parallel }^2)$$

(2)

$${\mathrm{K}}_{\mathrm{C}\mathrm{M}}=\left[\begin{array}{ccc}k(x_1,x_1)& \cdots & k(x_n,x_1)\\ ⋮& \ddots & \\ k(x_n,x_1)& & k(x_n,x_{\mathrm{n}})\end{array}\right]$$

(3)

where x denotes the data set of the point cloud, m(x) table mean function, $m\left(x\right)$ is usually taken as 0 in practical applications, ${\mathrm{K}}_{\mathrm{C}\mathrm{M}}$ denotes the covariance matrix, and $k(x_i,x_j)$ denotes the square exponential kernel function representing the similarity between point $x_i$ and point $x_j$.

After calculating the mean vector and covariance matrix of the Gaussian distribution, the function probability distribution at any point can be predicted. When predicting a new point $x_{n+1}$, the joint distribution of $f\left(x_{n+1}\right)$ and $f_{1:n}$ follows the Gaussian distribution in n + 1 dimension as given by Equation (4). The Gaussian process possesses a property referred to as closed nature, which signifies that any linear combination of variables within the process, or any conditional distribution derived from observed data, inherently retains its Gaussian form. Due to the closed nature of the Gaussian process, the marginalization as well as the conditionalization of the joint Gaussian distribution obey the Gaussian distribution as given by Equation (5):

$$\left[\begin{array}{c}{f}_{1:n}\\ f_{n+1}\end{array}\right]\sim \mathcal{N}(0,\left[\begin{array}{cc}{\mathrm{K}}_{\mathrm{C}\mathrm{M}}& \mathrm{K}\\ K^{\mathrm{T}}& k(x_{\mathrm{n}+1},x_{\mathrm{n}+1})\end{array}\right])$$

(4)

$$\mathrm{P}\left(f_{n+1}\right|x_{n+1},x_{1:n},f_{1:n})\sim \mathcal{N}(\mu \left(x_{n+1}\right),{\mathsf{\sigma }}^2\left(x_{n+1}\right))$$

(5)

where $\mathrm{K}=\left[\begin{array}{c}k(x_{n+1} , x_1)\end{array}k(x_{n+1},x_2)\cdots \begin{array}{c}k\left(x_{n+1},x_n\right)\end{array}\right]$ denotes the covariance vector between the new point $x_{n+1}$ and the known points, of dimension $n\times 1$, $\mu \left(x_{n+1}\right)={\mathrm{K}}^{\mathrm{T}}{{\mathrm{K}}_{\mathrm{C}\mathrm{M}}}^{-1}{f}_{1:n}$, ${\mathsf{\sigma }}^2\left(x_{n+1}\right)=k(x_{\mathrm{n}+1},x_{\mathrm{n}+1})-{\mathrm{K}}^{\mathrm{T}}{\mathrm{K}}_{\mathrm{C}\mathrm{M}}\mathrm{K}$.

(2)

Acquisition function

The Gaussian process possesses the capability to estimate both the mean and variance for each data point, providing a robust probabilistic framework for modeling the underlying data. Subsequently, the acquisition function addresses the critical task of selecting the next point for iteration, informed by the output of the Gaussian process. In this study, the expected improvement acquisition function is employed, which identifies the point with the highest anticipated improvement as the next candidate for evaluation. This method not only considers the optimal value of the objective function but also accounts for the uncertainty in the predictions, offering a more nuanced approach to optimization. The expected improvement function strikes a balance between exploration (searching new areas) and exploitation (refining known areas), thus helping to prevent the objective function from becoming trapped in a local optimum. The mathematical representation of the expected improvement acquisition function is provided in Equations (6)–(8).

$$I\left(x\right)=max\{0,f(x^{+})-f(x_{n+1}\left)\right\}$$

(6)

$$x_{n+1}=argmax\left(\mathrm{E}\right(max(0,f(x^{+})-f(x_{n+1}\left)\right)\left|D_t\right))$$

(7)

$$\begin{gathered} E\left(I\right)={\int }_{i=0}^{I=+\mathrm{\infty }}I{\displaystyle \frac{1}{\sqrt{2\pi }\sigma \left(x\right)}}exp\left(-{\displaystyle \frac{f\left(x^{+}\right)-\mu \left(x\right)-I}{2{\sigma }^2\left(x\right)}}\right)dI \\ =\sigma (x)\left[{\displaystyle \frac{f\left(x^{+}\right)-\mu \left(x\right)}{\sigma \left(x\right)}}\mathsf{\varphi }({\displaystyle \frac{f\left(x^{+}\right)-\mu \left(x\right)}{\sigma \left(x\right)}})+\mathsf{\phi }({\displaystyle \frac{f\left(x^{+}\right)-\mu \left(x\right)}{\sigma \left(x\right)}})\right] \end{gathered}$$

(8)

After solving the above formula, the expected improvement acquisition function is finally obtained as follows as Equation (9):

$$EI(x)=\left\{\begin{array}{c}(f\left(x^{+}\right)-\mu \left(x\right))\mathsf{\varphi }\left(Z\right)+\sigma \left(x\right)\mathsf{\phi }\left(Z\right) if \sigma \left(x\right)>0\\ 0if \sigma \left(x\right)=0\end{array}\right.$$

(9)

where $\mathrm{Z}={\displaystyle \frac{f\left(x^{+}\right)-\mu \left(x\right)}{\sigma \left(x\right)}}$, $\mathsf{\phi }\left(\mathrm{Z}\right)$ refers to the probability density function (PDF) of the normal distribution, and $\mathsf{\varphi }\left(\mathrm{Z}\right)$ refers to the cumulative distribution function (CDF) of the normal distribution.

Table 1. Process of Bayesian hyperparameter optimization.

Algorithm pseudocode

1: Input: Point cloud dataset x {1: n}, observation of the hyperparameter combination h = {P, R, V}, Total number of iterations T

2: for t = 1, 2, 3…T do:

3:   Calculate the extraction precision ${\mathrm{m}}_{\mathrm{t}}$ of the t iteration based on ${\mathrm{h}}_{\mathrm{t}}$

4:   Find ${\mathrm{h}}_{\mathrm{t}+1}$ based on acquisition function ${\mathrm{h}}_{\mathrm{t}+1}$ = argmax(h|F)

5:   Reconstructed algorithm model based on the new parameter ${\mathrm{h}}_{\mathrm{t}+1}$

6:   Update the Gaussian process with ${\mathrm{h}}_{\mathrm{t}+1}$ as new inputs for the next iteration

7: end for

8: Output the result of all iterations {${\mathrm{h}}_{1:\mathrm{T}}$,${\mathrm{m}}_{1:\mathrm{T}}$}

9: Find the optimal parameter ${\mathrm{h}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ when the extraction precision ${\mathrm{m}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ = min {${\mathrm{m}}_{1:\mathrm{T}}$}

depicts the process of Bayesian hyperparameter optimization. The hyperparameters combination is represented by h = {P, R, V}, where R represents the neighborhood radius, P represents the pixel gradient threshold, and V represents the normal vector threshold. The total number of iterations is T. ${\mathrm{h}}_{\mathrm{t}}$ represents the hyperparameter combination of the t-th iteration. In this study, mean squared error (MSE) is employed as the accuracy loss metric, which denotes the average squared distance between predicted and truth point clouds. ${\mathrm{m}}_{\mathrm{t}}$ denotes the best MSE value at iteration t. By iterative optimization, the optimal hyperparameter combination ${\mathrm{h}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ can be obtained.

3. Results

3.1. Data Background

The proposed autotuning hybrid method with BO is implemented and analyzed based on the point cloud dataset of national highways in Henan Province, as shown in Figure 2. This dataset is collected by the SSW vehicle-mounted laser system, which performs both forward and reverse measurements across the national highway network. The SSW vehicle laser system integrates a variety of sensors, such as the Inertial Measurement Unit (IMU), GNSS, cameras, and odometers, resulting in a highly accurate dataset for the enroute road segments in highway systems.

The detailed characteristics of the case dataset are as follows: For the purposes of this study, two representative road scenarios are selected from the dataset to validate the proposed method. The test sample data, depicted in Figure 3, is derived from road segments along Highways G311 and S102. The point cloud data captures a horizontal span that includes the road surface and extends to structures within a 15-m range on either side of the roadway, while the vertical scanning range covers up to 30 m above the road surface. This dataset encompasses both the spatial coordinates and the reflection intensity of each point, presented in LAS file format. Each sample dataset contains approximately 2 million points, providing an extensive and granular representation of the road environment.

3.2. Method Implementation Details

The implementation of the autotuning hybrid method with BO involves three phases: data preprocessing, hybrid road edge extraction model construction, and Bayesian hyperparameter optimization.

3.2.1. Results of Data Preprocessing

The point cloud data within this test sample area is impacted by noises from sources such as shadows, trees, and other environmental factors, presenting additional challenges for accurate road edge extraction. The primary objective of the data preprocessing phase is to effectively remove irrelevant points and isolate candidate points that are likely to correspond to road edges. The data preprocessing pipeline comprises three steps that are extensively adopted in the literature [3,7,41]. Figure 4 presents the visual progression of each step. The first step, elevation filtering, processes the original point cloud data to exclude a considerable volume of extraneous elements, such as tall buildings and trees, whose elevation profiles diverge from those of road edges. In the second step, RANSAC fitting is employed to delineate the road surface and discard outlier points, such as obstacles, that deviate from the road plane. Finally, in the third step, DBSCAN is utilized to execute fine-scale denoising by assessing the density distribution of point clouds, ensuring the removal of residual noises.

3.2.2. Results of Hybrid Road Edge Extraction Model Construction

The hybrid road edge extraction model comprises two modules: the 2D feature image-based module and the 3D characteristics-based module. In the 2D feature image-based module, 3D point clouds are projected into 2D feature images, and road edges are extracted using the Canny image processing algorithm. The hyperparameters to be determined in this module include the high and low thresholds of the pixel gradient. Initially, these hyperparameters are selected based on empirical values. The process of the 2D feature image-based method is depicted in Figure 5.

In the 3D characteristics-based module, the normal vector is computed as a spatial geometric feature and compared against a predefined threshold to identify road edge points. Following this, DBSCAN clustering is applied to denoise the extraction results, further optimizing their accuracy. The key hyperparameters in this module include the neighborhood radius and the normal vector threshold. Initial values of 0.5 m for the neighborhood radius and 0.1 for the normal vector threshold are adopted. Figure 6 presents the detailed process of precise road edge extraction in the 3D characteristics-based module.

3.2.3. Results of Bayesian Hyperparameter Optimization

The optimization of hyperparameter combinations play a crucial role in determining the performance of the road edge extraction model, which is addressed by BO. Initially, the error in point cloud extraction is employed as the objective function for BO. The hyperparameters optimized in the two modules include the pixel gradient threshold, the normal vector threshold, and the neighborhood radius. The pixel gradient threshold is further divided into the pixel low threshold and the pixel high threshold. The specific values of the hyperparameters during the iterative process are presented in

Table 2. Iterative process of optimization.

Iterations	Pixel Low Threshold	Pixel High Threshold	Neighborhood Radius	Normal Vector Threshold	Error
1	358.3	490.7	6.86	1.34	0.743
2	276.18	600	1.42	0.1	0.104
3	341.06	580.94	6.69	0.53	0.104
4	322.63	484.9	3.59	1.2	0.104
5	348.05	532.87	6.99	1.05	0.104
6	467.49	516.68	5.13	0.1	0.104
7	338.61	481.22	5.62	0.1	0.091
8	282.33	317.11	3.71	0.1	0.091
9	401.02	586.9	0.1	0.1	0.082
10	100	202.09	11.19	0.56	0.082
11	160.86	508.11	1.95	0.16	0.08
12	140.5	298.09	4.98	0.12	0.063
13	136.82	373.99	4.12	0.11	0.058
14	238.8	433.44	3.43	0.1	0.058
15	215.2	340.02	5.37	0.14	0.058
16	324.95	368.74	4.64	0.1	0.058
17	195.22	435.38	13.3	0.11	0.051

. The overall optimization process is illustrated in Figure 7, where it can be observed that the optimal parameters are obtained in the 17th iteration. These final parameters are pixel low threshold = 195, pixel high threshold = 435, neighborhood radius = 13.3, and normal vector threshold = 0.11.

3.3. Extraction Performance of Road Scenarios

In this section, we conduct a comprehensive evaluation of the proposed method’s performance, benchmarked against three methods across two road scenarios: straight and curved edges. The benchmark methods include the scanline-based approach, the feature image-based method, and the spatial characteristics-based method. The scanline-based approach is widely adopted in engineering applications of point clouds [7], while the latter two correspond to the distinct modules within the proposed hybrid framework. To ensure a more rigorous comparison, each module was independently optimized through BO to refine the hyperparameter settings.

The scanline-based method detects road edges by examining geometric variations along scanlines in the point cloud data [5]. It assumes that road edges exhibit continuity and can be delineated by linking points along a scanline with smooth transitions. The bend angle threshold identifies points exhibiting geometric shifts, typically initialized between 0.05 and 0.1. The Taubin smoothing process is applied to mitigate noises with an initial iteration count set to 20. The feature image-based method involves projecting point clouds onto the XOY plane to generate 2D feature images, followed by the application of the Canny image processing algorithm to extract road edges [8]. It assumes that road edges in the 2D projection of the point cloud display distinct gradient changes, which can be identified through gradient computation. The initial high pixel gradient threshold is set as 72% of the maximum gradient intensity, while the low threshold is set as 32%. The spatial characteristics-based method extracts road edges by detecting local geometric changes via identifying sharp variations in normal vectors within 3D point clouds [9]. It assumes that point clouds maintain relatively uniform density within localized areas, ensuring the reliability of normal vector computations. The initial neighborhood radius is set as 13.3, while the normal vector threshold is set as 0.11.

A total of six evaluation indicators, including detection quality (Q), completeness (R), accuracy (P), maximum distance (${\mathrm{D}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$), average distance (${\mathrm{D}}_{\mathrm{a}\mathrm{v}\mathrm{e}}$) and extraction efficiency (Ef), are used to assess the performance. Q, P, and R, respectively, reflect the overall accuracy, completeness, and local accuracy of the extraction method in processing the original point cloud data. ${\mathrm{D}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and ${\mathrm{D}}_{\mathrm{a}\mathrm{v}\mathrm{e}}$ are the maximum and average errors of road extraction based on sampling road edge points. The calculation formula is shown in Equations (10)–(15).

Q = TP/(TP + FP + FN)

(10)

$$\mathrm{R}=\mathrm{T}\mathrm{P}/(\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{N})$$

(11)

$$\mathrm{P}=\mathrm{T}\mathrm{P}/(\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{P})$$

(12)

$${\mathrm{D}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathrm{D}}_{1:\mathrm{M}}\right)$$

(13)

$${\mathrm{D}}_{\mathrm{a}\mathrm{v}\mathrm{e}}={\displaystyle \frac{\sum _{\mathrm{i}=1}^{\mathrm{M}}{\mathrm{D}}_{\mathrm{i}}}{\mathrm{M}}}$$

(14)

$$\mathrm{E}\mathrm{f}={\mathrm{T}}_{\mathrm{j}}$$

(15)

where TP denotes the correct extraction length of road edge. FP denotes the false extraction length of the road edge. FN represents the length of the unextracted road edge. M denotes the number of sampling road edge points. ${\mathrm{D}}_{\mathrm{i}}$ represents the distance from the i th sampling point to the road boundary. ${\mathrm{T}}_{\mathrm{j}}$ represents the computational time of method j.

First, we conduct the performance evaluation of the proposed method against three benchmark methods within the straight-road scenario. The detailed results of the corresponding evaluation indicators are provided in

Table 3. Performance of evaluation indicators in straight-road scenario.

Evaluation Indicator	Scanline-Based Method	Feature Image-Based Method	Spatial Characteristics-Based Method	Hybrid Method with BO
Maximum distance (Dmax)	2.41	2.43	2.52	1.83
Average distance (Dave)	0.30	0.35	0.43	0.27
Efficiency (Ef)	178.4	96.6	77.2	121.0
Quality (Q)	0.866	0.840	0.862	0.895
Completeness (R)	0.987	0.950	0.994	0.980
Accuracy (P)	0.876	0.878	0.867	0.912

, offering a comprehensive assessment of the methods’ performance. Though the spatial characteristics-based method retains a greater volume of point clouds and achieves the highest level of completeness, its effectiveness in accurately extracting road edges is comparatively weak. In terms of computational efficiency, the hybrid method with BO requires more time to obtain the optimal hyperparameter settings compared with either feature image-based or method. Nevertheless, it delivers superior performance in both detection quality and accuracy, achieving an accuracy rate of 91.2% in the straight-road scenario. This highlights the efficacy of integrating the strengths of 2D feature images and 3D spatial characteristics derived from point clouds, enabling the hybrid method to deliver superior road edge extraction in straight-road segments within highway systems.

Second, we evaluate the performance of the proposed method alongside three benchmark methods in the context of the curved road scenario. The corresponding evaluation indicators, presented in

Table 4. Performance of evaluation indicators in a curved road scenario.

Evaluation Indicator	Scanline-Based Method	Feature Image-Based Method	Spatial Characteristics-Based Method	Hybrid Method with BO
Maximum distance (Dmax)	4.29	4.13	4.77	3.18
Average distance (Dave)	0.39	0.44	0.56	0.34
Efficiency(Ef)	250.4	127.2	98.5	154.3
Quality (Q)	0.837	0.811	0.841	0.876
Completeness (R)	0.987	0.926	0.993	0.973
Accuracy(P)	0.847	0.867	0.846	0.898

, provide a detailed analysis of each method’s performance. The results shown in the table reflect patterns similar to those observed in the straight-road scenario. Although the hybrid method with Bayesian optimization (BO) compromises slightly on completeness relative to the spatial characteristics-based approach, it delivers superior performance in both detection quality and accuracy. This improvement can be attributed to the additional computational effort devoted to optimizing the hyperparameter configuration within the multi-dimensional feature space. The hybrid method with BO also achieves an accuracy close to 90% in the curved road scenario, positioning it as an effective and efficient complement to traditional manual road surveying for road edge extraction in highway systems.

Figure 8 presents a graphical visualization of the hybrid method with BO in two road scenarios. The hybrid method with BO exhibits superior performance compared with three benchmark methods in two road scenarios. Nevertheless, inaccuracies in road edge fitting remain inevitable. These errors are attributable not only to the inherent fitting capabilities of the methods per se but also to the quality of the MLS dataset, underscoring potential limitations that persist for all these methods. For instance, during point cloud data acquisition in highway systems, MLS equipment may be influenced by moving vehicles and dense vegetation alongside the roads, leading to the inclusion of false points within the dataset. Additionally, discrepancies in laser reflectivity across various road surface materials can lead to inconsistencies in the distribution of point cloud data. Environmental factors, including weather conditions, further exacerbate these challenges—intense sunlight, for example, can cause over-reflection in laser sensors, resulting in artifacts or data loss. These external environmental factors may contribute to the noises present in the MLS dataset.

4. Conclusions

This study proposed a novel autotuning hybrid method, enhanced by Bayesian optimization (BO), for road edge extraction in highway systems using MPL point cloud data. The proposed approach integrated two complementary modules, capitalizing on the advantages of both 2D feature images and 3D spatial characteristics, while leveraging BO for the automatic optimization of hyperparameters. In the feature image-based module, 2D feature images are generated by projecting point clouds onto the XOY plane. Concurrently, the spatial characteristics-based module extracts geometric and statistical features, such as normal vectors, directly from the 3D point clouds. BO is employed to optimize the hyperparameter set, including the pixel gradient threshold, neighborhood radius, and normal vector threshold, within a multi-dimensional feature space. Afterwards, a point cloud dataset from national highways in Henan Province, China, was utilized as the case study, evaluating performance across two typical road scenarios—straight- and curved-road edges. The experimental results revealed that the hybrid method outperforms three benchmark methods in terms of detection quality and accuracy. It could serve as an advanced decision-support tool, complementing traditional manual road surveying techniques by providing a more efficient and automated solution for road edge extraction in highway systems.

Admittedly, our proposed model comes with some limitations, and the following improvements are suggested: (i) the proposed hybrid method only considers four hyperparameters. In response to the variations in the scale and quality of labeled data, a broader spectrum of methods, such as neural network models, can be employed to conduct extensive comparative analyses; (ii) more surrogate models could be considered in Bayesian optimization to address various point cloud extraction tasks. The authors recommend that future studies could focus on these issues.