A new point cloud registration computation method of symmetric iterative closest point for quality detection

Abstract

This invention discloses a novel point cloud registration calculation method for symmetric iterative nearest points in quality inspection. The calculation method includes setting a point set in three-dimensional space. Hedianji The point set P and the point set Q are registered to solve for the six-degree-of-freedom rigid body transformation matrix, which includes a three-degree-of-freedom rotation matrix R and a three-degree-of-freedom translation vector t. A weighted point-surface model is constructed, and a symmetric iterative nearest-point objective function based on the weighted point-surface metric and an adaptive robust loss function is constructed based on the weighted point-surface metric and the adaptive robust loss function. The calculation of the symmetric iterative nearest-point objective function based on the weighted point-surface metric and the adaptive robust loss function includes alternately executing the corresponding point update step and the registration step until the convergence condition is met.

Description

A novel symmetric iterative point cloud registration calculation method for quality inspection Technical Field

[0001] This invention relates to the field of symmetric iterative nearest point, and more specifically to a novel point cloud registration calculation method for symmetric iterative nearest points used in quality inspection. Background Technology

[0002] With the rapid development of artificial intelligence and intelligent manufacturing, point cloud technology has been widely applied in various fields, including: coal industry, autonomous driving, rockfall monitoring, 3D reconstruction, robot automatic trajectory planning, lidar odometry and mapping challenges, digital twins and human-machine collaboration, digital preservation of cultural heritage, free-viewpoint video generation, and others. In point cloud registration technology, the Iterative Nearest Point (ICP) method and its variants have become the most commonly used solutions. This technique iteratively selects points in a point cloud set and matches them with the nearest neighbors in a target point cloud set, then calculates the transformation matrix that minimizes the distance between corresponding points, iteratively executing these steps until a termination condition is met. However, traditional ICP algorithms are extremely sensitive to outliers and local overlaps, exhibiting problems such as a narrow convergence domain, slow convergence speed, and susceptibility to local optima. To address these challenges, researchers have proposed various effective strategies to improve ICP registration performance.

[0003] To address the shortcomings of traditional ICP algorithms, researchers have proposed a series of improvements: Chen et al. proposed Point-to-Plane ICP (LICP), which improves convergence speed and may shrink the convergence domain by minimizing the distance from a point to a plane. Chetverikov et al. developed the Truncate Iterative Nearest Point (TrICP) method based on Euclidean distance registration, which effectively solves the problem of point set registration under partial overlap and noise interference. Segal et al. proposed Generalized ICP (GICP), which integrates point-to-point and point-to-plane strategies, significantly improving the fault tolerance of mismatches. The sparse ICP algorithm developed by Sofien et al. adopts the sparsity-induced paradigm, performs well in handling outliers and incomplete data, and greatly enhances the robustness of registration.

[0004] In terms of algorithm optimization, Li et al.'s KICP algorithm, based on kd-trees, significantly improved registration speed and accuracy, and has become a general framework for various point cloud alignment methods. Audenaert et al.'s Adaptive ICP (AICP) has been successfully applied in medical image segmentation. Yao et al.'s algorithm, based on the similarity of point cloud curvature features, exhibits excellent accuracy, robustness, and stability in unstructured environments, but its effectiveness is limited when features are not obvious. Zhang et al. combined Anderson acceleration and Welsch functions to develop a fast and robust ICP method based on point-to-point matching, and extended it to point-to-area registration scenarios.

[0005] To address specific application needs, Alexander et al. proposed an enhanced robust ICP initialization method using ellipsoidal matching defined by the point covariance matrix. Wang et al., addressing the problem of partially overlapping registration of noise-interference-free point clouds in large-scale measurements, innovatively introduced point-to-point distance constraints and a weighted enhanced distance (WELD) error metric, effectively avoiding local optima and suppressing planar region slippage.

[0006] In recent years, significant breakthroughs have been made in the study of symmetric objective functions: Rusinkiewicz proposed a point-to-surface symmetric objective function ICP (SICP), which improved convergence speed and registration accuracy without significantly increasing computational complexity. Li et al. further integrated novel symmetric functions with robust models to develop the Robust Symmetric ICP (RSICP) algorithm.

[0007] Besides the ICP series of algorithms, feature-based registration methods have also been extensively studied. Intrinsic Shape Features (ISS), proposed by Zhong, as a novel 3D shape descriptor, has been widely applied to point cloud registration. Fast Point Feature Histogram (FPFH), developed by Rusu et al., as an extension of Point Feature Histogram (PFH), performs excellently in 3D registration and has been successfully applied to point cloud alignment by many scholars. However, these feature-based registration methods have significant drawbacks: they are prone to failure when point cloud features are not significant, and they typically require high memory overhead. For example, the execution time of RANSAC-based registration techniques increases exponentially with the increase of outliers.

[0008] In recent years, deep learning-based registration algorithms have attracted widespread attention, such as: robust and efficient point cloud registration based on PointNet (PointNetLK), deep neural networks for 3D point registration, novel point cloud registration for autonomous driving (Global-PBNet), and robust point cloud registration frameworks based on depth map matching. While these algorithms offer superior performance, they still suffer from problems such as complex network structures and insufficient reproducibility of results, and lack rigorous mathematical models to guarantee stability, similar to the ICP series algorithms. Furthermore, registration for partially overlapping, outlier-containing, and incomplete point clouds requires further in-depth research. In particular, the problem of existing advanced methods easily failing when processing low-overlap point cloud data remains unresolved.

[0009] To address the limitations of existing algorithms, this invention proposes a novel improved ICP algorithm—the Symmetric Iterative Closest Point (WASICP) based on weighted point-surface metrics and an adaptive robust loss function.

[0010] First, an innovative weighted symmetric metric model considering normal vector rotation weights is proposed. This improvement not only accelerates the convergence speed but also enhances the convergence benchmark. Second, a robust loss function is used instead of the traditional one. The loss function enables the algorithm to effectively address registration challenges posed by partial overlap and outliers. Finally, a novel computationally simplified technique is designed. The effectiveness of the proposed algorithm is verified through simulation analysis on synthetic and real datasets, and comparison with classical benchmark methods and state-of-the-art registration techniques. Summary of the Invention

[0011] The purpose of this invention is to provide a novel point cloud registration calculation method for symmetric iterative nearest points for quality inspection, so as to solve the problems existing in the prior art.

[0012] To address the aforementioned problems, according to a first aspect of the present invention, a novel point cloud registration calculation method for symmetric iterative nearest points in quality inspection is provided, the calculation method comprising:

[0013] In the three-dimensional space of point cloud registration, the measurement point cloud set of the part to be measured is set. Reference point cloud set of standard parts The point set P and the point set Q are registered to solve for the six-degree-of-freedom rigid body transformation matrix, which includes a three-degree-of-freedom rotation matrix R and a three-degree-of-freedom translation vector t.

[0014] Construct a weighted point-surface model, and based on the weighted point-surface model, construct a symmetric iterative nearest point objective function based on weighted point-surface metrics and an adaptive robust loss function;

[0015] The calculation of the symmetric iterative nearest point objective function based on weighted point-surface metric and adaptive robust loss function includes alternating execution of corresponding point update steps and registration steps until the convergence condition is met. The symmetric iterative nearest point objective function based on weighted point-surface metric and adaptive robust loss function achieves high-precision registration between the part to be tested and the standard part, thereby completing the identification of dimensional deviations and defects in quality inspection.

[0016] Optionally, the mathematical model of the weighted point-surface model is:

[0017] (1)

[0018] Represents the point of motion The closest point to the target point Q The Euclidean distance between them This represents the positional difference of the points after the transformation. Indicates the weighting coefficient. This represents element-wise multiplication. Represents the combination of normal vectors of the transformed points, where

[0019] (2)

[0020] μ represents the adaptive attenuation coefficient. Represents the point of motion The closest point to the target point Q The Euclidean distance between them

[0021] According to expression (2) It is being updated and iterated upon.

[0022] Optionally, the formula for the symmetric iterative nearest point objective function based on weighted point-surface metric and adaptive robust loss function is as follows:

[0023] (3)

[0024] The adaptive robust loss function is expressed mathematically as follows:

[0025] (4)

[0026] μ represents the adaptive attenuation coefficient, which is determined according to expression (4). Update and iterate

[0027] The constraints on the rotation matrix R are expressed mathematically as follows:

[0028] (5)

[0029] I represents R 3×3 The identity matrix in space, Let be the determinant of the matrix.

[0030] Optionally, the weights and least squares are added to the corresponding point update step, based on the current transformation parameters ( , ), where Q is each point in point set P. Find the nearest neighbor ,in

[0031] (6)

[0032] in, This represents the positional difference of the points after the transformation. Indicates the weighting coefficient. This represents element-wise multiplication. Let represent the combination of normal vectors of the transformed points, and k represent the k-th iteration.

[0033] Optionally, weights and least squares are added to the registration step to minimize and The Euclidean distance between them yields the transformation matrix.

[0034] (7)

[0035] Optionally, for minute rotation amounts use ≈ , The linear similarity is approximately 1. Based on formula (8), the rotation matrix is ​​parameterized to obtain formula (9).

[0036] (8)

[0037] (9)

[0038] in, Substituting expression (9) into expression (7) yields expression (10).

[0039] (10)

[0040] in, ,

[0041] The mathematical expression using the weighted linear least squares method is shown in formula (11).

[0042] (11)

[0043] in, This represents the unknown quantity to be solved. The error equation is... Diagonal weight matrix , It is an m×6 dimensional coefficient matrix. The observation vector is m×6 dimensional, as shown in formula (12).

[0044] (12)

[0045] The solution to formula (11) can be expressed in the form of formula (13).

[0046] (13)

[0047] The beneficial effects of this invention are as follows:

[0048] 1. A novel improved ICP algorithm is proposed—symmetric iterative nearest point (WASICP) based on weighted point-surface metric and adaptive robust loss function. Compared with the benchmark method and existing advanced technology, it shows significant advantages in convergence speed, registration accuracy, and robustness to outliers and partially overlapping point sets.

[0049] 2. Without increasing time complexity, a symmetric mathematical model considering the rotation weight of the normal vector is constructed based on the Gaussian weight function, which has a higher registration success rate, faster convergence speed and better accuracy compared with the original algorithm;

[0050] 3. A robust loss method based on Cauchy weight function and adaptive loss framework is proposed, which enhances the robustness of symmetric objective function and makes it more suitable for registration of point sets containing outliers and partial overlap.

[0051] 4. Under the assumption of small incremental rotation, a simplified computational technique based on Rodrigues rotation parameterization is proposed;

[0052] 5. A systematic evaluation of WASICP and other well-known algorithms was conducted on the EPFL Statue Dataset, the Fast Global Registration Dataset, and the 3DMatch Dataset.

[0053] 6. Point cloud registration technology has significant advantages in part measurement and defect detection. First, point cloud data acquisition is highly efficient, allowing for the collection of large amounts of data in a short time, making it suitable for real-time inspection in large-scale production. Second, point cloud data accurately reflects the three-dimensional shape and surface condition of parts, greatly improving inspection accuracy. Finally, point cloud processing methods support automated operation, reducing human interference and improving the reliability of inspection results. Therefore, inspection methods based on point cloud registration algorithms have broad application prospects, especially in the context of automated and intelligent manufacturing. Although point cloud registration technology shows great potential in defect detection, it still faces challenges in practical applications such as data noise, registration accuracy, and poor algorithm robustness. Improving the robustness and real-time performance of point cloud registration algorithms is an important direction in current research. Therefore, this invention discloses a novel symmetric iterative nearest-point point cloud registration calculation method for quality inspection. Attached Figure Description

[0054] Figure 1 is a schematic diagram comparing the basic point-to-surface metric and the symmetric point-to-surface metric (two-dimensional presentation).

[0055] Figure 2 shows a comparison of Cauchy weights with different parameters.

[0056] Figure 3 shows a synthetic dataset generated using the bear model from the EPFL statue dataset. The convergence regions and convergence rates of different algorithms were compared and analyzed. This synthetic dataset includes various rotation and translation transformations. In the figure, the four intervals {[0%,25%), [25%, 50%), [50%, 75%), and [70%, 100%)} are represented by the four points {0.25, 0.5, 0.75, and 1}, respectively.

[0057] Figure 4 shows a partially overlapping point cloud pair constructed based on the monkey model in the EPFL statue dataset. A comparative analysis of various registration methods was conducted. In the figure, #S and #T represent the number of points in the source point cloud and the target point cloud, respectively. Above each data point are the root mean square error (r), execution time (t), and number of iterations (k) calculated according to formula (22). The logarithmic color scale is used to characterize the degree of deviation between the source point cloud after registration transformation and the real registration result.

[0058] Figure 5 shows point cloud data with outliers and partial overlap generated based on the owl model of the EPFL dataset. Various algorithms were compared and evaluated. In each sub-figure: the first column shows the fused data (when the distance between two points is less than 1×10). -5 The first column contains the point cloud input data (which is merged at the same time); the second column uses logarithmic color scales to show the error distribution between the estimated registration result of the source point cloud and the actual registration.

[0059] Figure 6 shows the registration results on multiple locally overlapping instances in the FGR dataset. Each result is labeled with the root mean square error (RMSE), running time (t), and number of iterations (k): the first column is the point cloud input information, and the last column is the RMSE error distribution map based on the logarithmic color scale.

[0060] Figure 7 shows the registration results of the Red Kitchen scene in the 3DMatch dataset. The last column presents the RMSE error distribution based on the logarithmic color scale. Detailed Implementation

[0061] The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, so as to better understand the purpose, features and advantages of the present invention. It should be understood that the embodiments shown in the drawings are not intended to limit the scope of the present invention, but are only for illustrating the essential spirit of the technical solution of the present invention.

[0062] In the following description, certain specific details are set forth for the purpose of illustrating various disclosed embodiments in order to provide a thorough understanding of the various disclosed embodiments. However, those skilled in the art will recognize that the embodiments may be practiced without one or more of these specific details. In other instances, well-known apparatuses, components, and techniques associated with this application may not have been shown or described in detail to avoid unnecessarily obscuring the description of the embodiments.

[0063] Throughout this specification, references to "an embodiment" or "an embodiment" indicate that a particular feature, component, or characteristic described in connection with the embodiment is included in at least one embodiment. Therefore, the appearance of "in an embodiment" or "an embodiment" in various places throughout the specification does not necessarily refer to the same embodiment. Furthermore, a particular feature, component, or characteristic may be combined in any manner in one or more embodiments.

[0064] In the following description, in order to clearly demonstrate the components and working method of the present invention, a number of directional terms will be used. However, terms such as "front", "back", "left", "right", "outer", "inner", "outward", "inward", "up", and "down" should be understood as convenient terms and not as limiting terms.

[0065] Furthermore, terms such as "horizontal," "vertical," and "sag" do not imply that a component must be absolutely horizontal or suspended, but rather that it can be slightly tilted. For example, "horizontal" simply means that its direction is more horizontal relative to "vertical," and does not mean that the component must be completely horizontal, but can be slightly tilted.

[0066] In the description of this application, it should also be noted that, unless otherwise expressly specified and limited, the terms "set up," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this application based on the specific circumstances.

[0067] First, we will introduce the Classical Iterative Closest Point (CICP) and the Symmetric Iterative Closest Point (SIPC).

[0068] In the classical iterative closest point (CICP), in three-dimensional space R... 3 In the given two sets of point clouds and The goal of registration is to solve for a six-degree-of-freedom rigid body transformation matrix, which includes a three-degree-of-freedom rotation matrix R and a three-degree-of-freedom translation vector t, so that the source point cloud P and the target point cloud Q are optimally aligned. The mathematical model of this registration goal is shown in Equation (14).

[0069] (14)

[0070] in, Represents the point of motion The nearest point in the target point set Q The Euclidean distance between them Special orthogonal group The indicator function. In other words, The constraint condition is applied to the rotation matrix R, and its mathematical expression is shown in equation (15).

[0071] (15)

[0072] Where I represents R 3×3 The identity matrix in space, Let be the determinant of the matrix. The classic ICP algorithm solves the problem shown in equation (14) iteratively by alternately executing equations (16) and (17) until the convergence condition is met:

[0073] Corresponding point update: based on current transformation parameters ( , ), for each Find the nearest point in Q. .

[0074] (16)

[0075] Registration steps: Solve for the new transformation matrix by minimizing the Euclidean distance between corresponding points. .

[0076] (17)

[0077] Here, k represents the k-th iteration. The most commonly used method for solving the registration step problem is singular value decomposition (SVD).

[0078] In Symmetric Iterative Closest Point (SIPC), the point-to-surface metric method is widely used in point cloud registration because it makes full use of the point cloud normal vector information. Assuming that the surface geometries of point clouds P and Q are basically the same (with only noise differences), we focus on analyzing the residual generated by the objective function when sampling a pair of neighboring points (p, q) on the surface. In the point-to-surface model, the error is defined as shown in Equation (18); while in the symmetric point-to-surface model, the error is expressed as Equation (19).

[0079] (18)

[0080] (19)

[0081] Here, n represents the normal vector. In point-to-plane measurement, the error is zero only when point p lies on the tangent plane of q. This situation is shown in Figure 1(a). However, the error of symmetric point-to-plane measurement is zero in all cases because... Inevitably and It is orthogonal (this property is visually demonstrated in Figure 1(b)), and this property can be easily extended to the three-dimensional case.

[0082] In 1992, Chen et al. proposed the point-to-surface metric method shown in Equation (20). Although this method improved the convergence speed of the ICP algorithm, it did not significantly expand its convergence domain.

[0083] (20)

[0084] In this function, the normal vector information of the point cloud p is not taken into consideration. Therefore, Rusinkiewicz proposed a symmetric point-to-surface metric method as shown in Equation (21) in 2018.

[0085] (twenty one)

[0086] In this model, point clouds P and Q are transformed to a neutral coordinate system. However, this model does not consider the rotation of the normal vectors in the registration coordinate system, which may lead to a decrease in registration accuracy. Although the original authors also proposed an accurate symmetry metric method, its implementation and linearization process are relatively complex. In 2022, Li et al. proposed a symmetry metric method that considers the rotation of the normal vectors, but did not incorporate the weighting factor of the normal vector rotation, which affects the convergence speed and convergence region. It is worth noting that ICP and its variant algorithms share a common feature: they all aim to align one point cloud with another through a transformation matrix.

[0087] Point-to-surface minimization has become the most mainstream technique in ICP variants. In the original LICP, the residual is zero only when the surface is locally flat. However, in real-world scenarios (such as mechanical parts scanned by a robot in a workshop), achieving perfect correspondence still faces significant registration challenges. Although SICP and RSICP have improved convergence speed and convergence domain, neither fully considers more realistic rotational normals, resulting in less than ideal convergence speed. Furthermore, SICP lacks robust model applications, and the proposed solution in RSICP is sometimes unsatisfactory. This invention proposes a novel weighted point-to-surface model that significantly improves convergence speed while maintaining the convergence domain, and introduces an adaptive robust loss model to effectively adapt to diverse datasets.

[0088] In one embodiment of the present invention, a weighted point-surface model is proposed, the mathematical expression of which is shown in equation (1).

[0089] (1)

[0090] Weighting coefficient The calculation method is detailed in equation (2), which uses an adaptive Gaussian weighting function. (Symbols) This represents element-wise multiplication. Similar to the ICP series algorithms, the method proposed in this study strictly guarantees the geometric transformation of the source point set. Compared with the original metric method, the developed symmetric metric model has three major advantages: (1) convergence speed is improved by more than 40%; (2) the convergence domain is expanded by about 30%; (3) registration accuracy is improved by 15-20%. It is worth noting that this model can achieve efficient point-to-surface ICP registration with only a 5% increase in computational complexity.

[0091] (2)

[0092] Where μ represents the adaptive attenuation coefficient, the specific definition and calculation method of which will be explained in detail later.

[0093] In one embodiment of the present invention, an adaptive robust loss function for SICP is proposed. During the registration correspondence update stage, the traditional ICP algorithm employs... The norm is used as the cost function. However, this method produces large residuals when dealing with outliers and partially overlapping registrations, leading to decreased registration accuracy or even failure. To address this issue, a robust loss function is typically used instead. The cost function achieves robust registration by assigning smaller weights to outlier pairs. For example, in robust ICP, an adaptive robust ICP model based on the Welsh function was proposed. This technique significantly improves the performance of traditional ICP in scenarios with partial overlap and low overlap rates, but it still has shortcomings under high outlier rates. The recently proposed adaptive weighted robust iterative nearest-point method uses a weight vector with an adaptive sparse neighborhood for weighted ICP registration. It achieves local suppression by autonomously learning the weight vector, thereby globally eliminating the influence of noise and outliers. However, some parameters still need to be manually adjusted to obtain the best registration accuracy.

[0094] In this study, we propose a novel symmetric robust loss function based on the Cauchy weight function and an adaptive loss framework. This method effectively aligns outliers and partially overlapping point sets without increasing time complexity. The objective function of our developed WASICP algorithm is shown in Equation (3).

[0095] (3)

[0096] in, The adaptive robust loss model proposed in this paper is expressed in the specific mathematical expression shown in equation (4).

[0097] (4)

[0098] Where μ represents the adaptive decay coefficient, and its value is updated using the algorithm in Table 1. Specifically: μ1 = 3 × median(0) represents three times the median distance between the corresponding points before the iteration, k represents the current iteration number, and T max The maximum number of iterations, The median distance between sampling points is represented by Stop1, which represents the 2-norm of the increment of the transformation matrix and the standard transformation matrix. τ is the convergence threshold. Literature indicates that existing methods typically improve robustness by eliminating points with large corresponding distances or normal errors. However, these methods are not only difficult to adjust parameters but also prone to getting trapped in local optima. Therefore, this paper proposes an adaptive weight calculation technique that integrates corresponding point distances and Cauchy weights. As shown in Figure 2, as the value of μ decreases, the algorithm's penalty for outliers gradually increases. Our proposed adaptive parameter μ continuously decays during iteration, while dynamically adjusting the penalty intensity based on sampling point information, thereby significantly improving the algorithm's robustness while avoiding over-penalization.

[0099] Table 1 Adaptive framework

[0100]

[0101] The constraints on the rotation matrix R are expressed mathematically as follows:

[0102] (5)

[0103] In one embodiment of the present invention, a main algorithm and linear approximation are proposed. Similar to the traditional ICP algorithm, the WASICP method proposed in this invention also includes corresponding update and registration steps. The difference lies in that the algorithm of this invention incorporates weight calculation and least squares problem into the solution process. The main registration process is as follows:

[0104] Corresponding point update: based on current transformation parameters ( , For each point in the point set Q, P is... Find the nearest neighbor .

[0105] (6)

[0106] Registration steps: Based on the known weights and the relationship between corresponding points, solve for the new transformation matrix. .

[0107] (7)

[0108] Since the rotation matrix is ​​known ,therefore Since this is also a known quantity, it can be transformed into a linear function of R. At this point, we can use the same optimization techniques as for point-to-plane metrics to solve the problem.

[0109] Weight and Robust Parameter Update: Adjust the weights according to formulas (2) and (4) and robust parameters Iterative updates will be performed.

[0110] Repeat the above steps until the convergence condition is met. Table 2 shows the complete pseudocode implementation of our WASICP development.

[0111] Table 2. Complete pseudocode for WASICP

[0112]

[0113] To simplify the optimization process, we use linear approximations of sinθ≈θ and cosθ≈1 for the small rotation increment θ, thus transforming the problem into a linear least squares system. First, we parameterize the rotation matrix based on the Rodrigues rotation formula shown in Equation (8).

[0114] (8)

[0115] Where θ represents the rotation angle and a represents the rotation axis. Observation shows that the last term of the above equation is a quadratic term of the rotation angle θ. Therefore, we assume cosθ≈1 and linearize it into the form shown in formula (9).

[0116] (9)

[0117] Where a' = asinθ. Substituting this expression into formula (7) will derive formula (10).

[0118] (10)

[0119] in, The mathematical formulation of this weighted linear least squares problem is shown in formula (11).

[0120] (11)

[0121] in, This represents the unknown quantity to be solved. The error equation is... Diagonal weight matrix , It is an m×6 dimensional coefficient matrix. This is an m×6 dimensional observation vector. Specifically:

[0122] (12)

[0123] Therefore, we can express the solution of formula (11) in the form of formula (13).

[0124] (13)

[0125] Experimental Example

[0126] 1. Evaluation Indicators and Experimental Environment

[0127] This section verifies the effectiveness of the proposed method through five sets of simulation experiments. The experiments provide a detailed analysis of convergence speed, convergence region, robustness, practical problem-solving ability, and memory efficiency. We also compare this method with eight classic registration algorithms, including: ICP, LICP, AICP, FICP, MCCICP, MCCLICP, SICP, and RSICP. Quantitative evaluation uses the root mean square error (RMSE) as the metric, and its specific calculation formula is shown in formula (22) in the figure.

[0128] (twenty two)

[0129] in, Represents the true transformation matrix. The output of the registration algorithm is shown. The experiment also recorded the number of iterations k and the running time t at the time of algorithm termination. The maximum number of iterations for all algorithms was set to 100, and the convergence threshold was 1e-5. The experimental platform was MATLAB 2023a, with an Intel Core i7-12700F processor (2.10GHz) and 16 GB of memory.

[0130] 2. Experiment on synthesized point cloud data

[0131] This section uses synthetic data from the EPFL model library and the Fast Global Registration (FGR) dataset to systematically analyze different registration methods from five dimensions: convergence region, convergence speed, outlier robustness, partial overlap registration capability, and scalability. It should be noted that the registration methods described below are for complex scenes such as living organisms and statues. Success in these complex scenes strongly demonstrates the outstanding advantages of the algorithm in robustness, noise resistance, and ability to handle complex situations. Industrial part registration often represents a subset of these complex situations (typically rigid and deformation-free), and its ability to handle more difficult problems implies better performance when dealing with relatively simpler problems. Therefore, the experimental data in this paper demonstrates the applicability of the novel symmetric iterative nearest-point point cloud registration calculation method for quality inspection.

[0132] (1) Analysis of convergence region and convergence rate

[0133] This section uses the bear model from the EPFL dataset for simulation analysis. For each working condition, the experiment is repeated 100 times independently, with the rigid body transformation matrix randomly generated in each iteration. :in For random rotation and translation amounts (δ represents the translation scaling factor, and dl is the diagonal length of the source point set's bounding box). As shown in Table 1, 12 specific working conditions were designed for the experiment. Gaussian noise with a standard deviation of σ = 0.005 × δ × dl was also added. The convergence threshold τ for all registration algorithms was set to 1e-5, and successful registration was determined when the registration error r < 0.1 × σ. This experiment quantifies the convergence domain performance through the registration success rate and evaluates the convergence speed based on the average number of termination iterations over 100 registrations.

[0134] Table 3

[0135]

[0136] Figure 3 shows the registration results under five different error metrics. It is important to note that since the experiment only introduced noise into the translation component and not the rotation component, the translation parameter has a significant impact on the registration results. As shown in Figure 3(a), when the translation distance is less than 25%, the ICP, LICP, SICP, RSICP, and WASICP algorithms all achieve a 100% success rate. As the translation distance increases, the success rates of ICP and LICP decrease more significantly. Nevertheless, the WASICP algorithm consistently maintains the second-highest success rate (second only to RSICP), demonstrating excellent robustness and effectiveness. It is noteworthy that when the translation distance reaches 50%-75%, the registration success rate of the traditional ICP algorithm drops to 0% (independent of the rotation angle); when the translation distance further increases to 75%-100%, the success rates of ICP and LICP also drop to 0%.

[0137] As shown in Figure 3(b), the comparison of convergence speeds among the algorithms indicates that the ICP algorithm consistently requires the most iterations. Under all 12 experimental conditions, the WASICP algorithm consistently outperforms the other comparative algorithms in convergence speed. This result fully validates the effectiveness of the weighted symmetric point-to-surface metric method proposed in this invention. Overall, WASICP combines superior registration accuracy with faster convergence speed.

[0138] (2) Local overlap and outlier analysis

[0139] The performance of each registration algorithm on partially overlapping point clouds was evaluated using the monkey model from the EPFL statue dataset. The experiment divided the complete model into two parts, retaining only the 20% overlap region: specifically, the first 60% of the points were selected as the source point cloud, and the last 60% as the target point cloud, with random rotation and translation transformations applied to the target point cloud. The registration results are shown in Figure 4. Among the nine comparison algorithms:

[0140] Experimental results show that the proposed method has significant advantages in solving the registration problem with low overlap rate.

[0141] Next, a more complex simulation experiment was conducted using the EPFL owl model. Based on local overlap, outliers of different proportions were introduced into the source point cloud, specifically 1%×m, 5%×m, 10%×m, 25%×m, and 50%×m (where m represents the number of points in the source point set). In terms of experimental setup, the first 60% of the points in the complete model were selected as the source point cloud, and the last 60% as the target point cloud. While adding outliers of different sizes to the source point cloud, random rotation angles and translation vectors were applied to the target point cloud. Figures 5(a)-(e) show the registration results at outlier proportions of 1%, 5%, 10%, 25%, and 50%, respectively.

[0142] Experimental results show that, under five different outlier ratio scenarios, the WASICP method outperforms RSICP by 1-2 orders of magnitude and other comparative methods by 5-7 orders of magnitude. As the outlier ratio increases, WASICP maintains robust registration performance with minimal impact. Notably, while the number of iterations for ICP, LCP, and FICP decreases with increasing outlier ratio, their registration accuracy also decreases accordingly. Overall, this technique demonstrates superior performance in handling local overlap and outliers, further validating the effectiveness of the proposed strategy.

[0143] (3) Scalability analysis

[0144] This section uses the Bimba, Children, Drago, Angle, and Bunny models from the FGR dataset to evaluate the scalability of each algorithm. This dataset contains 25 case studies, with each model providing 5 point cloud pairs with different overlap rates. This data, equipped with realistic transformation matrices, provides a valid benchmark for evaluating the algorithm's global optimization capability. The registration results are shown in Figure 6, with specific numerical information annotated at the top of each image. Experiments show that the proposed algorithm achieves a 2-4 order of magnitude improvement in registration accuracy compared to the original SICP, and the improvement is more significant than that of the classic ICP algorithm. Although the advantage in terms of iteration count is not particularly prominent, this is attributed to the method's effective avoidance of local optima stagnation and premature convergence problems.

[0145] Tables 4 and 5 list the registration root mean square error and execution time results for each algorithm, with the best value shown in bold and the second-best value underlined. Data analysis shows that WASICP significantly outperforms other algorithms for the Bimba and Bunny models; while slightly inferior to MCCICP and RSICP for the Children, Drago, and Angle models, it still significantly outperforms the other compared algorithms. It is worth emphasizing that all metrics in both tables are superior to the original SICP algorithm.

[0146] Table 4

[0147]

[0148] For the synthetic dataset generated based on FGR, the average root mean square error (××10) of different registration methods is listed. -3 The dataset contains locally overlapping point cloud pairs for 5 models, with 5 pairs for each model. The best values ​​are shown in bold, while the second-best values ​​are indicated by underline.

[0149] Table 5

[0150]

[0151] Root mean square error of median for different registration methods (×10) -3 The comparison results of the median time t (seconds) and the median time are as follows. This synthetic dataset is derived from locally overlapping point cloud pairs constructed by five FGR models (each model contains five pairs of data). The optimal values ​​in the table are shown in bold, and the second-best values ​​are marked with underlines.

[0152] (4) Real-world point cloud data testing

[0153] This section uses the 3DMatch real dataset for simulation experiments. This benchmark dataset contains eight scene fragments extracted from the RGB-D reconstruction dataset test set. Each fragment represents a surface 3D point cloud (stored in .ply format) generated by TSDF voxel fusion of 50 depth frames. This invention uses the first two sets of files to evaluate the performance of each registration algorithm. The experimental results of the nine comparative algorithms are shown in Figure 7, demonstrating that WASICP achieves excellent registration results.

[0154] The qualitative analysis results are summarized in Table 6, with the best results shown in bold and the second-best results underlined. Experiments show that compared to the other eight registration algorithms, this algorithm has a significant advantage in registration accuracy (although it requires more computation time). The ICP algorithm has the shortest execution time but lower registration accuracy, while the execution time of this algorithm is at a mid-range level—this result confirms the effectiveness of WASICP in processing real-world datasets.

[0155] Table 6

[0156]

[0157] The main advantages of the proposed WASICP method include: 1) proposing a weighted point-to-surface metric criterion; 2) developing an adaptive robust loss model; and 3) introducing a linear approximation solution method. Based on the EPFL statue model library, the FGR dataset, and the 3DMatch dataset, we conducted extensive experiments including outliers and partially overlapping scenes, comparing the proposed method with algorithms such as ICP, LICP, AICP, FICP, MCCICP, MCCLICP, SICP, and RSICP, thus verifying the advantages of the proposed method in terms of accuracy, robustness, and effectiveness.

[0158] The preferred embodiments of the present invention have been described in detail above. However, it should be understood that after reading the above teachings, those skilled in the art can make various alterations or modifications to the present invention. These equivalent forms also fall within the scope defined by the appended claims.

Claims

1. A novel point cloud registration calculation method for symmetric iterative nearest points for quality inspection, characterized in that, The calculation method includes setting a measurement point cloud set for the part to be measured in three-dimensional space. Reference point cloud set of standard parts The point set P and point set Q are registered to solve for the six-degree-of-freedom rigid body transformation matrix, which includes a three-degree-of-freedom rotation matrix R and a three-degree-of-freedom translation vector t. A weighted point-surface model is constructed, and a symmetric iterative nearest-point objective function based on the weighted point-surface metric and adaptive robust loss function is constructed based on the weighted point-surface metric and adaptive robust loss function. The calculation of the symmetric iterative nearest-point objective function based on the weighted point-surface metric and adaptive robust loss function includes alternately executing the corresponding point update step and the registration step until the convergence condition is met. High-precision registration between the part to be tested and the standard part is achieved through the symmetric iterative nearest-point objective function based on the weighted point-surface metric and adaptive robust loss function. The mathematical model of the weighted point-surface model is as follows: Represents the point of motion The closest point to the target point Q The Euclidean distance between them This represents the positional difference of the points after the transformation. Indicates the weighting coefficient. This represents element-wise multiplication. Represents the transformed combination of normal vectors, where μ represents the adaptive attenuation coefficient. Represents the point of motion The closest point to the target point Q The Euclidean distance between them, according to expression (2) The update iteration is performed; the formula for the symmetric iterative nearest point objective function based on weighted point-surface metric and adaptive robust loss function is as follows: The adaptive robust loss function is expressed mathematically as follows: μ represents the adaptive attenuation coefficient, which is determined according to expression (4). Update and iterate The constraints on the rotation matrix R are expressed mathematically as follows: (5) I represents R 3×3 The identity matrix in space, Let be the determinant of the matrix.

2. The calculation method according to claim 1, characterized in that, Add the weights and least squares to the corresponding point update step, based on the current transformation parameters ( , ), where Q is each point in point set P. Find the nearest neighbor ,in in, This represents the positional difference of the points after the transformation. Indicates the weighting coefficient. This represents element-wise multiplication. Let represent the combination of normal vectors after transformation, and k represent the k-th iteration.

3. The calculation method according to claim 2, characterized in that, The weights and least squares are added to the registration step, by minimizing... and The Euclidean distance between them yields the transformation matrix. （7）。 4. The calculation method according to claim 3, characterized in that, For minute rotation use ≈ 、 The linear similarity is approximately 1. Based on formula (8), the rotation matrix is ​​parameterized to obtain formula (9). in, Substituting expression (9) into expression (7) yields expression (10). in, The weighted linear least squares method is mathematically expressed as shown in formula (11). in, This represents the unknown quantity to be solved. The error equation is... Diagonal weight matrix ， It is an m×6 dimensional coefficient matrix. The observation vector is m×6 dimensional, as shown in formula (12). The solution to formula (11) can be expressed in the form of formula (13). （13）。

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