Hierarchical Point Cloud Analysis for 3D Defect Detection of Annular Welds

Abstract

Annular fillet welds on guide rods lie in stress concentration zones and are susceptible to fatigue failure under dynamic suspension loads, necessitating accurate quality inspection. However, their complex three-dimensional topography makes it difficult for conventional methods to balance detection accuracy and efficiency. This paper presents a geometry-driven 3D point-cloud analysis framework integrating weighted geometric template matching, unified height referencing, and decoupled dual-branch defect discrimination. The framework first locates the weld region via weighted template matching with a dynamic early-termination strategy and establishes a unified height datum through multi-stage filtering and RANSAC plane fitting. Defects are then decoupled by height characteristics: extreme height screening isolates oxide inclusions (ISO 6520-1 No. 303), while quantile dual-threshold layered analysis distinguishes insufficient throat thickness (ISO 6520-1 No. 5213) and excessive convexity (ISO 6520-1 No. 503). Experiments on 100 welds sampled across five production batches, evaluated under EN ISO 5817 quality level C, achieve zero false positives for oxide inclusions and insufficient-throat cases and an 88.2% detection rate for excessive convexity, delivering robust performance under the tested conditions.

1. Introduction

The guide rod serves as an essential component for force transmission and positioning between the suspension system and the vehicle body [1]. The annular welds at both its sides connecting the guide rod body and the bushing function as a stress-concentrated region where loads in three directions are primarily transmitted. These annular joints are circumferential fillet welds (FW), and their quality assessment is governed by international standards: EN ISO 6520-1 [2] for the classification of geometric imperfections and EN ISO 5817 [3] for the specification of quality levels and acceptance limits. However, improper welding parameters (e.g., heat input or wire feed speed) [4] and environmental disturbances (e.g., humidity, temperature fluctuations, surface contamination) can lead to defects such as depressions, undercutting, pores, solid inclusions, and insufficient weld dimensions. These defects not only diminish the strength of the weld connection and induce stress concentration but also increase the risk of weld cracking and failure of the guide rod connection under prolonged alternating loads [5]. In severe instances, this may result in loss of wheel control, potentially leading to traffic accidents [6,7].

Automated weld inspection must contend with multiple sources of uncertainty, including variable part positioning, specular reflections from metallic surfaces, and low-contrast boundaries arising from complex illumination conditions [8,9]. Intelligent visual inspection has thus become the predominant approach for nondestructive weld evaluation, with extensive research devoted to defect segmentation, classification, and quantitative characterization [10]. Significant progress has been made in two-dimensional image-based analysis of segmentation. Wang et al. [11] combined Otsu thresholding, edge detection, and density clustering to robustly isolate defect regions in high-noise X-ray weld radiographs. Tang et al. [12] developed MCFNet, a hybrid CNN-Transformer architecture designed to suppress heavy noise and scanning interference fringes in TOFD imagery. Yang et al. [13] proposed CvT-UNet, integrating CNN and Transformer components to address indistinct weld pool boundaries under dynamic lighting and impurity interference.

Concurrent advances have also been reported in defect classification and quantitative characterization. For defect classification, a CNN-based method utilizing three-directional weld images was applied to identify penetration defects in robotic arc-welded aluminum alloys [14]. Shu et al. [15] employed a stacked autoencoder to rapidly filter defect-free weld regions, then combined this with an improved SCS-Vibe background extraction algorithm to achieve efficient and accurate detection of typical aluminum alloy surface defects such as porosity, arc craters, and undercut. YOLOv5 was also leveraged for weld type classification and localization [16], followed by key feature point extraction via a spatial distance method for quantitative evaluation of weld geometry and imperfections [17].

Reconstructing weld and defect morphology overcomes the inherent limitations of planar image-based detection, specifically, the inability to capture depth, volumetric data, or curved-surface topography. Ou et al. [18] achieved accurate weld seam localization and boundary extraction via region-growing point cloud segmentation combined with point cloud normal filtering, reducing the average boundary extraction error by 0.263 mm compared to conventional methods in remanufacturing blade weld inspection. Wang et al. [19] employed the CIICEA algorithm for laser centerline extraction and the K-QCBFA algorithm for spatial curve fitting. Utilizing a defect detection criterion based on height-difference curves, their approach attained a recognition accuracy of 97.44% and enabled the classification of both convex and concave defects. Liu et al. [20] developed a 3D vision measurement method based on structured light reconstruction and a custom-designed GRP-PTv2 point cloud semantic segmentation network. Integrating a slicing algorithm with farthest-point sampling for high-precision weld toe localization, this method allows quantitative measurement of key weld dimensions (width and reinforcement height) with dimensional deviations consistently within 0.2 mm. Ji et al. [21] adopted an improved lightweight Transformer segmentation network for accurate laser stripe extraction and 2D weld region segmentation, supplementing the detection results with auxiliary geometric verification and 3D morphological reconstruction via structured-light depth calibration.

Furthermore, complementary 2D and 3D information further improves efficiency and precision [22,23]. Parikh et al. obtained full-scale depth morphology data of FSW weld surfaces using a 3D optical microscope, extracted defect contours from the depth data via morphological operations, and further conducted micro-scale quantification of geometric features including effective length, width, average depth and effective volume of defects based on the depth information, thus realizing the detection of surface defects such as groove-like depressions and surface underfill [24]. Hu et al. [25] integrated CBAM-enhanced YOLOv8n seg for 2D instance segmentation, aligned 3D point clouds via camera projection, and achieved pixel-level point cloud segmentation of irregular weld ROIs (Region of Interest). Liu et al. [26] detected pores, bulges, and humps by analyzing 2D height profile curvature and 3D surface curvature, then reconstructed nominal bead contours for full-dimensional defect evaluation.

Collectively, 2D methods perform effectively for localization and recognition on planar or near-planar weld surfaces, but lack depth quantification [27]; 3D reconstruction provides precise geometric measurement yet struggles with multi-class defect identification, and real-time processing [28]. By contrast, 2D/3D fusion frameworks exploit complementary modal cues to enable simultaneous defect detection and quantitative characterization for three-dimensional weld geometries. This paper presents a multi-stage framework for defect detection on automotive guide rod annular welds, integrating template matching, 3D point cloud processing, and decoupled feature extraction. A weighted similarity metric is introduced to optimize template matching, facilitating rapid and accurate localization of the weld region through a dynamic threshold strategy combined with an image pyramid scheme. Following the location of the region, the acquired 3D point cloud undergoes multi-stage filtering and RANSAC plane fitting to achieve high-fidelity 3D reconstruction, which provides precise weld height information. Building on the reconstructed geometry, a dual-branch detection algorithm is developed that incorporates decoupled defect features to enhance discriminability. One branch focuses on height extraction to identify oxide inclusions that protrude above the part surface, while the other employs quantile double-thresholding to separate morphological features from lower weld layers.

2. Methods

2.1. System Process

During acquisition, the guide rod is clamped at its middle to ensure stable orientation and parallel alignment with the test bench. A Shensight SR7140 3D laser profilometer (Shenzhen Shensight Technology Co., Ltd., Shenzhen, China; 405 nm blue Class 2M laser, 0.5 μm Z-repeatability, 8 μm X-resolution, 2500–8000 Hz, 3200 profiles/frame) captures the surface data [29,30]. Data were acquired at 4000 profiles/s with a robotic scanning velocity of 60 mm/s and a nominal stand-off distance of 140 mm. The spatial sampling interval is 8 μm along the laser line (X-axis, native sensor resolution) and 15 μm along the scanning direction (Y-axis, computed as scanning velocity divided by acquisition frequency).

Prior to each session, hand-eye calibration establishes the sensor-to-robot coordinate transformation. The laser beam is projected vertically onto the guide rod surface, with the laser line aligned perpendicular to the weld circumference. Two guide rods are clamped side by side with their longitudinal axes parallel to the robotic scanning direction, and a belt-driven transverse mechanism perpendicular to the rod axis transfers the sensor between the front and rear weld sides. For each pair of workpieces, the robotic arm first scans the front weld side of the first guide rod, then the belt mechanism moves the sensor to its rear side to complete the second scan. The same front-to-rear scanning sequence is then repeated for the second guide rod. The robotic arm moves it along a straight path parallel to the rod axis at constant velocity, stacking successive cross-sections to reconstruct the complete front and back weld morphologies under identical geometric conditions. The active 405 nm laser and a narrow-band optical filter reject ambient light without auxiliary illumination.

All acquisitions are performed in fixed single-exposure mode to ensure consistent intensity response. Before each session, the sensor is calibrated using a traceable gauge block (Z-axis) and a precision flat plate (plane), while RANSAC plane fitting serves as an in-situ correction to compensate for residual tilt and fixture misalignment. The full pipeline takes 200 ms per weld on an industrial PC (Intel Core i7 CPU, 64 GB RAM, Windows 11, no GPU acceleration); missing points in Figure 1 are due to fixture occlusion. The guide rods are welded via GMAW (ISO 4063 [31], process 135) with 1.2 mm low-silicon wire and pure CO₂ shielding at 160–220 A, 15–21 V, and ~60 cm/min. The CO₂-rich atmosphere promotes discrete Si/Mn deoxidation products protruding above the weld surface. These discrete Si/Mn deoxidation products are classified as oxide inclusions (No. 303) under EN ISO 6520-1. The subsequent processing pipeline employs feature-enhanced template matching for precise localization, followed by multi-stage filtering and RANSAC height referencing. Decoupled dual-branch feature extraction and hierarchical analysis then effectively discriminate insufficient throat thickness and excessive convexity.

2.2. Rapid and Accurate Localization of Welding Area

Edge point positions and gradient directions, derived from an enhanced Canny operator, serve as matching features for similarity calculations. This approach facilitates the rapid acquisition of accurate gradient directions and sub-pixel edge coordinates. To satisfy real-time processing requirements, a search strategy is implemented that integrates dynamic threshold judgment with an image pyramid algorithm to ascertain precise coordinates and rotation angles. Furthermore, weld seam area is rapidly extracted by a rectangular bounding box.

2.2.1. Improved Canny Edge Extraction

A Canny edge detection operator is adopted for gradient extraction, with the original 2 × 2 template replaced by a 3 × 3 template. In addition to the horizontal and vertical templates of the Sobel operator, gradient templates at 45° and 135° are incorporated [32]. The resulting gradient magnitude ti and gradient argument ui are defined by Equations (1) and (2):

$$t_i=\sqrt{t_x^2(i,j)+t_y^2(i,j)+t_{45°}^2(i,j)+t_{135°}^2(i,j)}$$

(1)

$$u_i=\arctan \left({\displaystyle \frac{t_y(i,j)}{t_x(i,j)}}\right)$$

(2)

where t_x(i, j), t_y(i, j), t₄₅(i, j), t₁₃₅(i, j) are the gradient magnitudes at pixel (i, j) in the corresponding directions.

Non-maximum suppression (NMS) is then applied to refine the gradient magnitude of each pixel. For any given pixel, its gradient direction serves as the reference to determine whether its gradient magnitude is the maximum within the corresponding neighborhood. For instance, when a pixel exhibits 45° gradient, its magnitude is compared with the adjacent pixels along the diagonal, and only the one with the greatest magnitude is preserved. By assigning zero gray values to non-maximum pixels, the wide edges of the guide rod are narrowed down to single-pixel edges, successfully eradicating edge blurring during the subsequent template matching phase.

On this basis, an adaptive dual threshold algorithm based on gradient statistical characteristics is applied to enhance edge detection and connectivity, thereby reducing the false edge detection rate of the traditional Canny algorithm. Since the standard deviation of an image quantifies the dispersion of gray values around the mean, the gradient mean M and standard deviation σ of the processed NMS gradient image are calculated to determine the adaptive double thresholds. The high threshold is defined as H = M + σ, which amplifies the gradient difference using variance to reflect the gray contrast characteristics of the guide rod surface. The low threshold is set as L = 0.4H, a ratio that balances edge detection integrity with the suppression of false edges.

Finally, a guide rod standard template is processed using the above Canny operator, converting the edge-detected image into a point set of edge points p_i = (x_i, y_i)^T,i = 1, 2, 3,…,n, each associated with a unique gradient direction vector d_i = (t_i, u_i)^T.

2.2.2. Similarity Measure Function

Geometric template matching computes a similarity measure using edge information from both the template and the target image [33], as defined in Equation (3).

$$\begin{array}{cc}{s}_0& ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n<}{d}_i^{\prime },e_{q+p^{\prime }}>\hfill \\ & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n(}{t}_i^{\prime }{v}_{x+x_i^{\prime },y+y_i^{\prime }}+u_i^{\prime }{w}_{x+x_i^{\prime },y+y_i^{\prime }})\hfill \end{array}$$

(3)

Here, n denotes the total number of valid edge points extracted from the template image after edge detection. The vector d_i^′ = R(θ)d_i represents the gradient direction of the i-th template edge point after rigid rotation, where t_i^′ and u_i^′ are its horizontal and vertical components, respectively. The vector e_q+p′ is the gradient direction at the corresponding position in the sub-image centered at position q = [x, y]^T in the target image, with $v_{x+x_i^{\prime },y+y_i^{\prime }}$ and $w_{x+x_i^{\prime },y+y_i^{\prime }}$ as its horizontal and vertical components. This metric quantifies the average consistency of gradient directions between the transformed template and the target sub-image. The similarity function for the sub-image at this position is illustrated in Figure 2.

To reduce the computational cost of square root operations, the gradient vectors of the template and the target image are first normalized to unit vectors, transforming Equation (3) into Equation (4). This normalization is equivalent to the normalized cross-correlation (NCC) measure widely used in robust template matching [34]:

$$s_1={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n<}{\overline{d}}_i^{\prime },{\overline{e}}_{q+p^{\prime }}>$$

(4)

However, the original similarity measure adopts an equal weight summation and fails to distinguish the importance of edge points. Thus, optimization is performed to introduce a weighting mechanism as:

$$\begin{array}{cc}{s}_2& ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{w}_i<}{\overline{d}}_i^{\prime },{\overline{e}}_{q+p^{\prime }}>\hfill \\ & ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{w}_i(}{\overline{t}}_i^{\prime }{\overline{v}}_{x+x_i^{\prime },y+y_i^{\prime }}+{\overline{u}}_i^{\prime }{\overline{w}}_{x+x_i^{\prime },y+y_i^{\prime }})\hfill \end{array}$$

(5)

where w_i = exp(−kσ_i²), with the subscript i indexing the i-th valid Canny edge point in the template image. This weighting strategy follows the principle of stability-weighted feature matching, as discussed in [35], wherein edge points with higher local stability are assigned greater contribution to the overall similarity measure. Each edge point p_i is associated with a unique gradient direction vector, and its weight coefficient w_i is independently calculated via the gradient variance σ_i² within its local neighborhood. This strategy incorporates local edge stability into the similarity computation.

$k={\left({\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n{\sigma }_i^2}\right)}^{-1}$ is set as the reciprocal of the mean gradient variance among all template edge points, as indicated in Equation (5). Accordingly, an edge point whose gradient variance equals the global average variance yields a weight coefficient w_i = e⁻¹ ≈ 0.3679. The weight decays exponentially with increasing gradient variance and converges to 1 for edge points with smaller variance. This adaptive formulation enables the weight scaling to spontaneously fit the inherent noise level of images, and eliminates manual presetting of fixed constants. Edge points with low gradient variance in smooth continuous edges, exhibit higher stability. Assigning greater weights to these points amplifies their contribution to feature extraction. Conversely, edge points with elevated gradient variance, typically arising in noise-affected regions, are more prone to introduce errors. Suppressing their weights mitigates interference and enhances matching accuracy. This weighting mechanism preserves the original algorithm’s invariance to illumination changes.

2.2.3. Stopping Criterion and Image Pyramid

A full-image traversal search incurs extremely high computational overhead. To accelerate the algorithm and satisfy the real-time inspection requirements, the early termination of similarity calculation for non-candidate regions is combined with an image pyramid strategy, thereby improving computational efficiency. Owing to its poor anti-interference and low generalizability, the traditional fixed-threshold termination method cannot handle complex conditions in circumferential weld seam detection. Therefore, a dynamic threshold mechanism T is introduced to optimize the termination criterion, as shown in Equation (6).

$$T=T_0\cdot \mathrm{mean}({〈{\overrightarrow{d}}_i^{\prime },{\overrightarrow{e}}_{q+p^{\prime }}〉}_{init})$$

(6)

$$s_{\mathrm{acc}}={\displaystyle \sum _{j=m+1}^n{w}_j}\cdot 〈{\overrightarrow{d}}_j\cdot {\overrightarrow{e}}_{q+j}〉(s_{\mathrm{acc}}<T)$$

(7)

where T₀ = 1.2 is adopted as a conservative engineering choice that provides a 20% safety margin relative to the mean partial similarity of the most stable template points, consistent with the early-termination principle in branch-and-bound template matching [33].The termination threshold T = T₀·mean (·) can adapt to varying working conditions, where the mean term is calculated from the five most stable template edge points (i.e., the top 5 points with the highest weight coefficients), as defined in Equation (6).

Template edge points are traversed in descending order of weight, prioritizing high-weight stable points with higher reliability during similarity calculation. The variable sacc denotes the accumulated weighted similarity score of the traversed points, which is calculated incrementally point by point as shown in Equation (7). If sacc remains below the dynamic threshold T after partial traversal, the traversal and calculation for the current sub-image are terminated immediately without processing all edge points, as the remaining low-weight points cannot make the total similarity reach the qualified threshold. Conversely, if sacc is consistently above T after all edge points have been traversed, the sub-image is classified as the target matching region. The template matching and positioning process for circumferential weld seam detection of the guide rod is shown in Figure 3, where Figure 3a is a schematic diagram of template matching and positioning of the guide rod, and Figure 3b displays the sub-pixel edge extraction result obtained by the improved Canny operator.

To fulfill real-time inspection requirements, a coarse-to-fine hierarchical search strategy based on an image pyramid is employed to decrease algorithm complexity. This strategy facilitates coarse target positioning in low-resolution images and precise matching in high-resolution images. The image pyramid is generated by assigning the gray mean of each 3 × 3 pixel neighborhood from the original image to the corresponding pixel position in the subsequent low-resolution layer. To efficiently position the weld seam region, a fixed-size rectangular bounding box is applied to encompass both the core weld seam area and the transition zone. This method effectively prevents the omission of edge features and the inclusion of irrelevant backgrounds, such as fixtures and non-welded components. Consequently, this operation swiftly extracts the effective weld seam region, accurately defines the region of interest (ROI), eliminates background interference, and significantly reduces the computational cost associated with subsequent 3D information extraction in Figure 3c.

2.3. Point Cloud Filtering and Height Map Generation

To establish the reference datum for the guide rod, random sample consensus (RANSAC) plane fitting is employed, and principal component analysis (PCA) is utilized to estimate the normal vector. The relative height of each point is calculated by integrating the reference plane information with the normal vector. This method effectively mitigates background point interference, reduces point cloud noise, and addresses issues related to non-uniform height references. Consequently, it provides accurate three-dimensional height data that is crucial for reliable identification of weld defects in subsequent analyses.

2.3.1. Preprocessing

Preprocessing is a crucial initial step in RANSAC datum fitting, aiming to remove noise points, reduce redundancy in the point cloud, and improve the accuracy and efficiency of subsequent RANSAC fitting. First, the original point cloud in Figure 4a undergoes pass-through (straight-through) filtering, with coordinate limits X ∈ [−30, +30] mm, Y ∈ [−30, +30] mm, and Z ∈ [−2, +8] mm relative to the weld-ROI centre, to remove extreme noise points that fall outside the effective spatial range of the weld, yielding the filtered point set P_min. Next, radius outlier removal is applied to P_min to eliminate isolated noise points, yielding the refined point set P_radius, as defined in Equation (8):

$$P_{radius}=\{p_i\in P_{\min }\left|k\le card\left(\{p_i\in P_{\min }{\left|‖p_i-p_j‖\right.}_2<r\}\right)\right.\}$$

(8)

where ‖p_i − p_j‖₂ denotes the Euclidean distance between points p_i and p_j, r represents the search radius (set to 0.1 mm), k is the minimum required number of neighbors (set to 5). The function card(·) counts the elements in the set [36]. Here, P_min denotes the point set after pass-through filtering, and P_radius denotes the point set after radius outlier removal. As illustrated in Figure 4b, point A is retained because its average distance to its neighbors is below the threshold, while point B is removed as an outlier.

Voxel-based uniform downsampling reduces point cloud density and lowers the computational cost of RANSAC iterations while preserving the geometric integrity of weld features [37], as shown in Figure 4c. The voxel grid size is set to 0.05 mm × 0.05 mm × 0.05 mm, and the resulting point set is given in Equation (9).

$$P_{sample}=\left\{{\displaystyle \frac{1}{N_v}}{\displaystyle {\sum }_{p\in V_v}p\left|V_v\in \mathbb{V}\right.,N_v=card\left(V_v\right)}\right\}$$

(9)

where $\mathbb{V}$ is the set of voxel grids, V_v denotes a single voxel grid, N_v is the number of points within a voxel, and the arithmetic mean of all points’ coordinates in that voxel is adopted as the representative sampling point.

2.3.2. RANSAC and Normal Vector Estimation

Three points are sufficient to define a plane; however, to enhance fitting robustness, six points are randomly sampled from the outer ring of the annular welds in Figure 4d. A distance threshold d_t = 0.02 mm is defined. Points whose distance d_i to the fitted plane satisfies the condition d_i ≤ d_t are classified as inliers, while the remaining points are considered outliers. The procedure is executed iteratively, retaining the current plane model whenever the number of inliers meets a predefined criterion, with a maximum of 1000 iterations. Whenever the inlier count of the current model exceeds the previously recorded maximum, the model parameters are updated. Iteration continues until the specified limit is reached, at which point the model with the highest inlier count is selected as the final plane.

Accurate estimation of surface normal vectors, essential for point cloud feature extraction, surface segmentation, and various geometric analyses, can be achieved through local plane fitting. Accordingly, local neighborhood normal vector estimation is performed on the weld point cloud corresponding to the reference plane of the guide rod’s base material. For each point p_i, a KD-tree is used to identify the k = 10 nearest neighbor points, and a local plane is fitted to these neighbor points via least squares, as defined in Equation (10):

$$T(\mathit{n},d)=\mathrm{argmin}{{\displaystyle \sum _{i=1}^k\left(\mathit{n}\cdot {\mathit{p}}_i-d_T\right)}}^2$$

(10)

Here, n is the unit normal vector of the local plane T, and p_i = (x_i,y_i,z_i) is the i-th point within the local neighborhood of the current point. The distance from the coordinate origin to the plane T is denoted by d_T.

To solve this optimization problem efficiently, principal component analysis (PCA) is adopted for surface normal estimation, following the classical robust framework in [38]. The key principle is as follows: for a set of 3D points approximately distributed on a local plane in Figure 4e, the point cloud exhibits minimum variance along the normal direction of the plane, and maximum variance along the two orthogonal directions within the plane. This distribution characteristic is accurately quantified by eigenvalue decomposition of the local covariance matrix of the point set.

First, a 3 × 3 covariance matrix M is constructed from the local neighborhood point set for the current point p_i, as defined in Equation (11):

$$\mathit{M}={\displaystyle \frac{1}{k}}{\displaystyle {\displaystyle \sum _{i=1}^k}(}{\mathit{p}}_i-\overline{\mathit{p}})\cdot {({\mathit{p}}_i-\overline{\mathit{p}})}^{\mathrm{T}}$$

(11)

where is the centroid of the k neighboring points. Centroid subtraction eliminates the translation offset of the point set, and retains only the spatial distribution characteristics of the point cloud relative to its geometric center.

Then, eigenvalue decomposition is performed on the symmetric positive semi-definite covariance matrix M, as shown in Equation (12):

$$\mathit{M}\cdot {\mathit{v}}_j={\lambda }_j\cdot {\mathit{v}}_j,j\in \left\{0,1,2\right\}$$

(12)

where λ_j is the j-th eigenvalue of the covariance matrix M, sorted in non-decreasing order as 0 ≤ λ₀ ≤ λ₁ ≤ λ₂; v_j is the unit eigenvector corresponding to λ_j, and the three eigenvectors are mutually orthogonal.

The eigenvalue λ_j directly quantifies the variance of the point cloud distribution along the direction of its corresponding eigenvector v_j. The smallest eigenvalue λ₀ corresponds to the direction of the minimum variance, which is exactly the normal direction of the local least-squares fitted plane. Therefore, the unit eigenvector v₀ associated with λ₀ is taken as the unit normal vector n of the local plane. To eliminate the direction ambiguity of the normal vector (two opposite directions obtained from decomposition), all normal vectors are uniformly oriented toward the 3D structured-light camera coordinate system in this work, ensuring consistency in subsequent height calculations.

Figure 5 presents simulated point cloud datasets with varying outlier levels for plane fitting in low-profile narrow workspaces. The benchmark dataset in Figure 5a comprises 300 inliers uniformly distributed on an ideal plane within X ∈ [0, 1] mm, Y ∈ [0, 1] mm and Z ∈ [0, 0.2] mm, with Gaussian noise (σ = 0.01 mm) added to simulate practical measurement errors; datasets in Figure 5b–d are constructed by adding 300, 500, and 700 outliers respectively to the benchmark to mimic gradually aggravated contamination, where all outliers share the same X–Y range as inliers with Z-values following a Gaussian distribution (μ = 0.1 mm, σ = 0.05 mm) to replicate clustered noise highly overlapping with valid signals in narrow spaces. In all panels, inliers identified by the proposed RANSAC algorithm are marked in navy blue, while rejected outliers are in brick red. Quantitative results show that the proposed method achieves a root mean square error (RMSE) of 0.010 mm and an angular deviation of 0.14° against the theoretical ground truth in the outlier-free scenario, which matches the preset noise level and verifies its intrinsic fitting accuracy; even in challenging outlier-contaminated scenarios, it maintains excellent performance with RMSE stably controlled within 0.015 mm and a maximum angular deviation of only 0.36°, and the negligible performance degradation across all test cases fully validates the good robustness and stability of the proposed method.

The standard deviation of point-to-plane distances, denoted RMSE, is adopted as the metric for evaluating plane fitting accuracy, as defined in Equation (13). This metric directly quantifies the dispersion of the point cloud along the normal direction of the plane, and provides an accurate assessment of reference plane flatness and height measurement reliability, compared with conventional coordinate standard deviation.

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n_{\mathrm{in}}}}{\displaystyle {\sum }_{i=1}^{n_{\mathrm{in}}}{d}_i^2}}$$

(13)

where n_in is the number of inliers, and d_i represents the distance from the i-th inlier to the fitted plane.

2.3.3. Height Measurement

To enable precise quantitative analysis of the three-dimensional weld morphology, the guide rod surface beneath the weld is selected as the height reference. This reference plane is established through point cloud preprocessing and RANSAC-based fitting, and is expressed as Ax + By + Cz + D = 0. The coefficients A, B, and C correspond to the three components of the reference plane’s normal vector n, while D denotes the plane intercept in Figure 4f. In this study, the plane coefficients (A, B, C) are normalized to unit norm (A² + B² + C² = 1) after RANSAC fitting, so that the point-to-plane distance is directly given by the numerator. Using the point-to-plane distance formula from analytic geometry, the height h_i of a point P_i(x_i, y_i, z_i) relative to the reference plane S_final is computed in Equation (14), where the normalization term $\sqrt{A^2+B^2+C^2}$ equals unity because (A, B, C) is a unit normal vector:

$$h_i={\displaystyle \frac{\left|A{x}_i+B{y}_i+C{z}_i+D\right|}{\sqrt{A^2+B^2+C^2}}}$$

(14)

Figure 6 presents the multi-stage hierarchical preprocessing, RANSAC robust plane fitting, and PCA-SVD based local surface normal estimation performed on the simulated point cloud data of the guide rod annular weld. The original noisy weld point cloud containing valid planar inliers and massive random outliers is displayed in Figure 6a. Boundary invalid points are pruned via pass-through filtering in Figure 6b to remove edge interference points. On this basis, isolated discrete noise is further eliminated in Figure 6c, and a uniformly density-simplified point cloud is obtained by voxel downsampling in Figure 6d. After the completion of multi-step point cloud preprocessing, a stable robust reference plane is fitted and constructed by the RANSAC algorithm, as shown in Figure 6e. The directional ambiguity of eigen-decomposed normal vectors is resolved by unifying their orientations against the normal vector of the RANSAC-fitted reference plane, so that height calculation errors are avoided. In addition, sparse sampling with uniformly scaled arrows is adopted for the clear visualization of normal vectors in Figure 6f. By this series of operations, point cloud denoising and redundancy compression are achieved while the weld geometric topology is preserved, a robust height reference plane is established, and high-precision 3D geometric information is obtained to support subsequent weld inspection tasks.

3. Results

3.1. Defect Type

In this experiment, 50 groups of guide rod workpieces, each containing two annular welds on the upper and lower surfaces, are randomly sampled across five production batches as test samples and subsequently independently examined by two qualified welding inspectors strictly following the ISO 17637 standard for visual testing [39]. Each inspector classifies all 100 annular welds independently without knowledge of the other inspector’s results. The two inspectors reach agreement on 94 of the 100 welds (94.0% raw agreement; Cohen’s κ = 0.88); the six remaining cases are adjudicated by an independent senior inspector (ISO 9712 [40] Level 3), whose re-examination with a calibrated digital fillet-weld gauge is adopted as the final ground truth. The dataset is divided into a calibration set and an independent test set. The calibration set comprises five normal guide-rod workpieces sourced from one production batch each, providing a total of ten annular weld height distributions. The repeatability of the height measurement was first verified on these same workpieces through three independent scanning trials with full re-clamping and re-acquisition per trial. This procedure was subsequently extended to all detected defective welds for classification repeatability assessment. The system was additionally deployed in a second installation environment equipped with different sensor units and fixturing hardware, where five normal guide rods were scanned to verify the consistency of the height measurement. The measured height profiles in this second environment were consistent with those obtained from the primary setup. These workpieces are used solely to derive the normal-weld height statistics and determine the detection thresholds; no defective specimen is used, and all class-specific thresholds are derived exclusively from these normal-weld data. The test set comprises the remaining 100 annular welds, which are completely independent of the calibration set and are not involved in any threshold determination. All performance metrics are computed on this independent test set, eliminating calibration-test data leakage. This study focuses on surface geometric defects; therefore, visual inspection results are adopted as the ground truth, and complementary volumetric NDT methods are not included. The morphology and geometric characteristics of different weld types can be clearly distinguished, shown in Figure 7.

The annular joint examined in this work is a circumferential fillet weld (FW) produced by metal active gas (MAG) welding, designated process 135 (GMAW) under ISO 4063. Imperfection classification and acceptability assessment follow established international welding standards. First, all detected indications are identified and named in accordance with EN ISO 6520-1, which standardises the nomenclature and reference codes for geometric imperfections in fusion-welded metallic materials. Acceptability is then evaluated against EN ISO 5817, a standard defining three quality levels (B, C and D) for fusion-welded joints, with level B setting the tightest acceptance limits. For this study, quality level C is adopted as the acceptance benchmark, consistent with baseline quality screening requirements for dynamically loaded chassis components. An imperfection is categorised as a defect, and thus unacceptable, only when its magnitude exceeds the corresponding limits specified in EN ISO 5817 level C; otherwise it is treated as an acceptable imperfection. This framework ensures that all inspection results are fully traceable to international welding standards.

Normal weld (Figure 7a): Smooth contour with natural transition to the base metal, no abrupt dimensional deviation, intact overall structure. The weld profile satisfies the EN ISO 5817 level C requirements for fillet weld geometry.

Oxide inclusions (Figure 7b) correspond to oxide inclusions (No. 303) as defined in EN ISO 6520-1. They appear as irregular point-like, strip-like or block-shaped solid particles embedded in the weld or protruding from its surface. In solid-wire MAG/GMAW, these discrete oxide particles form from deoxidation products such as silicates and manganese oxides, combined with CO₂-induced surface oxidation. They differ from the continuous flux-derived slag layer typical of SMAW or FCAW processes. The resulting bulge usually exceeds the nominal reinforcement height and may rise above the workpiece surface, providing the primary height-based feature for defect detection [41]. Per EN ISO 5817 level C, such protrusions beyond the permitted excess metal limit are classified as defects.

Excessive convexity (Figure 7c) is listed as imperfection No. 503 in EN ISO 6520-1. This imperfection arises from excessive molten metal outflow during welding, presenting as a localised build-up of deposited metal above the nominal weld face. The boxed area marks the protruding region [42], where reinforcement height deviates from the intended weld profile and produces an irregular cross-section. The fillet-weld dimensions were measured with a calibrated digital fillet weld gauge (display resolution 0.01 mm, maximum indication error ±0.03 mm) during visual testing. Repeated readings were taken at no fewer than five equally spaced cross-sections around the circumference and arithmetically averaged, yielding a mean leg length z = 2.75 mm. For an equal-leg 45° fillet weld in accordance with ISO 2553 [43], the nominal theoretical throat thickness is a = 0.707, z = 1.94 mm and the weld-face width between the two toes is $b=\sqrt{2}z=3.89$ mm. Under EN ISO 5817 quality level C, the permissible excessive convexity of a fillet weld is h ≤ 1 mm + 0.15b = 1.58 mm (maximum 4 mm), with a smooth transition required at the weld toe; the gauge-measured protrusion height exceeds 1.58 mm, and this indication is therefore classified as an unacceptable defect.

Insufficient throat thickness (Figure 7d) corresponds to imperfection No. 5213 in EN ISO 6520-1. It appears as a local reduction in the actual weld throat below the nominal value, caused by insufficient deposited metal or improper welding parameter settings. In the 3D point cloud, such features show a local drop in height below the weld threshold and a break in the connected domain of the core weld bead. EN ISO 5817 quality level C evaluates a fillet weld against its nominal throat thickness a rather than the base-material thickness. For imperfection No. 5213, a short reduction in the actual throat thickness A below the nominal throat a is permitted up to h ≤ 0.3 mm + 0.1a; substituting the digital-gauge nominal throat a = 1.94 mm gives the permissible reduction h = 0.49 mm (maximum 2 mm). Repeated digital-gauge readings confirm that the actual local throat reduction exceeds 0.49 mm, and this indication is therefore classified as unacceptable. Imperfections of this type are particularly harmful to critical load-bearing components [44].

3.2. Weld Seam Hierarchical Extraction

The height of the upper plane H_max of the annular guide rod component is determined from the RANSAC fitted reference height H₀ and the statistical upper bound of the normal weld height distribution. It serves as the nominal height of the component body and the critical threshold for identifying oxide inclusion defects, and also acts as the upper height benchmark for hierarchical extraction of weld point clouds.

First, the abnormal height points caused by oxide inclusions are removed to reduce their interference with the subsequent weld height statistics and point cloud filtering. Based on the height distribution histogram of normal welds after outlier removal, the lower and upper bounds of the normal weld height distribution are denoted as P_low and P_high respectively, defining the effective statistical interval [P_low, P_high]. Since the weld height data do not follow a normal distribution, parametric thresholding methods are inapplicable. A non-parametric quantile method is therefore adopted within the interval [P_low, P_high] to set dual thresholds, ensuring full coverage of the effective height range of all normal welds while meeting the requirements of accurate defect identification. Finally, point cloud filtering is performed using the derived dual thresholds to achieve hierarchical extraction of the 3D weld contour.

3.2.1. Low Threshold H_low: Complete Extraction of Full Weld Features

A low threshold H_low is designed to maximize the retention of valid normal weld point clouds while fully suppressing background noise from the guide rod reference plane. By preserving points with height between H_low and the upper plane of the guide rod workpiece, the complete weld point cloud P_weld is extracted encompassing the core weld fusion zone and edge transition zone. This point cloud accurately captures the full-dimensional morphology of the weld, including key geometric attributes such as weld width, edge contour, and base metal transition, thereby providing comprehensive data support for weld conformance inspection.

3.2.2. High Threshold H_high: Enhanced Extraction of Defect Features in the Core Zone

An upper threshold H_high is crucial for isolating the core deposited-metal layer that determines the effective weld throat, thereby providing the geometric basis for throat-thickness quality assessment. By setting directional thresholds, it not only ensures the consistent height of weld reinforcement in typical welds but also enhances the contrast of insufficient-throat defects. In cases of insufficient throat thickness, there is a noticeable break in the point cloud connectivity within the core region of the weld. Selecting point clouds with heights between H_high and the upper plane of the guide rod workpiece extracts the core weld point cloud P_core, focusing on the core zone of the deposited metal. This extraction yields high signal-to-noise ratio feature data essential for subsequent quantitative identification of weld continuity defects.

$$\left\{\begin{array}{c}{H}_{\mathrm{low}}=P_{0.03},P_{0.03}\in [P_{\mathrm{low}},P_{\mathrm{high}}]\\ H_{\mathrm{high}}=H_{\mathrm{low}}+(P_{\mathrm{high}}-P_{\mathrm{low}})/3\\ P_{\mathrm{weld}}=\left\{H_i\left|H_{\mathrm{low}}\le H_i<H_{\max }\right.\right\}\\ P_{\mathrm{core}}=\left\{H_i\left|H_{\mathrm{high}}\le H_i<H_{\max }\right.\right\}\end{array}\right.$$

(15)

In Equation (15), P_0.03 denotes the value below which 3% of the normal weld point cloud height data fall. The one-third proportional placement of H_high above H_low is derived from the cross-sectional geometry of a triangular fillet weld. For such a profile, the upper one-third of the weld height encloses approximately half of the total cross-sectional area. Extruding this area along the circumferential weld length yields the effective volume of deposited metal that sustains the throat thickness, thereby establishing the geometric basis for the threshold selection. The principle of height hierarchical extraction of annular weld point cloud is shown in Figure 8.

The numerical thresholds were derived solely from the five normal calibration guide rods (ten annular welds, one rod from each batch). After discarding residual points above 5.5 mm, the 0.5th and 99.5th percentiles of the pooled normal-weld heights define the effective interval [P_low, P_high] = [0.12, 4.98] mm. The low threshold H_low = P_0.03 ≈ 0.35 mm retains at least 97% of normal-weld points while suppressing reference-plane noise. The high threshold H_high ≈ 1.97 mm, which isolates the core deposited-metal layer; this value lies within 0.1 mm of the nominal theoretical weld throat (a = 1.94 mm), ensuring that connected-domain continuity within P_core directly reflects local throat integrity. The upper-plane threshold H_max = 5.05 mm bounds the normal upper-plane confidence interval [4.85, 5.05] mm. All thresholds are fixed before test-set evaluation, with no test-set data used in calibration.

The point cloud of the target weld region is obtained by screening with the preset height thresholds and then inversely converted into depth image pixels. Initially, points within the specified height range are isolated in Figure 9a. Subsequently, the point cloud of interest with pixel coordinates is extracted, encompassing all point data in P_target = {(u_i, v_i, x_i, y_i, z_i)| h_i ∈ [h_min, h_max]}. This hierarchically extracted annular weld point cloud is then correlated with the corresponding pixels, forming a pixel set U = {(u₁,v₁), (u₂,v₂), …, (u_k,v_k)}, as depicted in Figure 9b.

3.3. Defect Detection Analysis

In the 3D quantitative evaluation of annular weld quality, the volume parameter is commonly used to represent the total amount of deposited metal. However, this metric has inherent limitations: a single volume index reflects only the overall deposited metal quantity, but cannot resolve the local forming quality across different weld regions. It fails to capture composite defects where local excess metal accumulation coexists with local throat reduction elsewhere under the same total volume.

A stepwise defect discrimination framework with decoupled feature extraction is constructed based on the unified height reference and the extracted weld point cloud, as illustrated in Figure 10. One branch identifies oxide inclusions through extreme height screening, while insufficient throat thickness and excessive convexity are discriminated using images generated from the unfolded weld point cloud at different height thresholds.

3.3.1. Defect Discrimination

Figure 11 presents a three-dimensional morphological comparison of weld specimens, with Figure 11a–d depicting a normal weld, oxide inclusion, excessive convexity, and insufficient throat thickness, respectively. Marked differences in surface contour and height distribution across weld categories demonstrate that height features constitute the primary basis for differentiating defective welds from normal samples.

Oxide inclusion defects are identified as follows. First, the height distributions of multiple normal welds are statistically analyzed to determine the concentration range of their maxima. The analysis yields a confidence interval of 4.85 mm to 5.05 mm for the upper plane of normal welds. As shown in Figure 12a, the point cloud above H_max forms a smooth ring corresponding to the upper plane of the guide rod. If the extreme height exceeds this normal range and the abnormality persists, the oxide inclusion defect can be identified by quickly screening the height extreme value P_max (mm). As depicted in Figure 12b, the tested weld exhibits an extreme height of 7.15 mm, substantially above the component upper plane, and is therefore classified as an oxide inclusion defect. The height characteristics of the oxide inclusions are consistent with the two-dimensional representation after point cloud conversion, corroborating the recognition result.

To address the technical limitation that the inherent curvature of closed annular welds renders excessive convexity detection prone to misjudgment or missed detection, the extracted weld point cloud P_weld is projected onto a 2D reference plane and unwrapped into a rectangular representation. This unwrapping linearizes the curved weld path, enabling direct connected-domain analysis on a flattened geometry where excess metal accumulations manifest as distinct, isolated regions. The transformation is achieved via polar-to-Cartesian coordinate mapping combined with bilinear interpolation [45]. A morphological closing operation using a 3 × 3 circular structuring element with one iteration is then applied to the unfolded image to separate the excessive-convexity region from the main weld body, ensuring that the protruding metal forms a distinct connected component for reliable detection. For a normal weld, the core region contains a single connected domain (C₁ = 1), as shown in Figure 13a. Excessive convexity produces an elevated bulge due to localized metal accumulation, forming an independent additional connected domain in the flattened image. Consequently, when C₁ ≥ 2, the defect is reliably identified as excessive convexity, thereby circumventing the accuracy loss and computational overhead inherent in conventional volume-based methods, as illustrated in Figure 13b.

To overcome this, the core-weld point cloud P_core extracted by the proposed high-threshold hierarchical extraction method is used as the primary data foundation. Two indicators are then calculated: the number of connected domains C₂ and the area ratio R = S_max/S_total, where S_max denotes the area of the largest connected domain and S_total is the total area of the standard core weld. This yields a dual-index discrimination framework based on connected-domain count and area proportion. Specifically, if C₂ = 1, the core-weld connected domain remains continuous, and no local throat-thickness fracture is detected regardless of minor area-ratio fluctuations. Conversely, when C₂ ≥ 2, connectivity breakage within the core layer indicates insufficient throat thickness (No. 5213). The area ratio R = S_max/S_total quantifies the severity of material loss once a connectivity break has been identified, with R < 0.9 serving as a reference marker for pronounced throat reduction; however, R alone is not used as a defect-triggering threshold. This approach markedly improves the identification accuracy and robustness for marginal insufficient-throat defects, as illustrated in Figure 13c.

3.3.2. Statistics of Overall Inspection Results

Among the 100 annular welds, 33 are defective and 67 are normal. The defective samples include 5 oxide inclusions, 11 insufficient-throat cases, and 17 cases of excessive convexity, with one composite defect (combining excessive convexity and insufficient throat thickness, designated Type 3, 4) on the upper weld of workpiece No. 49. In binary classification, this composite defect is counted as one defective weld (one true positive); in multi-class evaluation, it is treated as a positive sample for both the excessive-convexity class and the insufficient-throat class under a multi-label counting rule. The method correctly detected both components; it is therefore counted as a true positive for both excessive convexity and insufficient throat thickness. The proposed method achieves zero false positives overall, missing only two minor excessive convexity defects.

Defect types are classified as follows: normal welds (Type 1), oxide inclusions (Type 2), excessive convexity (Type 3), and insufficient throat thickness (Type 4). As shown in

Table 1. Defect identification of weld seams.

	Upper Annular Welds Seam					Lower Annular Welds Seam					Integrated Discrimination
No.	Pmax/mm	C1	C2	R	Type	Pmax/mm	C1	C2	R	Type	Type	Detection Result
1	4.95	1	1	1	1	4.99	1	1	1.03	1	1-1	Yes-Yes
2	4.89	1	1	0.98	1	4.93	1	1	0.91	1	1-1	Yes-Yes
3	4.82	1	1	1.02	1	4.88	1	1	0.92	1	1-1	Yes-Yes
4	4.79	2	1	1.12	3	4.73	1	1	0.88	1	3-1	Yes-Yes
5	4.79	1	1	0.97	1	4.79	1	1	0.96	1	1-1	Yes-Yes
…											…
36	4.76	1	1	1.02	1	4.69	1	1	0.90	1	1-1	Yes-Yes
37	4.80	1	2	0.85	4	4.65	1	1	0.92	1	4-1	Yes-Yes
38	4.79	1	1	0.98	1	4.92	1	1	0.95	1	1-1	Yes-Yes
…											…
46	4.78	1	1	0.96	1	4.76	1	1	0.95	1	1-1	Yes-Yes
47	4.64	1	1	0.97	1	4.82	1	1	0.92	1	1-1	Yes-Yes
48	4.74	1	1	0.99	1	4.90	1	1	0.97	1	1-1	Yes-Yes
49	4.76	2	2	0.93	3,4	4.71	1	1	0.98	1	3,4-1	Yes-Yes
50	4.81	1	1	0.93	1	4.86	1	1	0.91	1	1-1	Yes-Yes

, the upper weld of workpiece No. 4 (C₁ = 2) is classified as excessive convexity, and the upper weld of workpiece No. 37 (C₂ = 2, R = 0.85) meets the criterion for insufficient throat thickness. For the composite-defect specimen No. 49, although its area ratio R = 0.93 does not fall below 0.9, the connected-domain count C₂ = 2 indicates a connectivity break in the high-threshold core layer; it is therefore classified as insufficient throat thickness under the C₂ ≥ 2 criterion, while C₁ = 2 simultaneously indicates excessive convexity.

Defect detection performance is evaluated using five metrics: precision (P_r), recall (R_e), F-measure (F_β), accuracy (A_c), and false detection rate (F_d). All metrics are derived from the confusion matrix (TP, FN, FP, TN) and are formally defined in Equations (16)–(20).

$$P_r={\displaystyle \frac{TP}{TP+FP}}$$

(16)

$$R_e={\displaystyle \frac{TP}{TP+FN}}$$

(17)

$$F_{\beta }={\displaystyle \frac{(1+{\beta }^2)P_r{R}_e}{{\beta }^2{P}_r+R_e}}$$

(18)

$$A_c={\displaystyle \frac{TP+TN}{TP+TN+FP+FN}}$$

(19)

$$F_d={\displaystyle \frac{FP}{TP+TN+FP+FN}}$$

(20)

Here, TP, TN, FP, and FN denote the numbers of true positives, true negatives, false positives, and false negatives, respectively. The weighting factor β balances precision P_r and recall R_e, which is set to 1 in this work. Accuracy A_c represents the overall proportion of correctly classified samples. Precision P_r measures the fraction of predicted defective samples that are truly defective, while recall R_e quantifies the fraction of actual defective samples successfully detected. With the exception of the false detection rate F_d, higher values of all metrics indicate better detection performance.

In binary classification experiments, positive examples refer to defective welds, and negative examples denote normal welds. Accuracy represents the ratio of correctly identified defective welds to all welds, while recall represents the ratio of correctly identified defective welds to all welds with defects. Evaluated on the entire test set, the method achieves an accuracy of 98.0% and a recall of 93.9%, as shown in

Table 2. Binary classification experiment.

Metrics	Pr	Re	Fβ	Ac	Fd
Value	1.000	0.939	0.969	0.980	0.000

.

In multi-class experiments, positive examples correspond to welds containing a specific defect type, while negative examples refer to welds with other defect types. All defect evaluation metrics exceed 88%, with accuracy (A_c), recall (R_e), and F-measure (F_β) reaching 100% for oxide inclusions and insufficient throat thickness, as shown in

Table 3. Multi-class experiment.

Type	Pr	Re	Fβ
Oxide inclusion	1.000	1.000	1.000
Insufficient throat thickness	1.000	1.000	1.000
Excessive convexity	1.000	0.882	0.938
Macro-average	1.000	0.961	0.979

.

95% confidence intervals for the reported proportions are computed using the Wilson score method. For binary classification: recall = 93.9% (31/33, 95% CI: 80.4–98.3%), false-positive rate = 0% (0/67, 95% upper bound: 5.4%). For multi-class metrics, the intervals vary by class frequency and are discussed in Section 4.4.

To verify the reproducibility of the detection workflow, all specimens are scanned three times with full re-clamping and independent data processing per trial. Trial-specific detection counts and consistency metrics are summarized in

Table 4. Trial-specific detection results for repeatability assessment.

Type	Ground Truth	Trial 1	Trial 2	Trial 3	Mean	Std. Dev.
Oxide inclusion	5	5	5	5	5.00	0.00
Excessive convexity	17	15	14	15	14.67	0.47
Insufficient throat thickness	11	11	11	11	11.00	0.00

.

Only one marginal excessive-convexity case at the decision boundary shows inter-trial variation; the second missed excessive-convexity case is consistently undetected across all three trials. The inter-trial Cohen’s κ coefficient for binary classification reaches 0.96, indicating almost perfect agreement. These results confirm stable and reproducible detection performance.

4. Discussion

4.1. Limitations of Data-Driven Approaches for Small-Sample Weld Inspection

Deep learning has driven notable progress in automated weld inspection. Two-dimensional convolutional architectures have demonstrated strong defect recognition capability on curated X-ray and optical weld datasets [15,16], and active-passive vision fusion frameworks have improved the robustness of weld seam region segmentation [25]. For volumetric defect characterization, 3D point cloud networks paired with multimodal data fusion have further expanded the dimensionality and granularity of inspection information [22,26,46].

However, such reported performance relies heavily on large, well-annotated training datasets, which are rarely available in high-mix, low-volume manufacturing. Label scarcity remains a core barrier to industrial deployment. For rare defects like oxide inclusions with incidence below 5% in stable production, building balanced, large-scale datasets is prohibitively costly for small-batch components. 3D point cloud models face further constraints. Their irregular, unordered structure requires more training data to learn robust geometric features than standard 2D grids. Expert-dependent 3D defect annotation is also far more labor-intensive [47], exacerbating class imbalance. Moreover, models trained on limited data perform poorly against sensor noise and domain shift in real production settings.

Few-shot and zero-shot methods reduce annotation requirements but have critical limitations for structured-light weld inspection. Zero-shot anomaly detectors fail to identify subtle topological defects within normal feature ranges; pre-trained foundation models lack calibrated 3D geometric inductive bias and suffer severe domain shift on metrology data [48]; few-shot metric learning degrades sharply when inter-class feature similarity is high. However, to circumvent the inherent limitations of the above data-driven paradigms, this work develops a geometry-driven weld defect inspection framework. It exploits the well-defined geometric properties of weld defects via calibrated 3D height measurement and physically interpretable statistical thresholding, requiring only five normal guide-rod calibration workpieces (one from each of five production batches; ten normal welds in total), from which all class-specific thresholds are derived, while delivering performance that warrants further investigation for industrial deployment, subject to validation on different workpiece types and operating conditions.

It should be noted that a direct quantitative comparison with existing weld inspection methods is not feasible in this study, as publicly available weld defect datasets are predominantly 2D X-ray or planar optical images, and no comparable 3D point-cloud dataset for annular fillet welds currently exists. Furthermore, the proposed method is a geometry-driven, rule-based approach requiring only minimal calibration samples, which is fundamentally different in paradigm from data-driven deep learning methods that require large annotated training sets. The internal ablation study (Table 4) therefore serves to quantify the independent contribution of each module within the proposed framework.

4.2. Ablation Studies and Module Contribution

Ablation experiments were conducted on the 100-weld dataset to quantify each module’s independent contribution. Results are summarized in

Table 5. Module ablation results on 100 welds.

Module Variant	Loc. Error (px)	Time (ms)	Height RMSE (mm)	TP	FN	FP
Full pipeline	1.2	82	0.012	31	2	0
Standard 2 × 2 Canny	3.7	95	—	28	5	4
Unweighted matching	2.1	88	—	29	4	2
Fixed threshold (no early stop)	1.4	217	—	30	3	0
No RANSAC (raw Z)	—	—	0.087	25	8	8
Single height threshold	—	—	—	23	10	6
Volume-based discrimination	—	—	—	26	7	4

, with detection counts verified against ground truth.

The full pipeline correctly identifies 31 of 33 defective welds with zero false positives. All oxide inclusions and insufficient-throat cases are detected, with only two marginal excessive convexity cases at the decision boundary missed, yielding a per-class recall of 88.2% for excessive convexity.

Replacing the proposed 3 × 3 multi-directional Canny operator with a standard 2 × 2 Sobel Canny increases mean localization error from 1.2 to 3.7 pixels, resulting in 5 false negatives, 4 false positives, and an overall recall of 84.8%. Removing the local-variance-weighted similarity metric degrades localization accuracy to 2.1 pixels, introduces 4 false negatives and 2 false positives, and reduces recall to 87.9%.

The dynamic early-stopping strategy primarily improves computational efficiency. Substituting a fixed threshold and disabling early termination has negligible impact on detection performance (90.9% recall, 3 false negatives from marginal excessive convexity, no false positives) but increases average matching time from 82 ms to 217 ms, a 165% rise in computational overhead.

RANSAC-based reference plane fitting is the most critical module for height measurement accuracy. Without robust plane fitting, raw Z-coordinate height values show a 7.3-fold rise in RMSE (0.087 mm vs. 0.012 mm) due to workpiece tilt and fixture misalignment, leading to 8 false negatives, 8 false positives, 75.8% recall, and a 12% false positive rate.

The dual-threshold hierarchical discrimination strategy decouples defect features across height scales. A single global threshold fails to resolve defects at different height layers, producing the lowest recall of 69.7% with 10 false negatives and 6 false positives. Compared with the proposed connected-domain topological analysis, global volume metrics cannot detect localized anomalies in composite defects with normal total volume, achieving only 78.8% recall with 7 false negatives and 4 false positives.

A measurement uncertainty budget is compiled following the Guide to the Expression of Uncertainty in Measurement (GUM), combining sensor repeatability (0.5 μm), the RANSAC plane-fitting residual (RMSE = 0.012 mm, Section 2.3.2), fixturing repeatability from the three-trial re-clamping tests (0.03 mm, Section 3.1), and quantile threshold statistical uncertainty (0.02 mm). The expanded uncertainty (k = 2) is ~0.08 mm, over 60 times smaller than the normal-weld height interval (P_high − P_low ≈ 4.86 mm), ensuring robust height-based layering and classification under typical operating conditions.

4.3. Industrial Implications

In industrial defect recognition scenarios involving three-dimensional workpieces, relying solely on 2D image annotations often fails to reliably distinguish defect types that arise from spatial location variations. Our algorithm, by contrast, operates without requiring highly reflective workpiece reference surfaces. It employs a spatial layering strategy to progressively extract defects at different height levels, enabling stable and accurate identification of various defect types. This approach demonstrates strong engineering robustness in practical inspection environments. In addition, the proposed 2D-to-3D-to-2D processing philosophy strikes a pragmatic balance between depth-limited 2D vision methods and data-hungry 3D deep networks: it uses 2D analysis for fast localization and topological characterization, 3D structured-light reconstruction for calibrated height metrology, and unfolded 2D profile analysis for final defect classification. This geometry-driven paradigm eliminates the need for large annotated datasets while maintaining inspection efficiency and accuracy, demonstrating the feasibility of a geometry-driven quality-control approach for annular welds under the tested conditions.

A key practical advantage of the quantile-calibrated height-threshold framework is its inherent tolerance against moderate variations in point-cloud density and surface imaging conditions. The large statistical margin between the normal-weld height interval and the expanded measurement uncertainty ensures that threshold-based layering remains stable even under reduced sampling density.

Full interpretability of decision logic further facilitates industrial deployment and operator acceptance, as all classification results are traceable to physically defined quantitative metrics: oxide inclusions via extreme height P_max outside the normal upper-plane confidence interval, excessive convexity via additional connected domains (C₁ ≥ 2) in the low-threshold unfolded profile, and insufficient throat thickness via core connectivity breaks (C₂ > 1) and reduced area ratio (R < 0.9) in the high-threshold core layer. This rule-based transparency enables straightforward on-site verification, threshold recalibration and root-cause analysis—advantages beyond the reach of black-box deep learning models without dedicated explainability frameworks.

4.4. Limitations and Future Work

All specimens share the same component type and are acquired within a fixed welding-parameter window. Aalidation across different component geometries and a wider range of operating environments remains to be conducted.

Oxide inclusion detection follows a deterministic height-threshold principle, whereby an inclusion is flagged only when its protrusion exceeds the upper bound of the normal weld height distribution; oxide particles flush with or below the weld surface would evade this height-based criterion. Conversely, local surface depressions or underfill may produce height profiles comparable to excessive convexity in certain cross-sections, potentially leading to misclassification between these two defect types.

Finally, although 95% Wilson-score confidence intervals and a measurement uncertainty budget have now been provided, the sample size remains modest.

Future work will address these limitations through extended cross-batch and cross-condition validation on different workpiece types, expansion of the dataset to narrow the confidence intervals for rare defect classes, refinement of the uncertainty budget for stricter quality levels (e.g., EN ISO 5817 Grade B), and benchmarking against established inspection methods, along with the integration of deeper geometric features for more subtle defect characterization.

5. Conclusions

This paper presents a full-process method for fast localization, three-dimensional point cloud processing, feature extraction, and defect evaluation of annular weld seams on automotive guide rods, in which weld imperfections are identified per EN ISO 6520-1 and evaluated against EN ISO 5817 quality level C acceptance limits. The main contributions are summarized as follows:

An improved geometric template matching method is developed, incorporating a weighted similarity metric and a dynamic threshold strategy. This approach addresses the challenge of simultaneously achieving high localization accuracy, real-time performance, and robustness to interference in annular weld inspection. It enables rapid and accurate weld region localization, thereby providing reliable spatial constraints for subsequent 3D point cloud analysis and defect detection.

A unified height reference is constructed through multi-stage point cloud preprocessing, RANSAC plane fitting and PCA-based normal estimation. A dual-branch decoupling strategy is then employed: one branch uses extreme height analysis to identify oxide inclusions that protrude above the part surface, while the other employs quantile dual-threshold layered analysis and connected-domain counting to discriminate excessive convexity and insufficient throat thickness. The proposed method achieves accurate localization and reliable defect classification for annular welds.

A dual-branch architecture with decoupled feature extraction is combined with a quantile-driven dual threshold strategy for hierarchical point-cloud characterization. Polar-to-Cartesian unwrapping linearizes the curved weld path, enabling connected-domain analysis to discriminate excessive convexity and insufficient throat thickness without resorting to computationally intensive 3D volume calculations. The results confirm the feasibility and potential of the method under controlled acquisition conditions, while further validation on larger multi-batch and multi-condition datasets is needed to establish industrial readiness.

Experimental evaluation on 100 annular welds sampled across five production batches demonstrates that the proposed method achieves zero false positives and high overall detection performance within the tested conditions. Oxide inclusions and insufficient-throat cases are detected with 100% recall, and excessive convexity with 88.2% recall, with only two marginal cases at the decision boundary missed. The method provides a transparent, geometry-driven alternative to data-hungry deep learning approaches for small-batch weld quality inspection under test conditions.