Air-ground multi-source point cloud robust registration method based on structure entropy sampling

Abstract

The invention discloses an air-ground multi-source point cloud robust registration method based on structure entropy sampling, and the method comprises the steps: S1, reading a source point cloud and a target point cloud, and obtaining the space coordinates, normal vectors and curvatures of the source point cloud and the target point cloud; s2, performing standardized preprocessing on the source point cloud and the target point cloud; s3, performing structure entropy sampling on the source point cloud and the target point cloud after standardization processing, and constructing a sampling point set for registration; s4, adopting a point-to-point registration method with robustness to calculate an optimal rigid transformation matrix between the source point cloud and the target point cloud obtained through structure entropy sampling in the step S3, and realizing point cloud registration; s5, performing spatial restoration on a registration result in the step S4, and constructing a rigid transformation matrix under an original coordinate; and S6, outputting the transformation matrix and the registration point cloud after spatial restoration. The method has the advantages that the key points are optimized through structural entropy driving sampling, the anti-noise capability is high, the registration precision is high, and the registration requirement of small-error space-ground fusion data can be met.

Description

Technical Field

[0001] This invention belongs to the field of 3D point cloud registration technology, specifically relating to a robust registration method for multi-source point clouds based on structural entropy sampling. Background Technology

[0002] With the widespread deployment of various sensors such as UAV platforms and ground-based LiDAR, acquiring multi-source point cloud data from both air and ground has become increasingly convenient. However, due to differences in the viewing angles of different sensors, variations in ranging principles, and external environmental disturbances, the acquired point clouds often suffer from registration errors, noise, and outliers, severely impacting the accuracy of 3D reconstruction and measurement. Especially in sub-resolution level detail matching, traditional iterative nearest-point algorithms are prone to getting trapped in local optima, heavily reliant on initial values, and sensitive to outliers, leading to unstable registration results. Therefore, it is essential to introduce more robust and efficient data preprocessing and sampling strategies to improve the convergence and overall accuracy of registration algorithms.

[0003] In recent years, to improve the robustness and efficiency of point cloud registration, academia and industry have proposed a variety of improvement strategies. Some methods are based on region features and keypoint detection, achieving more reliable initial registration by extracting surface geometry or color descriptors; others are deep learning-driven feature learning methods, which automatically extract representations for adaptive registration through networks, but these methods have high requirements for training data and generalization performance; hybrid sampling strategies attempt to combine uniform sampling with importance sampling to balance structure preservation and efficiency.

[0004] However, the aforementioned methods often fail to achieve a balance between decentralized processing and adaptive sampling in multi-source heterogeneous data environments and on noisy, incomplete, small-scale error point clouds. This can even lead to the amplification of small errors or registration failure. Therefore, improvements are still needed in registration stability, global accuracy, and avoiding getting trapped in local optima when registering small error data. Summary of the Invention

[0005] The purpose of this invention is to provide a robust registration method for multi-source point clouds based on structural entropy sampling. Even under conditions of small initial errors, occlusion, sparse sampling, and other interferences, it can still achieve high-precision and high-efficiency registration between point clouds from different sources. It is especially suitable for scenarios such as air-to-ground fusion and small-error model registration.

[0006] Therefore, the present invention adopts the following technical solution:

[0007] A robust registration method for multi-source point clouds in the air and ground based on structural entropy sampling includes:

[0008] S1, read the source point cloud data and the target point cloud data respectively and extract the spatial coordinates, normal vectors and curvature to obtain the coordinate set V0 of the source point cloud and the coordinate set V1 of the target point cloud;

[0009] S2, standardizes the source point cloud and target point cloud obtained from S1, including scale normalization and decentralization, to eliminate scale bias and spatial translation differences of point clouds from different sources.

[0010] S3: Perform structural entropy sampling on the standardized source point cloud and target point cloud obtained in S2, filter out key points, and construct a set of sampling points for registration;

[0011] S4 employs a robust point-to-point registration method to calculate the optimal rigid transformation matrix between the source point cloud and the target point cloud obtained by the structural entropy sampling in S3, thereby achieving highly robust point cloud registration.

[0012] S5, multiply the registration result obtained in S4 by the original point cloud scaling factor Scale0 again, and add the geometric center obtained in S2. This maintains the point cloud structure consistent with the original coordinate system, completing the spatial restoration of the registration result; the translation vector t in the rigid transformation matrix obtained in S4 is restored to the original scale: t′=t·Scale0, and the rigid transformation matrix in the original coordinate system is constructed:

[0013] S6. The final output includes the transformation matrix T′ and the spatially restored registration point cloud. The transformation matrix can be used for subsequent attitude analysis or map stitching, and the point cloud data is saved as a standard format file for easy system integration.

[0014] The normalization and decentralization operations described in step S2 above include:

[0015] S21, calculate the maximum and minimum values ​​of the input source point cloud and the target point cloud in the x, y, and z directions respectively, and calculate their maximum norm as the scaling factor Sclae:

[0016] Sclae0=max(||max(V0)-min(V0)||)

[0017] Scale1=max(||max(V1)-min(V1)||)

[0018] Where Scale0, max(V0), and min(V0) are the scaling factor, maximum boundary value, and minimum boundary value of the source point cloud, respectively; Scale1, max(V1), and min(V1) are the scaling factor, maximum boundary value, and minimum boundary value of the target point cloud, respectively.

[0019] S22, scales the input source and target point cloud coordinates according to their corresponding scaling factors, so that all coordinates fall within the unit space range:

[0020]

[0021] Where V′0 represents the coordinate set of the source point cloud after scaling by scaling factor Scale0; V′1 represents the coordinate set of the target point cloud after scaling by scaling factor Scale1;

[0022] S23, calculate the geometric center position of all points in the scaled source point cloud. Geometric center position of all points in the target point cloud

[0023]

[0024] S24, mean removal processing: using all the geometric center positions obtained in S23, translate the source point cloud and the target point cloud as a whole, so that their geometric centers move to the origin:

[0025]

[0026] After the above processing, the source point cloud and the target point cloud are transformed to a uniform scale and their centers of gravity are aligned, which helps to improve the numerical stability of subsequent sampling and registration.

[0027] The structural entropy sampling described in step S3 above includes the following steps:

[0028] S31. For each point, construct a local neighborhood containing its K nearest neighbors, where K can be set to 12 to 64.

[0029] S32, construct the covariance matrix based on the local neighborhood of each point, perform eigenvalue decomposition, and define the curvature of the point as the ratio of the minimum eigenvalue to the sum of eigenvalues:

[0030]

[0031] Where λ0, λ1, and λ2 are eigenvalues, and λ0 is the smallest eigenvalue;

[0032] S33, for each point i and its neighboring point j, calculate the following three types of difference information:

[0033] (1) Difference term of normal vector angle: used to measure the degree of change in the surface orientation of two points:

[0034]

[0035] Where n i n jLet these be the normal vectors of the two points;

[0036] (2) Curvature difference squared term: reflects the degree of difference between the local curvature structures of two points:

[0037] C ij =(Curvature i -Curvature j ) 2

[0038] (3) Spatial distance: Euclidean distance between two points, which can be used to assess the density of the distribution between the two points:

[0039] D ij =||p i -p j ||

[0040] Where p represents the coordinates of the point;

[0041] S34, Using the above three types of difference information as structural entropy values, construct the structural entropy scoring function H for this point. i , representing the structural complexity between this point and its neighboring points:

[0042]

[0043] The higher the score, the richer and more representative the point is in terms of normal, curvature, and space compared to its neighboring points;

[0044] S35, Calculate all point ratings H i The mean μ and standard deviation σ are used to select points that meet the following conditions as key structural points:

[0045] μ-δσ <H i <μ+δσ

[0046] The proposed sampling method based on structural entropy preserves morphological representativeness while eliminating redundancy and outliers.

[0047] In step S3 above, a value of K of 32 and a value of δ of 1.0 are recommended.

[0048] The robust point-to-point registration method described in step S4 above includes:

[0049] A robust optimization function is introduced to define the residual cost between point pairs to enhance the robustness of registration to outliers, as shown in the following formula:

[0050]

[0051] Where e is the Euclidean distance between the registration point pairs, v is an adjustable scale parameter; the robust optimization function value ρ(e) is the penalty coefficient of the paired point pairs. The larger the penalty coefficient, the smaller the proportion of the point pair in the subsequent calculation of the minimization error function, and vice versa.

[0052] Combining the Anderson Acceleration optimization method, the registration point pairs are updated iteratively, with the Euclidean distance of the registration point pairs representing the error. The optimal rigid transformation matrix T is solved iteratively to minimize the cumulative error of the error function. During the iteration process, the rigid transformation matrix T from the source point cloud to the target point cloud is continuously solved and updated.

[0053]

[0054] Where R is the rotation matrix and t is the translation vector;

[0055] The iterative process is as follows: Apply the current rigid transformation matrix T to perform a rigid transformation on the source point cloud to generate intermediate registration results; establish new registration point pairs for the transformed source point cloud and target point cloud; recalculate the robust optimization function value to confirm the proportion of the error in the error function of the new registration point pair; calculate the Euclidean distance between the current registration point pairs. If the error function value satisfies the minimization condition, the iteration terminates; otherwise, proceed to the next iteration. Use the Anderson Acceleration technique to update the rigid transformation matrix T by combining historical iteration information and the current estimate to accelerate convergence; finally, output the source point cloud after the last transformation as the registration result, and retain the optimal rigid transformation matrix.

[0056] Compared with the prior art, the present invention has the following beneficial effects:

[0057] 1. This invention integrates normal vector, curvature and spatial distribution information through a sampling mechanism driven by structural entropy, and uses the structural entropy function to characterize the geometric uncertainty of points, and selects key structural points for registration, thereby enhancing the adaptability to scenes with occlusion and density inconsistencies.

[0058] 2. This invention unifies the scale and center of point clouds from different sources through mean removal and normalization preprocessing, making subsequent registration more stable and accurate, and improving the generalization performance of the algorithm in heterogeneous point cloud scenarios such as air-ground fusion.

[0059] 3. This invention improves the registration robustness at outliers by using a robust optimization function and introduces the Anderson acceleration method to speed up the optimization registration efficiency, thus balancing accuracy and convergence speed.

[0060] 4. This invention does not rely on semantic tags, markers or external matching priors, and can complete accurate registration using only the geometric properties of the point cloud itself, which has low dependency characteristics and good versatility.

[0061] 5. The algorithm of this invention can be directly embedded into existing point cloud mapping systems, SLAM systems or registration platforms as a pre-registration module or structure point extraction module, which is easy to deploy and highly adaptable to engineering. Attached Figure Description

[0062] Figure 1 This is a flowchart of a robust registration method for multi-source point clouds in air and ground according to an embodiment of the present invention;

[0063] Figure 2 This is a schematic diagram (50,000 points) of uniform sampling of air-to-ground fused point cloud data in an embodiment of the present invention;

[0064] Figure 3 This is a schematic diagram (31028 points) of the structural entropy sampling of air-to-ground fused point cloud data in an embodiment of the present invention;

[0065] Figure 4 This is a schematic diagram of the robust optimization function used in the embodiments of the present invention (v = 1.0);

[0066] Figure 5 and Figure 6 These are schematic diagrams showing the effect of the first group of air-to-ground fused point cloud data before and after registration using the method of the present invention in this embodiment of the invention.

[0067] Figure 7 and Figure 8 This is a schematic diagram showing the effect of the second set of air-to-ground fused point cloud data before and after registration using the method of the present invention in an embodiment of the present invention. Detailed Implementation

[0068] The method of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0069] See Figure 1 The present invention provides a robust registration method for multi-source point clouds based on structural entropy sampling, comprising the following steps:

[0070] S1, point cloud data loading and information extraction.

[0071] The source and target point cloud data are read separately via a file interface, and all point cloud data are represented as a 3×N matrix for easier subsequent processing. Spatial coordinates, normal vectors, curvature, and optional color channel information are extracted from the point cloud data to obtain the coordinate set V0 of the source point cloud and the coordinate set V1 of the target point cloud.

[0072] S2, Normalization and Mean Removal Preprocessing.

[0073] Because point cloud data from different acquisition sources differ in scale units, coordinate references, and measurement ranges, direct registration may result in significant errors. Therefore, this invention standardizes the source and target point clouds obtained in S1 through normalization and mean removal operations. The specific steps are as follows:

[0074] S21, Determine the scaling factor and standardize the scale.

[0075] Calculate the maximum and minimum values ​​of the source and target point clouds obtained by S1 in the x, y, and z directions, respectively, and calculate their maximum norm as the scaling factor:

[0076] Scale0=max(||max(V0)-min(V0)||)

[0077] Scale1=max(||max(V1)-min(V1)||)

[0078] Where Scale0, max(V0), and min(V0) are the scaling factor, maximum boundary value, and minimum boundary value of the source point cloud, respectively; Scale1, max(V1), and min(V1) are the scaling factor, maximum boundary value, and minimum boundary value of the target point cloud, respectively.

[0079] S22, scale the source and target point cloud coordinates obtained from S1 according to their corresponding scaling factors, so that all coordinates fall within the unit space range:

[0080]

[0081] Where V′0 represents the coordinate set of the source point cloud after scaling by scaling factor Scale0; V′1 represents the coordinate set of the target point cloud after scaling by scaling factor Scale1.

[0082] S23, calculate the geometric center position of all points in the scaled source point cloud. Geometric center position of all points in the target point cloud

[0083]

[0084] S24, mean removal processing: using all the geometric center positions obtained in S23, translate the source point cloud and the target point cloud as a whole, so that their geometric centers move to the origin:

[0085]

[0086] After the above processing, the source point cloud and the target point cloud are transformed to a uniform scale and their centers of gravity are aligned, which helps to improve the numerical stability of subsequent sampling and registration.

[0087] S3, Keypoint Sampling Based on Structural Entropy. To extract keypoints with strong geometric representation and high information content from the point cloud, a scoring strategy based on structural entropy is used to quantitatively evaluate each point in the standardized source and target point clouds obtained in S2, and keypoints with strong representation and stable structure are selected to construct a sampling point set for registration. This includes the following steps:

[0088] S31. For each point, construct a local neighborhood containing its K nearest neighbors. K can be set to 12 to 64, with a recommended value of 32.

[0089] S32, construct the covariance matrix based on the local neighborhood of each point, perform eigenvalue decomposition, and define the curvature of the point as the ratio of the minimum eigenvalue to the sum of eigenvalues:

[0090]

[0091] Where λ0, λ1, and λ2 are eigenvalues, and λ0 is the smallest eigenvalue.

[0092] S33, for each point i and its neighboring point j, calculate the following three types of difference information:

[0093] (1) Difference term of normal vector angle: used to measure the degree of change in the surface orientation of two points:

[0094]

[0095] Where n i n j Let be the normal vectors of the two points.

[0096] (2) Curvature difference squared term: reflects the degree of difference between the local curvature structures of two points:

[0097] C ij =(Curvature i -Curvature j ) 2

[0098] (3) Spatial distance: Euclidean distance between two points, which can be used to assess the density of the distribution between the two points:

[0099] D ij =||p i -p j ||

[0100] Where p represents the coordinates of the point.

[0101] S34, Using the above three types of difference information as structural entropy values, construct the structural entropy scoring function H for this point. i, representing the structural complexity between this point and its neighboring points:

[0102]

[0103] A higher score indicates that the point is more informational and representative of its neighbors in terms of normal, curvature, and space.

[0104] S35, Calculate all point ratings H i The mean μ and standard deviation σ are used to select points that meet the following conditions as key structural points:

[0105] μ-δσ <H i <μ+δσ where δ is recommended to be 1.0.

[0106] The provided sampling method based on structural entropy preserves morphological representativeness while eliminating redundancy and outliers. For details, see [link to relevant documentation]. Figure 2 and Figure 3 contrast( Figure 2 To perform uniform downsampling, Figure 3 (For cross-entropy sampling).

[0107] S4, robust point-to-point registration. A robust point-to-point registration method is used to calculate the optimal rigid transformation matrix between the source and target point clouds obtained through S3 structural entropy sampling, achieving highly robust point cloud registration. The specific operation is as follows:

[0108] Introducing robust optimization functions such as Figure 4 Define the residual cost between point pairs to enhance the robustness of registration to outliers.

[0109]

[0110] Where e is the Euclidean distance between the registered point pairs, and v is an adjustable scaling parameter. The robust optimization function value ρ(e) is the penalty coefficient for the paired point pairs. The larger the penalty coefficient, the smaller the proportion of the point pair in the subsequent calculation of minimizing the error function, and vice versa.

[0111] Combining the Anderson Acceleration method, this approach iteratively updates registered point pairs, using the Euclidean distance of each pair as the error. The optimal rigid transformation matrix T is then solved iteratively to minimize the cumulative error of the error function. During the iteration process, the rigid transformation matrix T from the source point cloud to the target point cloud is continuously solved and updated.

[0112]

[0113] Where R is the rotation matrix and t is the translation vector.

[0114] The iterative process is as follows: Apply the current rigid transformation matrix T to perform a rigid transformation on the source point cloud to generate intermediate registration results; establish new registration point pairs for the transformed source point cloud and target point cloud; recalculate the robust optimization function value to confirm the proportion of the error in the error function of the new registration point pair; calculate the Euclidean distance between the current registration point pairs. If the error function value satisfies the minimization condition, the iteration terminates; otherwise, proceed to the next iteration. Use the Anderson Acceleration technique to update the rigid transformation matrix T by combining historical iteration information and the current estimate to accelerate convergence; finally, output the source point cloud after the last transformation as the registration result, and retain the optimal rigid transformation matrix.

[0115] S5, Registration result restoration: The registration result obtained in S4 is multiplied again by the original point cloud scaling factor Scale0, and the geometric center obtained in S2 is added. This maintains the point cloud structure consistent with the original coordinate system, completing the spatial restoration of the registration result. The translation vector t in the optimal rigid transformation matrix obtained in S4 is restored to the original scale: t′=t·Scale0, constructing the rigid transformation matrix in the original coordinate system.

[0116] S6, Output Results. The final output includes the transformation matrix T′ and the spatially restored registered point cloud. The transformation matrix can be used for subsequent attitude analysis or map stitching, and the point cloud data is saved in a standard format file for easy system integration.

[0117] Furthermore, the 4×4 transformation matrix output by this method, including rotation and translation components, is suitable for air-to-ground fusion point cloud data with inconsistent sources, uneven density, and different viewpoints. It can also be used for subsequent point cloud registration, attitude estimation, pose tracking, and other tasks.

[0118] Applying the algorithm of this invention to actual air-to-ground fusion data, the effects of the first set of air-to-ground fusion point cloud data before and after registration using the method of this invention are as follows: Figure 5 , Figure 6 As shown, the effects of the second set of air-to-ground fused point cloud data before and after registration using the method of this invention are respectively as follows: Figure 7 , Figure 8 As shown in the figure, the method of this invention can register small-error spatial-to-ground fusion data.

Claims

1. A structure entropy sampling based robust registration method for air-ground multi-source point cloud, characterized in that, Comprise: S1, respectively read the source point cloud data and target point cloud data and extract the spatial coordinates, normal vector and curvature, get the source point cloud coordinate set V0 and target point cloud coordinate set V1; S2, the source point cloud and target point cloud obtained by S1 are standardized pretreated, including scale normalization and decentralization, to eliminate the scale deviation and space translation difference of different source point clouds; S3, the source point cloud and target point cloud obtained by S2 are structure entropy sampled, the key points are screened out, and the sampling point set for registration is constructed; S4, a point-to-point registration method with robustness is adopted, the optimal rigid transformation matrix between the source point cloud and target point cloud obtained by S3 structure entropy sampling is calculated, and high robustness point cloud registration is realized; S5, multiply the registration result obtained in S4 by the original point cloud scaling factor Scale0, and add the geometric center obtained in S2 Thus, the point cloud structure is consistent with the original coordinate system, and the spatial restoration of the registration result is completed. Restore the translation vector t in the rigid transformation matrix obtained in S4 to the original scale: t' = t • Scale0, and construct the rigid transformation matrix in the original coordinates: S6, the final output result includes transformation matrix T' and spatial restored registration point cloud; The transformation matrix can be used for subsequent pose analysis or map stitching, and the point cloud data is saved as a standard format file for system integration.

2. The structure entropy sampling based robust registration method for air-ground multi-source point clouds according to claim 1, characterized in that The normalization and decentralization operation in S2 includes: S21, the boundary maximum and minimum values of the input source point cloud and target point cloud in x, y and z directions are calculated respectively, and the maximum norm is calculated as the scaling factor Scale: Scale0=max(||max(V0)-min(V0)||) Scale1=max(||max(V1)-min(V1)||) Wherein, Scale0, max(V0), min(V0) are the scaling factor, boundary maximum and boundary minimum of the source point cloud respectively; Scale1, max(V1), min(V1) are the scaling factor, boundary maximum and boundary minimum of the target point cloud respectively; S22, the coordinates of the input source point cloud and target point cloud are scaled respectively according to the corresponding scaling factor, so that all coordinates fall within the unit space range: Wherein, V'0 represents the coordinate set of the source point cloud after scaling according to the scaling factor Scale0; V'1 represents the coordinate set of the target point cloud after scaling according to the scaling factor Scale1; S23, respectively calculate the geometric center positions of all points in the source point cloud after the scaling processing and the geometric center positions of all points in the target point cloud S24, mean value processing, using the geometric center position of all obtained by S23, respectively, the source point cloud and target point cloud are translated as a whole, so that the geometric center moves to the origin: After the above processing, the source point cloud and target point cloud are converted to a unified scale and the center of gravity is aligned, which helps to improve the numerical stability of subsequent sampling and registration.

3. The structure-entropy sampling based robust registration method for air-ground multi-source point clouds according to claim 1, wherein, The structure entropy sampling in S3 includes the following steps: S31, for each point, a local neighborhood containing its K nearest neighbors is constructed, K can be set to 12-64; S32, based on the local neighborhood of each point, a covariance matrix is constructed, eigenvalue decomposition is carried out, and the curvature value of the point is defined as the ratio of the minimum eigenvalue to the sum of eigenvalues: Wherein, λ0, λ1, λ2 are eigenvalues, and λ0 is the minimum eigenvalue; S33, for each point i and its neighbor point j, the following three kinds of difference information are calculated respectively: (1) Normal vector angle difference term: used to measure the change degree of the surface orientation of two points: where n i , n j are the normal vectors of the two points; (2) Curvature difference square term: reflects the difference degree between the local curvature structures of two points: C ij = (Curvature i -Curvature j ) 2 (3) Spatial distance term: Euclidean distance between two points, which can evaluate the tightness of the distribution between two points: D ij =||p i -p j || Where p represents the coordinates of the point; S34, using the above three kinds of difference information as the structure entropy value to construct the structure entropy score function H of the point i , indicating the structural complexity between the point and its neighborhood points: The higher the score, the more information-rich and representative the point is in terms of normal, curvature, and space relative to its neighborhood points. S35, count all the point scores H i the mean μ and the standard deviation σ, and screen out the points that satisfy the following conditions as the key structure points: μ - δσ < H i < μ + δσ The provided structure entropy-based sampling method not only preserves morphological representation, but also excludes redundancy and outliers.

4. The structure entropy sampling based robust registration method for air-ground multi-source point clouds according to claim 3, characterized in that, The recommended K value is 32, and the δ value is 1.

0.

5. The structure-entropy sampling based robust registration method for air-ground multi-source point clouds according to claim 1, wherein, The robust point-to-point registration method described in S4 includes: A robust optimization function is introduced to define the residual cost between point pairs to enhance the robustness of registration to outliers, as follows: Where e is the Euclidean distance between the registration point pairs, and v is the adjustable scale parameter; the robust optimization function value ρ(e) is the penalty coefficient of the paired point pairs, the greater the penalty coefficient, the smaller the proportion of the point pair in the subsequent error function minimization calculation process, and vice versa; Combined with the Anderson Acceleration optimization acceleration method, the registration point pairs are updated through iteration, the Euclidean distance of the registration point pairs is the error, and the optimal rigid transformation matrix T is solved through iteration to minimize the error accumulation value of the error function; the rigid transformation matrix T from the source point cloud to the target point cloud is constantly solved and updated during the iteration process: Where R is the rotation matrix and t is the translation vector; The iteration process is as follows: the current rigid transformation matrix T is applied to the source point cloud to generate an intermediate registration result; new registration point pairs are established for the transformed source point cloud and the target point cloud; the robust optimization function value is recalculated to confirm the proportion of the error of the new registration point pairs in the error function; the Euclidean distance between the current registration point pairs is calculated, and if the error function value meets the minimization condition, the iteration is terminated; otherwise, the next iteration is entered, and the Anderson Acceleration technology is used to update the rigid transformation matrix T by combining historical iteration information and current estimated value to accelerate convergence; finally, the source point cloud after the last transformation is output as the registration result, and the optimal rigid transformation matrix is retained.

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