Fault identification method of transmission corridor based on 3D R-tree integrated point cloud data segmentation

2.1. Data acquisition and preprocessing

For the identification of hazards in transmission corridors, point cloud data of the transmission corridors and their surrounding environments are collected [13]. Unmanned aerial vehicles (UAVs) can quickly access designated areas without direct human intervention and are unaffected by adverse environmental conditions, enabling the detection of hazardous areas with cost-effectiveness, high efficiency, and operational flexibility [14]. They can acquire high-precision point cloud data in real time and achieve millimeter-level measurement accuracy [15], which facilitates the creation of detailed three-dimensional (3D) models of the transmission corridor and its surroundings. Therefore, this paper integrates UAV and LiDAR technologies to construct a point cloud data acquisition platform for transmission line corridors and their surrounding environments.

The platform built in this study comprises four main components: (1) an UAV serving as the flight platform; (2) a lightweight LiDAR scanning system; (3) a high-precision position and orientation system (POS), consisting of a differential GNSS receiver and an inertial measurement unit (IMU), directly and rigidly connected to the LiDAR to accurately obtain position and attitude information at each scanning instant; and (4) an integrated control and synchronization unit, which ensures strict time synchronization among the LiDAR, POS system, and UAV flight control system while controlling real-time data acquisition and storage.

The specific configuration and technical parameters of the point cloud data acquisition platform constructed in this study are as follows:

(1) UAV Platform: A DJI Matrice 300 RTK unmanned aerial vehicle is employed, with a designed maximum payload of 2.7 kg, a flight endurance of ≥ 55 minutes, and an IP45 protection rating. To enhance stability in the strong electromagnetic environment of transmission corridors, key electronic components (such as the compass and GPS antenna) are additionally equipped with custom electromagnetic shielding covers.

(2) LiDAR System: The platform is equipped with a Velodyne VLP-16 LiDAR scanner. It operates at a wavelength of 905 nm, with a scan frequency of 10 Hz, a vertical field of view of ±15° (30° total), and a 360° horizontal field of view. It generates approximately 300,000 points per second, with a nominal ranging accuracy of ±3 cm (@100 m). The LiDAR is mounted to the UAV airframe via a rigid vibration-damping bracket, and its installation pitch angle is precisely calibrated to optimize scanning coverage of the corridor's conductors and vegetation.

(3) POS System: A NovAtel SPAN-CPT integrated navigation system is used, combining a GNSS receiver with a tactical-grade IMU. In this experiment, real-time kinematic (RTK) differential positioning is implemented. A reference station established at a ground control point with known coordinates provides correction data via a data link, achieving centimeter-level real-time positioning (horizontal: ±1 cm, vertical: ±2 cm) and high-precision attitude measurement (roll/pitch accuracy: 0.05°, heading accuracy: 0.15°).

(4) Synchronization and Control Unit: Based on Pixhawk 4 flight controller hardware and custom-developed time-synchronization firmware, hardware trigger pulses ensure clock synchronization errors among the LiDAR, IMU, and GNSS receiver are less than 1 microsecond. Raw point cloud data, IMU raw data, and GNSS observation data streams are recorded in real time on an onboard 512 GB solid-state drive.

Prior to field operations, professional flight planning software is used to design parallel scanning flight lines based on preset scan angles, point cloud density requirements (> 200 pts/m²), and terrain relief. The flight altitude is set to 80 m (above ground level), the flight speed to 5 m/s, and the side overlap is maintained at no less than 60 %. During flight, the operator monitors the POS system's positioning status and point cloud coverage preview in real time. After acquisition, vendor-specific software is used for integrated post-processing. This involves rigorous calculation and fusion of the raw laser ranging data, high-frequency IMU attitude data, and differential GNSS trajectory data, ultimately generating dense point clouds (in LAS format) with real-world 3D geographic coordinates (WGS84/UTM). The positioning and orientation module utilizes airborne GPS data and base station GPS data for differential processing to determine the platform’s position, while the IMU measures the attitude to determine the UAV's orientation in flight. The data synchronization control module is primarily responsible for controlling data acquisition, synchronization, and recording among the laser scanner, IMU, GPS, and other hardware components.

2.2. Systematic error control and accuracy verification

To mitigate systematic errors inherent in the UAV LiDAR system, a rigorous calibration procedure is performed before and after each field mission at a dedicated calibration field with known three-dimensional coordinates. This calibration determines the lever-arm offsets and angular misalignments (boresight angles) between the LiDAR sensor and the IMU. The calibration process involves flying over the calibration field at multiple altitudes and orientations, allowing for the estimation and correction of systematic ranging and attitude biases.

To verify the absolute georeferencing accuracy of the acquired point cloud, multiple ground control points (GCPs) are established within the survey area using a high-precision GNSS receiver in RTK mode. The coordinates of these GCPs are measured with centimeter-level accuracy (horizontal: ±1 cm, vertical: ±2 cm) and serve as independent validation references. After generating the point cloud, the coordinates of the GCPs extracted from the point cloud are compared against their surveyed coordinates. The root mean square errors (RMSEs) in the planimetric and vertical directions are calculated to quantify the absolute accuracy. The experimental results demonstrate that both the planimetric and vertical RMSEs are less than 10 cm, meeting the accuracy requirements for subsequent hazard identification along transmission corridors.

In addition to absolute georeferencing accuracy, the relative precision of the point cloud is assessed by evaluating the consistency of point-to-point distances on rigid structures (e.g., transmission towers) across overlapping flight swaths. The standard deviation of these distances is computed to quantify relative registration errors. The observed relative registration error is maintained below 5 cm, ensuring reliable extraction of geometric features such as tree canopy boundaries and conductor sag profiles. This level of precision is essential for accurate hazard identification, as the subsequent analysis primarily depends on relative spatial relationships between vegetation and transmission infrastructure.

2.3. Field data collection measures

In practical data collection, to address the complex electromagnetic environment around transmission corridors and ensure flight safety and data quality, the following measures are implemented in this study. First, the DJI Matrice 300 RTK is an industrial-grade UAV platform with an IP45 protection rating and electromagnetic interference resistance; nevertheless, additional electromagnetic shielding materials are installed on sensitive components such as the compass and GPS modules to further mitigate potential interference from high-voltage transmission lines. Second, flight paths are carefully planned to maintain a safe distance from energized lines in compliance with regulations, and a combination of manual and automatic control modes is employed when traversing high-interference zones. Finally, all data are collected under conditions of low wind speed and clear weather to maximize the stability of the flight platform. The LiDAR equipment used in the experiment is lightweight-modified and underwent integrated calibration with the UAV to ensure that its accuracy remains unaffected by minor fluctuations in flight attitude.

The collection of point cloud data from transmission line corridors and their surrounding environments via airborne LiDAR field flights is a complex process. First, a field survey of the transmission corridor and the target data acquisition area is conducted. Based on the survey results and flight mission requirements - and considering factors such as the inherent scanning angle and scanning frequency of the LiDAR system, flight swath width, flight line overlap, and local weather conditions – specialized flight planning software is used to design the flight route. Next, ground base stations are established for dynamic GPS positioning. Before and after data acquisition, the relevant equipment must be calibrated to correct systematic errors and improve the accuracy of the collected point cloud data. Finally, coordinate resolution is performed by integrating airborne GPS data, ground base station GPS data, IMU attitude data, and raw laser ranging data, yielding laser point cloud data of the transmission corridor and its surroundings with real three-dimensional spatial coordinates in the acquisition area.

2.4. Point cloud denoising and tower component segmentation

During the actual inspection of transmission corridors, the accurate classification of tower components is challenging due to the diversity of tower designs and ancillary elements, as well as interference from the extensive point clouds of the tower frame [16]. To address these issues, this paper proposes a segmentation method for the suspension layer of electric towers based on vertical layered projection features, which effectively removes the majority of point clouds from the tower’s skeletal structure that do not contain relevant components.

To fully utilize the symmetry and projection features of an electric tower and to extract its geometric structural parameters, realigning the tower point cloud is necessary [17]. First, the tower is projected onto the YOX plane; the realignment process is illustrated in Fig. 1. This paper defines the direction in which the cross-arm extends as the principal direction of the tower, as shown by the red arrow in Fig. 1(a). By calculating the covariance matrix and eigenvectors of the projected tower point cloud, the eigenvector corresponding to the largest eigenvalue is identified as the principal direction of the tower plane.

Fig. 1Schematic diagram of the tower point cloud realignment process

a) Before realignment

b) After realignment

Subsequently, the angle $\theta$ between the principal direction of the tower plane point cloud and the $Y$-axis is calculated according to Eq. (1) [18]:

where, $v_1$ and $v_2$ represent the components of the principal direction vector of the tower plane point cloud along the $Y$-axis and $X$-axis, respectively.

The tower is then rotated about the $Z$-axis to align the principal direction of the tower head parallel to the $Y$-axis. The coordinates of each point after tower realignment are calculated as follows:

where, $x_p$, $y_p$ and $z_p$ are the coordinates of the tower points after realignment.

The tower redirection results are sensitive to point cloud density and accuracy. A higher point cloud density of the tower leads to a more accurate estimation of its principal direction. When the tower point cloud is sparse or exhibits incompleteness, significant deviation in the principal direction estimation may occur, resulting in inadequate realignment. The accuracy of the point cloud collected by the UAV LiDAR system is within the centimeter level, which reduces deviation in principal direction estimation and contributes to data denoising. During point cloud denoising, traditional methods may mistakenly remove slender structures such as power lines due to their low point density, compromising the integrity of subsequent corridor spatial modeling. To address this, this study introduces a filtering strategy based on local density preservation within the denoising algorithm. By setting density thresholds and evaluating structural continuity, it removes discrete noise points while preserving low-density feature point clouds – such as spatially continuous power lines.

To improve the accuracy of tree and structure extraction, ground points are removed using a progressive morphological filter. This method iteratively applies morphological opening operations with increasing window sizes to separate ground and non-ground points based on elevation differences. After ground removal, the remaining point cloud consists mainly of vegetation, towers, and power lines, which facilitates subsequent segmentation and classification.

2.5. 3DOR-tree index

The core idea of the proposed indexing method is to combine the space-partitioning efficiency of an octree with the dynamic adaptability of an R-tree. This hybrid structure, referred to as the 3DOR-tree, enables fast spatial queries while maintaining a balanced tree organization. Spatial indexing of point cloud data is a data structure that describes and organizes spatial relationships within point clouds to facilitate efficient querying, accessing, and processing [19]. Such an index structure can significantly improve the operational efficiency of point cloud data, which is particularly crucial when handling large-scale point clouds. When applied to identifying tree hazards in transmission corridors, point cloud data from the corridor and its surroundings can be spatially indexed according to location and shape, forming a hierarchical data structure. This structure improves data query and access efficiency and can rapidly locate areas where potential tree hazards exist.

The three-dimensional R-tree index is a tree data structure used for spatial data storage and querying, effectively adapting to the distribution characteristics of spatial data and providing robust and efficient spatial query capabilities. Its fundamental concept is to organize and index data in three-dimensional space using a tree structure. Specifically, the R-tree divides space into a series of overlapping minimum bounding boxes (MBBs) and uses these bounding boxes as nodes for indexing.

R-tree index generation can be categorized into dynamic and static methods. The dynamic generation method aligns more closely with the requirements of multidimensional spatial data management; however, each point can be inserted into the index only after complex operations such as node selection and splitting. This method is challenging to implement for the point cloud data from hundreds of millions of transmission corridors and their surrounding environments, which necessitates a more efficient index creation approach. An octree offers an efficient way to organize three-dimensional spatial data. It recursively subdivides 3D space into eight equal subspaces and continues partitioning each non-empty subspace until a termination condition is met, thereby constructing a hierarchical data structure. This structure effectively adapts to the spatial distribution characteristics of point cloud data from transmission corridors and their surroundings and provides rapid spatial query and location functions. Therefore, this paper employs the R-tree-integrated octree (3DOR-tree) method to establish a spatial index for point cloud data of transmission corridors and their surrounding environments, which optimizes the index structure, reduces unnecessary node splitting and merging operations, and improves index maintenance efficiency. Specifically, the method first uses an octree to partition the point cloud data into different subnodes according to spatial location, building the corresponding octree structure. The octree nodes are then inserted into the R-tree index as R-tree entries, constructing an R-tree data structure that incorporates octree information. The construction process is illustrated in Fig. 2.

Fig. 2Construction of the 3DOR-tree index for the transmission corridor and its surroundings

When constructing the 3D R-tree spatial index for point cloud data of transmission corridors and their surrounding environments, the fan-out parameters of the 3D R-tree must first be determined - specifically, the maximum number of entries $I_{max}$ and the minimum number of entries $I_{min}$ allowed per node. In this paper, an entry refers to a point in the denoised point cloud data of the transmission corridor and its surroundings. Next, the three-dimensional space containing the point cloud data is recursively partitioned using an octree until the number of points in each subspace (i.e., each octree child node) meets a convergence condition, namely that the point count in each leaf node is less than or equal to $I_{max}$. For each child node (octree leaf) generated during octree partitioning, if its point count is between $I_{min}$ and $I_{max}$, the spatial extent (minimum bounding cube) and its contained points are directly inserted into the R-tree as an R-tree leaf node. If an octree leaf node contains fewer than $I_{min}$ points, those points are placed into a temporary local array. Points in this array are then sequentially traversed and recombined into multiple leaf nodes that satisfy the $I_{min}$-to-$I_{max}$ condition, and these leaf nodes are inserted individually into the R-tree. If a node produced during octree partitioning does not satisfy the fan-out parameter condition (i.e., its point count is greater than $I_{min}$ but less than $I_{max}$), its points are added to a global point group. After octree partitioning is complete, points in the global point group are traversed and each is inserted as a single-point entry into the R-tree.

The process for creating a 3D spatial point cloud data index of the transmission corridor and surrounding environment is described as follows.

Input: Point set (denoised point cloud data of the transmission corridor and its surrounding environment), R-tree fan-out parameters $I_{min}$ and $I_{max}$.

Output: 3D R-tree index structure.

The proposed 3DOR-tree construction algorithm follows a hybrid strategy of “organize first, index later”. Its core workflow can be summarized into three main stages:

(1) Octree-based Spatial Coarse Partitioning: The octree is used to recursively and uniformly partition the entire point cloud space into multiple subspaces until the number of points within each subspace reaches a manageable scale.

(2) Node Classification and Packaging: For each octree leaf node, it is categorized into one of three types based on its point count. Subsequently, these nodes (or their points) are packaged into suitable "data blocks" ready for direct insertion into the R-tree.

(3) R-tree Batch Construction and Finalization: The data blocks packaged in the second stage are inserted in batches into the R-tree framework. Finally, a minimal number of unpackaged scattered points are processed to form the final balanced index structure.

This algorithm fundamentally avoids the high cost of node splitting inherent in traditional dynamic R-trees when constructing indexes for massive point clouds, by transforming the unordered insertion of billions of points into the batch processing of a limited number of regular data blocks. The detailed step-by-step description follows.

Step 1: Initialization.

(1) Compute the minimum bounding box (MBB): Determine the MBB for all points (i.e., tuples stored in R-tree leaf nodes) from the denoised 3D point-cloud data of the transmission corridor and its surroundings. This is done by scanning all points in the denoised corridor point cloud and finding their minimum and maximum values along the $x$-, $y$-, and $z$-axes ($x_{min}$, $y_{min}$, $z_{min}$, $x_{max}$, $y_{max}$, $z_{max}$). These extrema define the cube of minimal volume that encloses all points of the denoised 3D point-cloud data, i.e., the minimum enclosing box.

(2) Define the root-node range of the octree: Taking the lower-front-left corner of the minimum enclosing box ($x_{min}$, $y_{min}$, $z_{min}$) as the origin, compute the smallest cubic region that contains the entire denoised point-cloud set of the corridor and its surroundings. This cubic region serves as the spatial range of the octree’s root node.

(3) Create local point arrays $A_1$ and $A_2$, and initialize a global point array. These arrays are used for temporary storage of point data during processing.

Step 2: Octree Space Partitioning and Point Assignment.

Check the tuple count in the current octree node (which may be the root or a child node). If the number of tuples exceeds the maximum fan-out parameter $I_{max}$ of the R-tree, perform octree-based spatial partitioning: the space of the current node is evenly subdivided into eight subspaces, and each tuple is assigned to the corresponding child node according to the position of its minimum bounding box (MBB) center point; then proceed to Step 3. If the tuple count is less than or equal to $I_{min}$, stop further node subdivision.

Step 3: Processing Points That Do Not Satisfy the Fan-out Parameters.

(1) Clear the point array $A_1$. Traverse the child nodes of the octree; if a child node contains fewer points than the R-tree’s minimum fan-out parameter $I_{min}$, add those points to $A_1$. At this stage, the total number of points in $A_1$ is denoted as $M$.

(2) Perform different operations depending on the value of $M$:

If $M<I_{min}$, transfer all points from array $A_1$ to array $A_2$.

If $I_{min}<M\le I_{max}$, pack all points in $A_1$ as a single leaf node and insert it into the R-tree.

If $I_{max}<M\le 2I_{max}$, split the points in $A_1$ into two leaf nodes and insert them into the R-tree.

If $2I_{max}<M$, divide the first $(q-1)I_{max}$ points in $A_1$ evenly into $(q-1)$ leaf nodes and insert them into the R-tree, where $q=\frac{M}{I_{max}}$. The remaining points, denoted as $M^{\text{'}}$, satisfy the relationship given in Eq. (3):

If $M^{\text{'}}=I_{max}$, insert the remaining points as a single leaf node into the R-tree.

If $I_{max}<M\text{'}\le 2I_{max}$, split them into two leaf nodes and insert both into the R-tree.

Step 4: Recursive Space Partitioning.

If the number of points in a child node processed in Step 3 exceeds $I_{max}$, the spatial partitioning and point assignment operations described in Step 2 are recursively applied to that node.

Step 5: Insert Nodes Satisfying the Fan-Out Parameters.

Traverse the child nodes of the octree; if a child node contains a number of points between $I_{min}$ and $I_{max}$, its points are inserted into the R-tree as a leaf node.

Step 6: Process Remaining Points.

After all octree branch splits are completed, the points in array $A_2$ are inserted individually into the R-tree as entries.

Step 7: Termination.

The algorithm concludes, generating a three-dimensional R-tree spatial index that contains all point cloud data of the transmission corridor and its surrounding environment.

In summary, the 3DOR-tree construction algorithm introduces the octree as an efficient spatial pre-processor to quickly organize the point cloud into batches based on spatial location. These spatial batches (or their recombined forms) are then fed as whole units to the R-tree for batch index construction. This division of labor leverages the strengths of both structures: it preserves the excellent dynamic querying and balancing capabilities of the R-tree while overcoming its inherent inefficiency during large-scale data insertion. This makes it particularly well-suited for managing large-scale, elongated, belt-shaped point cloud data such as that from transmission corridors.

The core advantage of the 3DOR-tree index proposed in this paper lies in its integration of the complementary characteristics of the octree and the R-tree, which makes it superior to either structure alone in terms of index construction, update efficiency, and tree structure balance.

In terms of node operation optimization, traditional dynamic R-trees may trigger recursive splitting from leaf to root nodes for each point insertion when handling massive point clouds, resulting in substantial computational overhead. While the pure octree is efficient in spatial partitioning, its static and uniform subdivision tends to generate numerous empty nodes when dealing with linearly distributed and non-uniform point clouds such as those of transmission corridors, leading to storage waste and increased query depth. The 3DOR-tree addresses this issue through a cooperative division of labor: the octree first acts as an efficient spatial allocator, rapidly organizing the point cloud into cubic units with good spatial locality. These units (or their compliant subsets) are then inserted as bulk data units into the R-tree as a whole. This strategy transforms billions of individual point insertion operations into a significantly smaller number of batch node insertions, fundamentally reducing the frequency of node splitting and merging within the R-tree. Moreover, the pre-partitioning by the octree ensures that spatially adjacent points remain close in physical storage, greatly enhancing cache hit rates for subsequent spatial queries.

Regarding the maintenance of tree balance during dynamic updates, the 3DOR-tree demonstrates strong adaptability. When new point cloud data is acquired during inspections, the system first locates the corresponding octree spatial unit. Update operations (insertion or deletion) are strictly confined to the local R-tree subtree associated with that unit. The inherent balancing algorithms of the R-tree ensure that the subtree is rebalanced after the update. Since the top-level framework of the octree remains stable, this “local disturbance, local rebalancing” model avoids global index reconstruction. Furthermore, by strictly adhering to the R-tree’s fan-out parameters, the system automatically prevents node overflow or underflow, thereby maintaining the overall index structure’s performance without degradation over long-term and continuous dynamic updates.

In summary, the 3DOR-tree is not a simple superposition of two structures but achieves a unified approach of “rapid spatial organization” and “efficient dynamic balancing” through hierarchical collaboration between the octree and the R-tree. This makes it particularly suitable for managing large-scale, periodically collected point cloud data in transmission corridors.

Through the aforementioned operations, a three-dimensional spatial point-cloud data index for the transmission corridor and its surrounding environment is established, yielding a 3D R-tree index structure.

In summary, the three-dimensional R-tree-integrated octree method is employed to construct the index structure for point cloud data of the transmission corridor and its surrounding environment. This approach primarily uses the spatial partitioning strategy of the octree to allocate points from the denoised corridor and surrounding environment point cloud to the same or adjacent nodes, thereby improving index generation efficiency. By inserting points in batches using nodes as insertion units, the time-consuming process of point-by-point insertion is avoided. Concurrently, a dynamic generation method is applied to construct the R-tree, which ensures optimal spatial adaptability while maintaining a balanced tree structure and efficient space utilization.

By defining query conditions (such as the minimum safe distance between the transmission corridor and surrounding trees), the constructed three-dimensional R-tree index of the transmission corridor and its environment can be used to rapidly locate potential areas containing concealed tree hazards. Subsequently, regions that do not satisfy the query parameters are quickly filtered out, effectively narrowing the search scope for personnel and thereby improving the efficiency and accuracy of subsequent hazard identification in transmission corridors.

2.6. Cylinder-constrained Euclidean clustering for point cloud segmentation

To accurately extract the transmission corridor point cloud from cluttered environments, a cylinder-constrained Euclidean clustering algorithm is proposed. The key idea is to restrict the clustering search space to a cylindrical volume defined by the corridor’s linear extent, thereby filtering out central tree hazard points while preserving the corridor structure. After locating potential areas with tree hazards, it is necessary to extract the point cloud of the transmission corridor and its surroundings, effectively segmenting the data to obtain a relatively pure point cloud. Pure point cloud data accurately reflect the geometric shape of the transmission corridor and its environment, which facilitates measuring the distance between the corridor and potential tree hazards and determining the presence of safety risks. Euclidean clustering is a segmentation method based on Euclidean distance measurement, primarily used for point cloud data segmentation [20-21]. Applying this method to hazard identification in transmission corridors enables accurate and efficient extraction of pure point clouds of the corridor and its surroundings.

The clustering method processes a set of $\mathcal{l}$ points from the point-cloud data, initially regarded as $\hslash$ separate partitions. It defines an affinity measure between points in the $\mathcal{l}$ -dimensional space, then iteratively merges the two closest partitions into one and recomputes inter-partition distances until all distances exceed a fixed threshold. Applying Euclidean distance in cluster analysis allows data similarity to be assessed based on point-to-point distance: a shorter distance implies higher similarity, leading to grouping such points into one cluster; a larger distance indicates insufficient similarity, meaning the points belong to different clusters [22]. The traditional Euclidean clustering algorithm proceeds as follows: from the localized point cloud within a potential tree-hazard area, select a query point $s$, define a distance threshold $g$, and use the 3D R-tree index of the corridor and its surroundings to retrieve the $\hslash$ nearest points. Then compute the Euclidean distance $g_{ij}$ between each of these points and the query point.

The threshold $g$ in this paper is determined according to the average point-cloud spacing and the regulatory safe clearance between trees and power lines. Its value strongly influences cluster completeness and segmentation accuracy; a sensitivity analysis will be presented in later experiments. If $g_{ij}$ is smaller than $g$, the corresponding point is added to cluster $Q$. This process repeats until no new points are added to $Q$. Subsequently, a new query point can be chosen, and the procedure is repeated to identify other clusters in the point-cloud data of the transmission corridor and its surroundings [23].

However, due to the complex grid-like environment of the transmission corridor, the standard Euclidean clustering algorithm alone is insufficient to separate the corridor point cloud and its surrounding environment from closely spaced objects within the data. Therefore, this paper proposes a new segmentation algorithm for point cloud extraction, namely the Euclidean clustering algorithm with cylinder k-point constraints. Its core lies in defining a three-dimensional cylindrical constraint consistent with the spatial orientation of the corridor and embedding it as a spatial filter into the clustering process.

The geometric definition of the cylinder is based on the central axis fitted from the corridor point cloud. This axis is determined by two endpoints, $p_1$ and $p_2$. Centered on this axis, a cylindrical space is constructed with a radius $R$ (typically half the safe width of the corridor plus a margin) and a length covering the entire axis. The criteria for determining whether any point $q$ lies within this constrained cylinder include: first, the perpendicular distance from point $q$ to the central axis must not exceed the radius $R$; second, the projection of point $q$ along the axial direction must fall within the range defined by the endpoints $p_1$ and $p_2$.

During the clustering process, this cylindrical constraint is used to dynamically filter neighboring points. Specifically, when expanding a cluster from a seed point, the algorithm quickly retrieves all candidate neighboring points within a Euclidean distance threshold via the 3DOR-tree index. However, not all candidate points are incorporated into the current cluster; only those that simultaneously satisfy the aforementioned geometric criteria for being inside the cylinder are merged. This strategy strictly confines the spatial exploration range of clustering to the physical passage of the corridor, thereby effectively excluding interfering objects on both sides, above, and below, and ensuring that the extracted point cloud clusters purely correspond to target structures such as transmission lines and towers.

For tree-hazard point clouds, which are typically located near the center of the transmission corridor, the resulting high density and disordered distribution make it impractical to traverse the entire corridor point cloud directly to locate the points at both ends of the corridor. Therefore, a density-gradient-based inverse search is employed to find the point cloud at one end of the corridor and to identify the $K$ nearest points at that end [24]. The entire corridor and its surrounding point cloud are constrained within the spatial range of the corridor itself. Consequently, this approach effectively filters out the central tree-hazard point cloud.

If, within the point cloud space of the transmission corridor and its surrounding environment, a set of $n$ points constitutes the point cloud data $\left\{x,i=\mathrm{1,2},\cdots ,n\right\}$, the central density estimation function for a point $x$ can be defined as:

where, $u$ denotes the bandwidth, $d$ is the spatial dimension, and $k\left(x\right)$ is the kernel function.

The kernel function $k\left(x\right)$ is given by:

where, $v_{k,d}$ is a normalization constant that enforces the constraint that the integral of $k\left(x\right)$ equals 1. The point with the highest probability density, obtained by differentiating Eq. (5), is given by:

where, $\nabla \widehat{F}\left(x\right)$ represents the gradient of $F\left(x\right)$, used to find that makes the point $F\left(x\right)$ maximum (i.e., the point with the highest probability density). Here, $k^{\text{'}}$ represents the derivative of $k\left(x\right)$.

By defining $\mu \left(x\right)=-k^{\text{'}}\left(x\right)$, the following relationship can be obtained:

The gradient vector at the current point in the point cloud data can be obtained from Eq. (7) as:

The gradient direction calculated by Eq. (8) indicates the orientation toward the lowest point cloud density region, thereby guiding the subsequent search for corridor endpoint $k$-points. By iteratively repeating this process, the direction with the lowest point cloud density can be determined. In complex point cloud scenes, the transmission corridor often appears as an elongated structure with relatively low point density. Identifying this lowest-density direction allows the corridor's characteristic structure to be effectively discerned, thereby establishing a foundation for subsequent point cloud extraction and tree-hazard identification.

In the Euclidean clustering algorithm with cylinder $k$-point constraints, determining the initial $k$ points – i.e., selecting a set of points that represent the position of one end of the transmission corridor – is a crucial step. These points are frequently used as starting seeds for the subsequent clustering process to facilitate locating and delineating the corridor. The following describes the procedure for finding these initial k points:

(1) In the point cloud of the transmission corridor and its surrounding environment $\left\{x,i=\mathrm{1,2},\cdots ,n\right\}$, an unmarked point is randomly selected as the initial center point $B$.

(2) All points within a radius $r$ of $B$ are identified as neighboring points and are considered to belong to the same transmission corridor cluster. These points are stored in a set $G$. Each point in $G$ is assigned to a class cluster $Q$, and the count of points in $Q$ is incremented by one. This process helps to identify the most frequently occurring class clusters, i.e., the most probable transmission corridor clusters, in subsequent steps.

(3) For the center point $B$, the vector from $B$ to each point in $G$ is computed, and these vectors are summed to obtain a shift vector $B_{shift}$.

(4) Then, $B$ is updated as $B=B-B_{shift}$, moving $‖shift‖$ units toward the sparse region of the point cloud. Here, $shift$ denotes the magnitude of the shift vector.

(5) The above steps are repeated until the magnitude of $shift$ is less than a predefined threshold, at which point all relevant points in the transmission corridor and its surrounding environment are searched and labeled.

After completing the above steps, different point-cloud clusters are obtained. The access frequency of all points in the point cloud $\left\{x,i=\mathrm{1,2},\cdots ,n\right\}$ is examined, and the cluster with the lowest frequency is selected as the $k$-point cloud at one end of the transmission corridor.

The parameter k represents the number of seed points used to initialize clustering and locate the ends of the corridor. To enhance the algorithm's robustness and generalizability across different environments, the value of $k$ is not fixed but is adaptively determined based on the local characteristics of the point cloud data.

The adaptive strategy primarily relies on local point cloud density and scene complexity. First, during the preprocessing stage, the algorithm estimates the local point cloud density along the flight trajectory in the corridor's direction. The base value of k is proportional to this density and is standardized by considering the theoretical cross-sectional area of the corridor, ensuring it reflects the structural richness within the current scanning area. Second, a scene complexity factor is introduced for adjustment: the undulation of terrain or the layering variation of vegetation is quantified by calculating the standard deviation of elevation in the candidate region. In areas of high complexity, $k$ is appropriately increased to obtain a more stable and representative initial seed set, thereby countering noise and local irregularities. Finally, to ensure computational efficiency and stability, the value of k is constrained within an empirically reasonable range. This range is determined through extensive experimental validation across various typical corridor environments, balancing the representativeness of seed points with the avoidance of unnecessary computational overhead.

Through the above approach, the geometric definition of the cylindrical constraint and the adaptive rule for k jointly ensure that the clustering algorithm can accurately and stably segment the main structure of the transmission corridor from complex and variable inspection point clouds, laying a solid foundation for subsequent hazard distance calculations.

By identifying the extremum among these k points at one end, the cylindrical range of the corridor at that end can be determined, as given by:

The cylindrical range defined by Eq. (9) is subsequently applied as a spatial constraint in the Euclidean clustering process, restricting the search for corridor point clouds within this specified range and thereby excluding interfering objects outside the corridor. The Euclidean clustering process for the corridor and its surrounding point cloud data under cylindrical constraint is described as follows:

(1) For a query point $s$, its neighborhood within the point cloud data of the transmission corridor and its surroundings is searched to obtain a set of proximate points.

(2) It is then determined whether the number of points in cluster $Q$ is smaller than a predefined threshold $K$. If this condition holds, the distance from each point to $s$ is computed and compared with a distance threshold $g$. Points with distances smaller than $g$ are placed into $Q$. If the condition does not hold, the distance to point $s$ is calculated only for those points satisfying a cylindrical spatial constraint; points with distances below $g$ are then assigned to $Q$.

(3) Next, it is checked whether the number of points in $Q$ exceeds the total number of points in the transmission corridor and the surrounding environment. If not, a new point (other than $s$) is selected from $Q$ as the next query point, and the procedure returns to the step 1 to further expand cluster $Q$.

(4) When no more points can be added to cluster $Q$, the clustering process terminates after all points are assigned to a cluster, completing the extraction of the transmission corridor and its surrounding point cloud. The resulting clusters constitute the point-cloud collection of the corridor and its environment, with each cluster potentially containing data of transmission lines, towers, vegetation, and other infrastructure within the corridor. After extracting the corridor point cloud, the remaining non-corridor points (primarily vegetation and ground) are further processed to isolate individual tree crowns. Height-based filtering is applied to remove ground points (detailed in Section 3.2), followed by Euclidean clustering with an adaptive distance threshold to segment individual trees. Specifically, points belonging to the same tree crown are grouped based on spatial proximity and vertical continuity. The centroid and highest point of each cluster are then computed to represent the tree’s position and height, which are used for subsequent hazard-distance calculation.

2.7. Tree hazard identification

In this study, the random sample consensus (RANSAC) algorithm combined with least-squares fitting is applied to the pure point cloud of the transmission corridor and its surroundings extracted in Section 2.4.1, in order to fit a spatial equation representing the corridor and determine its spatial location. Tree hazards are then identified by computing the distance between the tree-crown point cloud and the fitted transmission corridor.

The point cloud of the transmission corridor serves as the observation data for fitting the spatial equation. First, the RANSAC algorithm is employed to identify the two most distant points within the corridor point cloud along its linear direction. These two points are used as endpoints to define an initial straight-line model. This model then scans the corridor point cloud to locate points that lie close to the line; these are designated as inliers. All inliers are subsequently used to compute a new line model. The fit between the inliers and the model is evaluated to determine whether the linear model is satisfactory. If the criterion is not met, the process is iterated until the model converges. Once the RANSAC algorithm finds a sufficiently good model (i.e., one with enough inliers), least-squares optimization is applied to refine it by minimizing the sum of squared perpendicular distances from all inliers to the line. As a result, a more accurate linear model is obtained, which better represents the actual spatial location of the transmission corridor.

In the implementation, the above process is crucial for determining whether the model can reach a predefined fitness threshold $\gamma$, estimating the minimum number $\psi$ of inliers required for the model to be accepted, and deciding the number of iterations $T$ needed for the algorithm to obtain a suitable model. The initial value of $\gamma$ can be set based on the average point spacing or empirical values, and then adjusted according to the model quality during iteration. The parameter $\psi$ can be determined empirically or through experiments. The number of iterations $T$ must be determined by the following formula.

In the algorithm implementation, determining whether the model can reach a predefined goodness-of-fit threshold $\gamma$, estimating the minimum number of inliers $\psi$ required for the model to be accepted, and deciding the number of iterations $T$ needed for the algorithm to obtain a suitable model are crucial. The initial value of $\gamma$ can be set based on the average point spacing or empirical values and adjusted according to the model quality during iteration, while $\psi$ can be determined empirically or through experimentation.

The theoretical value for the number of iterations $T$ can be derived from a probabilistic model. Assume that fitting a linear model for the transmission corridor requires a minimum sample size of $m$ (typically $m$ =2 for a straight line). Let $\rho$ be the probability that a randomly selected point from the corridor point cloud is an inlier, i.e., the inlier ratio. To ensure that after $T$ iterations, the probability of at least one random sample consisting entirely of inliers (i.e., successfully finding a valid model) is not less than a set confidence level $\beta$ (usually 0.99 or 0.95), the following derivation is required. The probability that a single sample consists entirely of inliers is ${\rho }^m$, and the probability of failure for a single sample is $1-{\rho }^m$. Therefore, the probability that all $T$ iterations fail is ${\left(1-{\rho }^m\right)}^T$.

By requiring this failure probability to be less than $1-\beta$, the minimum number of iterations $T$ that must satisfy the condition can be solved as:

Based on Eq. (10), the iteration count is dynamically adjusted during the RANSAC fitting process to ensure the algorithm maintains a high confidence level in finding a valid model. The inlier ratio $\rho$ in the formula is an unknown parameter before algorithm execution. This paper employs a data-driven adaptive strategy to estimate it: in the initial stage of the algorithm, an exploratory sampling and model validation are performed, and the initial inlier ratio is calculated based on the validation results:

where, $\theta$ is the number of inliers for the current model, and $\delta$ is the total number of points involved in the fitting. During the main loop of RANSAC, $\rho$ is dynamically updated to the inlier ratio corresponding to the best model found so far. This strategy allows the iteration count $T$ to adaptively adjust according to the actual data conditions, optimizing computational efficiency while ensuring confidence in model discovery.

After obtaining the spatial linear equation of the transmission corridor point cloud through the above method, the distance between the corridor point cloud and non-corridor point clouds (primarily canopy point clouds) can be calculated, thereby completing the identification of tree hazards.

After obtaining the spatial linear equation of the transmission corridor point cloud through the above method, the distance between the corridor point cloud and non-corridor point clouds is calculated, with three-quarters of the tree height set as the height threshold $l$. The entire point cloud data of the transmission corridor and its surroundings are scanned to locate tree points higher than $l$, which are regarded as canopy point clouds. For each point in the canopy point cloud, its distance to the spatial linear equation of the corridor is computed and compared with a predetermined safe clearance distance $d_{safe}$ (the minimum safe distance between trees and the transmission corridor). If the distance at any point is less than $d_{safe}$, that point is marked as a tree-hazard point, thereby completing the identification of tree hazards along the transmission corridor. The value of $d_{safe}$ is defined based on the “Regulations on the Protection of Power Facilities” and the growth characteristics of trees. It directly influences the accuracy of hazard identification and the false-negative rate, as discussed in Section 3 of this paper.

The minimum safe distance $d_{safe}$ is defined through a three-layer fusion approach: first, strict adherence to the minimum clearance distance specified in power-facility protection regulations for the specific voltage level (e.g., 4.2-7.0 m for 220 kV lines); second, addition of a vegetation growth margin based on estimated annual tree growth (approximately +1.0 m/year for fast-growing subtropical species); and third, compensation for UAV LiDAR measurement errors (±5-10 cm accuracy requiring +0.3-0.5 m robustness margin), resulting in a comprehensive threshold that balances regulatory rigidity, dynamic prediction, and data uncertainty.

To ensure the scientific validity and applicability of the tree-hazard identification criteria, this study establishes a clear standardization process for determining the core parameter – the minimum safe-distance threshold between trees and the transmission corridor. This threshold incorporates the following factors: first, strict adherence to the minimum clearance distance specified for the specific voltage level in the power-facility protection regulations applicable to the experimental area; second, the addition of a reasonable safety margin based on estimated annual tree growth; and finally, consideration of the measurement-accuracy error of the UAV LiDAR point cloud data to ensure robustness in threshold setting. Accordingly, the value of $d_{safe}$ for the experimental corridor is determined as $X$ meters. Simultaneously, the height threshold for canopy-point-cloud screening is set to three-quarters of the tree height to focus on the upper tree portions that may pose an actual threat. By comparing the computed distance between canopy points and the corridor spatial equation with $d_{safe}$, objective and standardized hazard identification is achieved.