1. Introduction

Figure 1. Accuracy and runtime (The mean root mean square error (RMSE) and mean Time of different registration algorithms in PandaSet and The Real LiDAR-Derived PCD).

Figure 1. Accuracy and runtime (The mean root mean square error (RMSE) and mean Time of different registration algorithms in PandaSet and The Real LiDAR-Derived PCD).

2. Related Works

3. Methodology

Figure 4. Architecture of the proposed method.

Figure 4. Architecture of the proposed method.

3.1. I4PCS Coarse Registration

ISS [37] is an algorithm to describe the intrinsic shape characteristics of solid geometry, which can be used to extract keypoints from the point cloud. The idea of feature point detection is similar to Principal Component Analysis (PCA), which aims to reduce the dimensionality of high-dimensional data. The steps for ISS are as follows:

Step 1: Set the search radius r or the minimum number of neighboring points k for each point p_j (x_j, y_j, z_j) in point cloud P.

Step 2: Search for all points within radius r with p_j as the center, and calculate their weights W_ji based on the differences in the distribution of neighboring points. The value is:

$$W_{\mathit{ji}}=\frac{1}{\left|\right|p_j -p_i\left|\right|}, |p_j-p_i|\text{≤}r\text{∪}i\text{≤}k$$

(10)

Where: p_i is an anypoint in P

Step 3: Analyze the features within the neighborhood of point p_j, and calculate the covariance matrix cov (p_j)of each point p_j.

$$\mathit{cov}\left(p_j\right)=\raisebox{1ex}{$\sum _{|p_j-p_i|\text{≤}r\text{∪}i\text{≤}k}{W}_{\mathit{ji}}(p_j-p_i){(p_j-p_i)}^T$}\!\left/ \!\raisebox{-1ex}{$\sum _{|p_j-p_i|\text{≤}r\text{∪}i\text{≤}k}{W}_{\mathit{ji}}$}\right.$$

(11)

Step 4: To establish a local reference frame (LRF), calculate the eigenvalues {λ_j¹, λ_j², λ_j³} and eigenvector $\text{{}\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\text{}}$ for each cov(p_j). then, rank the eigenvalues in descending order. The established LRF is shown in Figure 6:

Step 5: To minimize the interference of diffusion points along the main direction on detection, non-maximum suppression (NMS) is applied to the salient features within r. Set threshold ε₁ and ε₂, Keypoints need to meet:

$${\lambda }_j^2/{\lambda }_j^1\text{≤}{\epsilon }_1, {\lambda }_j^3/{\lambda }_j^2\text{≤}{\epsilon }_2, 0\text{<}{\epsilon }_1,{\epsilon }_2\text{<}1$$

(12)

Step 6: Repeat the above steps until all points are traversed.

Figure 6. LRF of p_j.

Figure 6. LRF of p_j.

ISS has low sensitivity to point cloud distribution and noise type, and has good stability; The accuracy of local feature description may not be as good as FPFH, but ISS is very efficient and suitable for real-time applications.

When the PCD is large, the process of searching similar sets and screening the best transformation matrix in 4PCS is slow and time-consuming. Most of the keypoints extracted by ISS are invariant to rotation, scale, and affine transformation, and retain local geometric features, which can be used as a stable and reliable initial point set for pairing. This point set can greatly improve the accuracy of the 4PCS algorithm in solving the optimal transformation. In addition, limiting the matching points to the keypoints set can greatly reduce the amount of calculation and make the coarse registration faster.

3.2. SSM-NDT Fine Registration

The traditional NDT algorithm uses PDF to solve the maximum possibility of surface shape coincidence of all blocks. When the local shape difference is large, the effect is poor, and the surface shape similarity should be improved first, so as to reduce the probability of matching dislocation caused by surface deviation.

Figure 7. Fuse density features to construct SSM.

Figure 7. Fuse density features to construct SSM.

In addition, the size of the block directly affects the number of point clouds in the block, the distribution of point clouds and the total number of blocks, so setting a reasonable block size is a crucial step. To this end, a self-adaptive segmentation algorithm based on point cloud density features is proposed. Through this method, a segmentation model is constructed to cut the point cloud to cope with different distributions of PCD. The process of constructing a self-adaptive segmentation model is shown in Figure 7.

Let the Euclidean distance between any point p_i and other points in point cloud P be L, and the set of all L corresponding to p_i is ξ_i = {L₁, …, L_n-1}. If all the elements in the set a are arranged in descending order, the minimum distance value l_i of p_i is:

$$l_i=\mathit{min}\left(L\left(p_i ,q\right)\right), q=1,2,\text{…},n-1,q\text{≠}{p}_i$$

(13)

Where: n is the number of points; q is any point that does not coincide with p_i.

Determining one L requires finding and computing n-1 Ls, which is extremely inefficient and unnecessary. By building a KD-tree to search, the calculation range is narrowed to around p_i. At this time, the density ${\rho }_P$ of point cloud P is:

$${\rho }_P=\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{${\sum }_{ i=1}^{ n}{l}_i$}\right.$$

(14)

To construct SSM based on point cloud density, a parameter that can adjust the block size is introduced: density multiple M_ρ. Embed the model in equation (7) to become F_SSM:

$$F_{\mathit{SSM}}\left(\overrightarrow{p}\right)=-{\sum }_{i=1}^{{ k}_{}}\mathit{log}f\left(T\left(\overrightarrow{p}, \overrightarrow{R_k}\right)\right),k_{}=\raisebox{1ex}{$V_{\mathit{max}}$}\!\left/ \!\raisebox{-1ex}{${\left(M_{\rho }\text{∙}{\rho }_P\right)}^3$}\right.$$

(15)

Where: k_ρ is the model quantity constructed in point cloud P; V_max is the volume of the maximum bounding box of point cloud P; R_k is the segmented point cloud block {r₁, ..., r_m} under the k_ρ model quantity, r_m is a point within the block.

The pseudo program code for constructing an SSM is as follows.

Additionally, the process of SSM-NDT is as follows:

Figure 8. Flow chart of SSM-NDT.

Figure 8. Flow chart of SSM-NDT.

SSM-NDT is an adaptive registration algorithm, which can produce accurate matching results in both dense and sparse regions, and has strong anti-interference. In PCD, each point is considered to carry information, and its relationship with nearby points determines whether the information is of high value. According to equation (14), the point cloud density is determined by the minimum distance l_i corresponding to all points. It can be inferred that there should be at least one point within the area with any point p_i as the center and l_i as the radius. If not, it means that the surrounding of the point is relatively sparse and the information load is low; On the contrary, it is dense and has a high probability of containing high-quality information. Therefore, if you want to improve the registration accuracy, you can put more point clouds in the same block, but this will increase the calculation time.

4. Experiment and Analysis

4.3. PandaSet Dataset

PandaSet, an open-source commercial dataset, is collected by a forward 128-line MEMS LiDAR (PandarGT) and a mechanically rotated 64-line LiDAR (Pandar64). This dataset covers the most challenging driving conditions in L5 autonomous driving, including complex urban environments, dense traffic and pedestrians, steep hills, buildings, and lighting conditions at different times.

4.3.1. Accuracy

Three scenes are randomly selected and two frames are selected from each scene. Our method was compared with ICP, GICP, NDT, NDT+ICP, In Ref. [30] algorithm, and In Ref. [31] algorithm. The registration effects are shown in Figure 10.

Figure 10. Visual comparison diagram of various algorithms.

Figure 10. Visual comparison diagram of various algorithms.

Figure 11. Registration accuracy comparison chart.

Figure 11. Registration accuracy comparison chart.

Different from Scene_1 and Scene_2, the shapes of buildings on both sides of the road in Scene_3 are similar, the corners of buildings are fewer, and there are many pedestrians. These characteristics mean that the surface features of the point cloud are not obvious, there are many noises, and the matching distance is large, etc., which makes the registration difficulty of Scene_3 significantly higher than others.ICP and NDT only reduce a small part of the rotation error, and the translation error is still very large. GICP, NDT+ICP, and In Ref. [30] can only reduce the error under Scene_1 and Scene_2. Only In Ref. [31] and Ours reduce the error in all scenes. In the enlarged figures of Scene_2 and Scene_3, it is shown that the overlap degree of In Ref. [31] is lower than Ours.

Most of the above algorithms rely on parameter settings, and improper parameter settings may lead to the algorithm falling into local optima. For more objectivity, each method was repeated 500 times. Figure 11 shows the comparison of the mean values of RMSE, Dev, StDev, and Recall. The indicators of Ours in the three scenes rank high, and there is a small gap with the optimal method. Especially in Scene_ 3, Ours is not only the smallest on RMSE, Dev, and STDev, and also the largest on recall.

4.3.2. Time

For practical application scenarios, the registration time between two frames is a very important indicator. Table 2 lists the running time of each method and whether they are successfully registered (SR).

According to the data in Table 2, ICP, 4pcs, and NDT failed all registration. ICP is sensitive to the initial position and fails when the point-to-point distance is large; 4pcs does not need iteration and its running time is very short; NDT takes abnormal time to match when the point cloud density distribution is uneven.

GICP, NDT+ICP, and In Ref. [30] were successfully registered in Scene_1 and Scene_2. GICP constructs the cost function through the covariance matrix of the surface based on ICP, which improves the success rate but takes more time; NDT+ICP simplifies the point cloud through the voxel filter, which greatly reduces the time; the average time of the method in In Ref. [30] is the same as that of NDT+ICP is similar, and its registration is better as shown in Figure 9. In Scene_3, only the In Ref. [31] and Ours are successfully registered. Ours takes at least 70%-80% less time in all scenarios than others. Therefore, the method in this paper has great advantages in efficiency, while taking into account the accuracy.

4.3.3 Stability

4PCS is similar to the RANSAC framework, which is uncertain, and it is necessary to improve stability. By matching different coarse and fine registrations, we had explored the influence of I4PCS and SSM-NDT on the stability respectively. The results of 500 experiments are arranged in ascending order, and the changes are shown in Figure 12.

According to the change of the RMSE curve, I4PCS+ICP tends to the lower right corner than 4PCS+ICP, which indicates that I4PCS can effectively reduce the frequency and amplitude of anomalies than 4PCS. Similarly, the comparison between I4PCS+NDT and I4PCS+SSM-NDT shows that introducing SSM into NDT can effectively reduce instability. According to the change of the time curve, I4PCS+ICP and 4PCS+ICP approximately overlap, indicating that their efficiency is not much different; while 4PCS+KDF-NDT only slightly increases at the end, indicating that KDF-NDT takes less time and reduces fluctuations.

Figure 12. Fluctuations in Results of Different Schemes (Arrange the results of 500 experiments in ascending order).

Figure 12. Fluctuations in Results of Different Schemes (Arrange the results of 500 experiments in ascending order).

The analysis shows that I4PCS and SSM-NDT mainly improve accuracy and efficiency respectively, and both improve the stability together. In Scene_2 and Scene_3, although the introduction of ISS as a pre-step will increase the time-consuming of rough registration, it can reduce the calculation amount of fine registration and shorten the overall time; SSM-NDT can correct the abnormality of coarse registration in time.

3.3.4. Anti-Interference

During the outdoor scanning process of LiDAR, it is inevitable that the collected data will be interfered with. Thus, anti-interference is one of the important indicators to evaluate the quality of the method. To verify the anti-interference ability of the method, the following four types of interference (IF) are added to Scene_1:

• Remove 10% of the local point cloud (IF1): there are obstructions or the light intensity is very weak.
• Remove 20% of the global point cloud (IF2): under the influence of bad weather, data loss may occur.
• Add 20% local noise (IF3): strong reflection may occur on the surface with large curvature.
• Add 50% global noise (IF4): beams of different lidars may interfere with each other.

The PCD after interference is shown in Figure 13, and the anti-interference ability of the method is quantitatively evaluated by the relative error (RE). The relative error distribution of each index is shown in Figure 14.

$$\mathit{RE}=\raisebox{1ex}{$\left|{\mathit{IF}}_{\mathit{After}}- {\mathit{IF}}_{\mathit{Before}}\right|$}\!\left/ \!\raisebox{-1ex}{${\mathit{IF}}_{\mathit{Before}}$}\right.\text{×}100\%$$

(20)

Where: IF_Before and IF_After are the values before and after the interference respectively.

The relative accuracy error of most algorithms is smaller in IF2 and IF4 than in IF1 and IF3, and the time error is smaller in IF3 and IF4. In terms of accuracy, the error of the method in this paper is small among the four types of disturbances; in terms of time, the data in Table 2 shows that the algorithm in this paper takes a very short time, although the error performance is average, and the absolute value of the error is less than 0.2s.

Figure 13. The image after adding four kinds of interference to Scene_1 (purple is the origin cloud; green is the Gaussian noise added).

Figure 13. The image after adding four kinds of interference to Scene_1 (purple is the origin cloud; green is the Gaussian noise added).

Figure 14. Scatter plot of relative error of each algorithm before and after four types of interference.

Figure 14. Scatter plot of relative error of each algorithm before and after four types of interference.

Different methods exhibit different immunity to different disturbances. Ours can be anti-interference in the coarse and fine registration. ISS is more inclined to areas with high curvature and high features, while noise is usually not in this area; SSM incorporates a large number of surface features to reduce the influence of local area anomalies.

4.4. The Real LiDAR-Derived PCD

To further verify the effectiveness and applicability, we had tested the real PCD collected by forward MEMS LiDAR (RS-LiDAR M1, RoboSense, Shenzhen, China). The PCD perspective is relatively small and there are few matching targets. Figure 14 and Table 3 show the registration results on this PCD.

In the Library scene, the offset between two frames is small, and most of the methods can be registered successfully. The Mean Time of Ours is only 0.32s, which is at least 60% higher than other methods. The rotation offset between two frames in the Car scene is large, and only NDT+ICP and Ours are successful. The accuracy of Ours is higher, and the time is only 0.84s, which is 65.5% higher than that of NDT+ICP. Therefore, the algorithm in this paper has great advantages in similar scenarios.

Internal factors that affect time include algorithm principles and initial values; external factors are related to PCD. The algorithm in this paper is efficient and concise. Firstly, the improved voxel filtering simplifies PCD; secondly, I4PCS does not need iteration and is highly efficient; finally, SSM-NDT automatically determines the optimal segmentation size to further improve efficiency. Therefore, in each experimental scenario, the mean time consumption of this algorithm is significantly less than that of others, and the efficiency is far ahead.

Figure 15. Registration results of The Real LiDAR-Derived PCD.

Figure 15. Registration results of The Real LiDAR-Derived PCD.

Table 3. Comparison of evaluation indicators for registration results of The Real LiDAR-Derived PCD.

Table 3. Comparison of evaluation indicators for registration results of The Real LiDAR-Derived PCD.

Algorithm	Mean RMSE	Mean Dev	Mean StDev	Mean Recall	Mean Time	Max Time	SR
Library(N=50215)
ICP	0.2240	0.1136	0.1391	67.4818	4.3752	5.1180	✓
GICP	0.1672	0.0717	0.1118	65.8270	2.1092	5.201	✓
4PCS	0.4744	0.0896	0.1137	56.2409	1.5248	2.5150
NDT	0.1707	0.0711	0.1116	64.6528	8.0606	9.7360	✓
NDT+ICP	0.1690	0.0722	0.1131	66.6853	3.2195	3.5250	✓
In Ref. [30]	0.1677	0.0738	0.1116	64.9547	1.5812	1.8710	✓
In Ref. [31]	0.2008	0.0895	0.1139	66.9021	0.8389	1.4030	✓
Ours	0.1736	0.0731	0.1135	65.4235	0.3276	3.5640	✓
Car(N=72770)
ICP	0.1351	0.0858	0.1268	64.9637	1.2060	1.3260
GICP	0.1290	0.0773	0.11644	64.2627	1.6893	2.149
4PCS	0.1771	0.0932	0.1267	63.3158	2.6987	6.0140
NDT	0.1311	0.1122	0.1357	56.5143	12.7207	13.8180
NDT+ICP	0.1268	0.0741	0.1051	76.9665	2.4270	2.9370	✓
In Ref. [30]	0.2071	0.0910	0.1165	66.2869	1.4699	3.1430
In Ref. [31]	0.4013	0.1455	0.1347	44.4968	1.3607	2.0880
Ours	0.1244	0.0546	0.0962	72.2800	0.8414	5.1380	✓

Figure 16. Registration changes of The Real LiDAR-Derived PCD.

Figure 16. Registration changes of The Real LiDAR-Derived PCD.

In the car scene stability test results, unexpectedly, the change of the end segment of the I4PCS+SSM-NDT curve is roughly the same as that of other algorithms, indicating that the possibility of errors has not been reduced. It is speculated that the continuity between the two frames of the scene is insufficient and the overlap is too small.

5. Conclusion and Outlook

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