Feature Consistent Point Cloud Registration in Building Information Modeling

Point Cloud Registration contributes a lot to measuring, monitoring, and simulating in building information modeling (BIM). In BIM applications, the robustness and generalization of point cloud features are particularly important due to the huge differences in sampling environments. We notice two possible factors that may lead to poor generalization, the normal ambiguity of boundaries on hard edges leading to less accuracy in transformation; and the fact that existing methods focus on spatial transformation accuracy, leaving the advantages of feature matching unaddressed. In this work, we propose a boundary-encouraging local frame reference, the PyramidFeature(PMD), consisting of point-level, line-level, and mesh-level information to extract a more generalizing and continuous point cloud feature to encourage the knowledge of boundaries to overcome the normal ambiguity. Furthermore, instead of registration guided by spatial transformation accuracy alone, we suggest another supervision to extract consistent hybrid features. A large number of experiments have demonstrated the superiority of our PyramidNet (PMDNet), especially when the training (ModelNet40) and testing (BIM) sets are very different, PMDNet still achieves very high scalability.

This work is focused on point cloud registration in building information modeling. In practice, one of the key tasks to solve is how to align pre-built 3D models against the scanned points of real buildings. The aligning results help generate complete and accurate digital models, contributing largely to measuring and applications such as simultaneous localization and mapping (SLAM), 3D reconstruction [21][22][23][24], localization [25][26][27][28], and pose estimation [29][30][31][32].
Existing point cloud registration methods have achieved state-of-the-art performance on common datasets. However, most of them fail in BIM scenarios due to various challenges, noise, and data distribution from varying architectural styles, to name a few. Classic methods [33][34][35][36][37] search for hard correspondence which shows little robustness against noise since Euclidean distance is quite sensitive to offset. Feature-based methods [38][39][40][41] extract local or global reference frames in higher dimensional space to achieve registration. Most of the proposed features take not only point-level information but also line structures, 1 Introduce a boundary-encouraging point cloud feature, PMD, to represent local geometry with higher generalization for registration, as well as solve the normal ambiguity problem. 2 Introduce feature matching loss to the feature extractor to produce consistent hybrid representation. 3 Our PMDNet shows state-of-the-art performance and higher generalization on samples from different distributions. Moreover, high performance still can be observed even when the clouds become more sparse as the distance increases.

Classic Registration Methods
Among classic algorithms (Figure 1a)), ICP [33] and its variants [34,35] try to search the correspondences to minimize the distance (usually Euclidean distance) loss between the projections and the destination points or planes. However, the optimization tends to fall into local optimum because the objective function has multiple local extremums [52], especially in BIM scenarios which usually consist of millions of points. Thus, these methods usually include both coarse and fine registration, where the former aims to provide a better initialization for the latter. To enhance the robustness against noise and outliers, the following researchers view registration as a probabilistic distribution problem, where the input point clouds are treated as two distributions from the same probabilistic model [36,37]. Despite saving efforts to establish the correspondences, they still require a proper initialization because of their non-convex loss function.

Feature-Based Registration Methods
Instead of establishing Euclidean-based correspondences, feature-based methods extract the local reference frame (LRF) for each point in the input clouds to form feature-based correspondences. These descriptors must be distinct from each other, invariant to transformation, and robust to noise and outliers. Despite such high demands it takes, researchers proposed unique descriptors. PFH [38] calculates the invariant pose of a centroid and its neighbors in high dimensional space. FPFH [39] improves PFH with time efficiency by reducing the dimension of histograms but preserves the local feature. SHOT [40] instead focuses on normals by encoding the normal histogram in different coordinates, generated by concatenating all the local histograms. Furthermore, using Hough voting, PPF [41] calculates the 6-D pose of an LRF in a centroid-off manner. It ignores all the global coordinates and only consists of related information of normals, translation, and angles. However, mesh structures, which outmatch points and lines regarding robustness and continuity, are barely included.

Learning-Based Registration Methods
Early methods estimate a good initial transformation for the ICP baseline [38,53,54]. Nevertheless, recent works utilize deep neural networks to calculate a global or local reference frame for each point and then iteratively solve the transformation (Figure 1b) [55,56] firstly introduced the CNNs to the point cloud tasks, followed by numerous deep learning methods to achieve registration [42][43][44]. Among those, PointNetLk [45] uses PointNet to calculate the global features of input clouds and minimize the distance between them. DCP [46] tries to search soft correspondence with the transformer and solve the registration by SVD. These works achieved state-of-the-art performance on ModelNet and other common datasets. However, being specific to objects, they fail to generalize. On another, we notice an ambiguity (see Section 3.1) during the calculation of point normals.

Registration in BIM
In architecture, registration is usually performed via multiple approaches (between images, images and clouds, or clouds). Due to the huge number of points, many methods learn planar geometry instead of local reference frame from regular structures. Turning to planar geometries helps reduce the calculation and improves the overall accuracy due to declined influence of outliers. Taking plane-based registration as an example, plane structures are extracted from both input clouds via RANSAC-based methods [57][58][59], Hough transform [60], or clustering [61]. Afterward, correspondence is estimated using the extracted planes, whose accuracy largely depends on the planar segments and their normals. Ref. [62] homogenizes the as-built and as-planned models by extracting similar cross-sections and thus solves the registration problem, suggesting that the geometric shape of a room, if captured accurately, can be a distinguishing feature. Similarly, ref. [63] proposes a four-degree of freedom (DOF) registration problem and then decomposes it into two steps: (1) horizontal alignment achieved by matching the source and target ortho-projected images using the 2D line features and (2) vertical alignment achieved by making the height of the floor and ceiling in the source and target points equivalent. Ref. [64] segments the input clouds into plane pieces and clusters the parallel ones to eventually determine a transform matrix. However, considering geometries are extremely sensitive to density, their performance falls rapidly as the cloud density decreases because it is hard to extract the planar information on sparse clouds.

Feature Consistent Registration
Due to the normal ambiguity, hand-crafted features fail to calculate the explicit normal of boundaries on hard edges, which provides more information for registration than other points. We solve this problem by introducing another local reference frame, PMD, which learns point and mesh structures in the meantime. Also, previous works pay more attention to spatial registration accuracy, leaving behind feature-matching precision, which may help improve the representation in high dimensions.
We propose PMDNet to solve the issues mentioned above, including a feature extractor that generates a hybrid PMD feature, a parameter network to establish soft correspondence, and a solver to estimate transformation.
More specifically, PMDNet learns a soft correspondence between input clouds via hybrid PMD feature distances in an iterative way. The source cloud is transformed during each iteration i by the estimated transformation in the last iteration i − 1. Then, hybrid features are extracted from both the source and reference cloud. In the meantime, PMD-Net uses a parameter network to learn the annealing parameters, α, β, to refine the soft correspondence. The transformation is generated by Sinkhorn [65] algorithm. Finally, we calculate two losses in the current iteration, L f e and L tr , and back-propagate them to the feature extractor and the transformation estimator. Figure 2 illustrates the pipeline of the PMDNet. We use RPMNet [66] as the backbone.

PMD Feature: Local Reference Frame to Encourage Boundaries
Given two sets of points, X = {x j |j = 1, . . . , J} ∈ R J×3 serving as src and Y = {y k |k = 1, . . . , K} ∈ R K×3 denoted as re f . The objective is to estimate a rigid transformation G =Ĝ(R,t) ∈ SE(3) withR ∈ SO(3) andt ∈ R 3 , such that: Normal Ambiguity. Given a centroid point p, the normal of p is calculated within a local reference frame consisting of p and its neighbors p 1 , p 2 , . . . , p n . As Figure 3 shows, mesh Ω 1 , Ω 2 with normals n 1 , n 2 intersect at p. Here comes the ambiguity; p seems to have multiple normals, depending on which mesh its neighbors are selected.
To fix normal ambiguity, we suggest refining the PPF feature with angles of intersected meshes. PPF originally are 4D point pair features describing the surface between the centroid point x c and each neighbor x i in a rotation-invariant manner: where n c , n i are the normals of centroid point x c and its neighbors x i . Hereby we introduce another three components to PPF, that is: where x 1 , x 2 , x 3 are the closest neighbors of x c , and Ω(p 1 , p 2 , p 3 ) denotes a mesh consisting of three points, p 1 , p 2 , p 3 . Together, x c , x 1 , x 2 , x 3 form a triangular cone where x c is the apex, and x 1 , x 2 , x 3 locate on the bottom surface. The three newly added components describe the angle of the apex. PMD solves normal ambiguity because it considers not only the angle of points, but also the angle of intersected meshes. However, the selected points x 1 , x 2 , x 3 do not always belong to an existing mesh. Thus, we suggest that the introduced angles, ∠x c in PMD, should be considered along with ∠(n i , ∆x c,i ), ∠(n c , n i ) in PPF, so as to make a trade-off between them. In this case, PMD appears to be more generalizing than the PPF or other features in that the aforementioned five angles correct each other. As is proposed in RPMNet, PMD is concatenated with x c and ∆x i , forming a 13D descriptor to capture both global and local features.

L f e : Feature-Aware Loss towards Feature Consistency
Deep learning methods mainly consist of a couple of modules, including a feat extractor and an SVD for computing transformation. Previous works compute the loss regarding SVD, ignoring the extractor, leading to in-explicit connection information between them. RPMNet [66] uses a Euclidean distance loss and a second loss to encourage the inliers. Ref. [67] implements Cross-entropy between estimated and ground-truth correspondence. PointNetLk [45] calculates the loss with estimated transformationĜ and ground-truth G GT . In these cases, the gradient may remain still when back-propagated to the feature extractor, leading to poor updates.
RPMNet defines the total loss L tr as the weighted sum of the Euclidean distance L el between the estimated and reference clouds, and a second loss L inlier to encourage inliers.
However, the L tr is not feature-aware. Thus, we introduce another loss, L f e (Equation (7)), to calculate the difference between two features. L f e is transformation-free and only works in feature extractor: where F is the extracted 13D feature, Γ(F , G) transforms the input F with a given transformation G. ⊕ calculates the difference between two features. The overall loss is now the weighted sum of L f e and L tr :

Annealing Parameter Network
Instead of hard correspondence, we use soft correspondence to predict transforms: F is the hybrid feature in high dimensional space generated by the extractor. • α serves as a threshold to preserve inliers and punish outliers. • β is an annealing parameter to ensure convergence.
Considering α, β are usually distinct on various datasets, RPMNet uses a parameter network that takes both source and reference point clouds as input to predict α, β in an iterative manner. To be specific, the two clouds are concatenated into P J+K,3 . After that, another column in which all the elements are either 0 or 1 is added to P. Finally, P is fed to a PointNet baseline with softmax activation in the final layer to ensure the predicted parameters are always positive.

Experiments
We use ModelNet40 [68] as a common registration problem to train our PMDNet. It contains 12,311 samples of 40 categories. Figure 4 illustrates part of its samples. The clouds used for PMDNet are furthermore subsampled to 512 points for reducing the calculation. We conducted extensive experiments to evaluate the detailed performance of PMDNet against other methods.

J j=1
where F is the extracted 13D feature, Γ(F , G) transforms the input F with a given transformation G. ⊕ calculates the difference between two features.
The overall loss is now the weighted sum of L f e and L tr :

Annealing Parameter Network
Instead of hard correspondence, we use soft correspondence to predict transforms: F is the hybrid feature in high dimensional space generated by the extractor. • α serves as a threshold to preserve inliers and punish outliers. • β is an annealing parameter to ensure convergence. Considering α, β are usually distinct on various datasets, RPMNet uses a parameter network that takes both source and reference point clouds as input to predict α, β in an iterative manner. To be specific, the two clouds are concatenated into P J+K,3 . After that, another column in which all the elements are either 0 or 1 is added to P. Finally, P is fed to a PointNet baseline with softmax activation in the final layer to ensure the predicted parameters are always positive.

Experiments
We use ModelNet40 [69] as common registration problem to train our PMDNet. It contains 12,311 samples of 40 categories. Figure 4 illustrates part of its samples. The clouds used for PMDNet are furthermore subsampled to 512 points for reducing the calculation. We conducted extensive experiments to evaluate the detailed performance of PMDNet against other methods.  In each experiment, we follow approximately the same process. Firstly, the raw cloud of 2048 points is input. Then, a rigid transformation matrix M ∈ SE(3) is generated with random rotation between [0, 45 • ] and random translation between [−0.5, 0.5] about each axis. Afterward, a copy of the raw serves as the reference cloud (Y), and another copy is used as the source (X), which will be transformed afterward by M. Both X and Y are furthermore shuffled and subsampled randomly to 512 points. Finally, The source (X) and reference (Y) clouds are input to PMDNet to get the estimated transformationĜ =Ĝ(R,t). which would be evaluated against G GT = G GT (R GT , t GT ) = M −1 using the metrics aforementioned. Figure 5 illustrates the whole process. copy is used as the source (X), which will be transformed afterward by M Y are furthermore shuffled and subsampled randomly to 512 points. Final (X) and reference (Y) clouds are input to PMDNet to get the estimated tr Ĝ =Ĝ(R,t). which would be evaluated against G GT = G GT (R GT , t GT ) = M metrics aforementioned. Figure 5 illustrates the whole process.
The parameters of our PMDNet is listed in Table 1.
All the competing methods are evaluated using their pretrained mod Net40.

Metrics
All the metrics are based on the rotation and translation errors: ) returns the angle of rotation provide both mean square error(MSE) and mean absolute error(MAE) fo with previous works [46]. All the metrics are listed in Table 2. Cham f erD universally used in registration problems, however, it is extremely sensit considering CD inlier ≈ 0 and CD outlier ≫ 0. Hence, we clip this distance wi of d = 0.1 for mitigation. In addition, considering that cloud computing tec nowadays widely used in many fields but are often constrained by resou efficiency is also an essential metric to evaluate the likelihood of a model bein mobile devices. Figure 5. Experiment process.
The parameters of our PMDNet are listed in Table 1. All the competing methods are evaluated using their pre-trained models on ModelNet40.

Metrics
All the metrics are based on the rotation and translation errors: where {R GT ,R} and {t GT ,t} denote the ground-truth and estimated rotation and translation, respectively. ∠(A) = arccos( tr(A)−1 2 ) returns the angle of rotation matrix A. We provide both mean square error (MSE) and mean absolute error (MAE) for consistency with previous works [46]. All the metrics are listed in Table 2. Cham f er Distance (CD) is universally used in registration problems, however, it is extremely sensitive to outliers considering CD inlier ≈ 0 and CD outlier 0. Hence, we clip this distance with a threshold of d = 0.1 for mitigation. In addition, considering that cloud computing technologies are nowadays widely used in many fields but are often constrained by resources [69], time efficiency is also an essential metric to evaluate the likelihood of a model being deployed to mobile devices.

Ablation Experiment
In this subsection, we compare the contribution of different components of our PMD feature to determine the best choice in afterward experiments. According to its definition, we set up five controlled groups with various components to determine their influence on the overall performance of clean, noise, and unseen data. Table 3 illustrates all the detailed results. Most groups show bare differences in noise scenarios, however, their performance on clean and unseen vary largely from one to another. PMDNet A achieves the best of all, reaching up to over 90% Recall, outperforming PMDNet E (which is the same definition with PPF) by over 10% Recall on clean and unseen. Comparing PMDNet B and PMDNet C , it is easy to find that, ∠x c alone fails to improve the representation of local geometry. ∠x c has worsened the learning, leading to an obvious decline in accuracy. It goes the same with ∠(n i , ∆x c,i ), ∠(n c , n i ) despite ∠(n i , ∆x c,i ), ∠(n c , n i ) only lead to a slight decline on clean and unseen. Only when ∠(n i , ∆x c,i ), ∠(n c , n i ) and ∠x c are both included at the same time, can they together improve the registration greatly. In previous sections, we have made a reasonable explanation for this when introducing the PMD (Section 3.1). We suggest the added angles in PMD, ∠x c , and the original ones in PPF, ∠(n i , ∆x c,i ), ∠(n c , n i ) be considered together, so they can compensate each other to achieve higher performance and generalization. Because ∠x c only works for those boundaries on multiple hard edges where ∠(n i , ∆x c,i ), ∠(n c , n i ) fails to calculate an explicit normal. This explains the huge gap between PMDNet A and PMDNet B , PMDNet C , PMDNet s E. Table 3. Ablation results. PMDNet A , PMDNet B , PMDNet C , PMDNet D , PMDNet E is the same as introduced in Table 4. Bold and underline denote best and second best performance.

ID Scene MAE(R)↓ MAE(T)↓ CCD(1e
In the following experiments, we keep the components in PMDNet A since it outperforms other groups. All the following PMDNet refers to PMDNet A .

Generalization Capability
First, we provide the performance on clean data of each method in Table 5, along with the qualitative results of our method in Figure 6. In this case, all the methods are trained, tested, and evaluated on the whole ModelNet40 dataset.  We can see DeepGMR and FGR achieve the best performance on clean data, outmatching PMDNet and RPMNet by approximately 1%, which means the differences between DeepGMR, FGR, RPMNet, and our PMDNet are quite bare. Actually, except for ICP and IDAM, the remaining methods roughly reach the same accuracy and Recall, proving their performance on basic clean data.
Then, we evaluate the performance of unseen data, to test the generalization capability of each competing method. The training set only consists of the first twenty categories of ModelNet40, and all the competing methods are evaluated on the remaining twenty categories. This experiment is quite challenging for the generalization of point cloud features because all the data used in the evaluation is never seen during training. The more generalizing the feature is, the more promising results it outputs. Table 6 shows the comparison of all the candidate methods. Our method achieves the best performance and greatly outmatches the second place, RPMNet, and other methods, reaching a 100% Recall. DeepGMR, FGR, and IDAM show a large decline (over 70% to less than 10%) here compared with their performance on clean data. Figure 7 illustrates the qualitative results of our method. Be advised that, ICP is not listed here since it is not a learning-based method.  Generally, our method achieves state-of-the-art performance both on unseen categories and clean data.

Gaussian Noise
In this experiment, we evaluate the robustness of each competing method in presence of Gaussian noise. After subsampled, each pair of inputs, the source cloud, and the reference cloud is randomly noised with N (0, 0.01) and clipped to [−0.05, 0.05] to prevent extreme outliers, respectively. In this case, the one-on-one correspondence in dense clouds is corrupted due to the noise. That's why we have been using sparse clouds from the beginning. Table 7 illustrates the results. RPMNet and DCP v2 are the top 2 best methods in this experiment, reaching less than 1deg loss and over 90% Recall. Taking ICP (6.5 MAE(R), 0.05MAE(t), 77% Recall) as a baseline, we divide the methods into two categories, those worse than ICP (DeepGMR, FGR, and IDAM) and those better than ICP (RPMNet, DCV v2, and PMDNet). Despite being less accurate than DCP v2 or RPMNet, PMDNet still achieves acceptable results, reaching 80% Recall. We attribute the vulnerability of PMDNet to our PMD feature. It solves the normal ambiguity of boundaries by introducing mesh angles. However, noise causes the coordinates of points to shift-on the one hand, the original boundaries are off the surrounding meshes; and on the other hand, points that are originally not boundaries become the boundaries.

Registration Comparing on BIM Scenarios
In this section, we test the performance of each competing method on BIM scenarios, more specifically, on clouds of uniform density and clouds with varying density, using their pre-trained models. We select 30 CAD models(.dwg) and uniformly subsample a cloud(.ply) on each of them, consisting of points varying from 0.1 million to 1.0 million. During evaluating, the ground-truth rotation R GT and translation t GT is fixed to [45 • , 45 • , 45 • ] and [1, 1, 1], respectively. Other settings are still the same as introduced above.

Clouds of Uniform Density
In this experiment, to test the basic performance on BIM scenarios, only the dataset is changed from ModelNet40 to BIM clouds, the density of which is left unchanged. Table 8 illustrates the results of each competing method. PMDNet outperforms any other method. Compared with others, the error of PMDNet is only about 20% of that of others, and the CCD is even one ten thousand of that of the second best. Meanwhile, It is worth noting that PMDNet has once again achieved a 100% Recall, twice as high as that of DCP v2. Figure 8 visually illustrates the input and output of PMDNet.
We perform another robust experiment to examine the robustness of candidate methods in noisy BIM scenarios. In this section, a random noise from N (0.01, 0.01) is added and clipped to [-0.05, 0.05] like aforementioned. Table 8. Results on clean density-uniform BIM scenarios. Bold and underline denote best and second best performance, respectively.  src, re f , pred clouds are colored in green, red, and blue, respectively. Table 9. Results on noise density-uniform BIM scenarios. Bold and underline denote best and second best performance, respectively. PMDNet fails on three scenarios, contributing to large MAE(R) and ERM, but achieves great performance on all the other samples. PMDNet † shows the metrics on successfully predicted scenarios. For each cloud P ∈ R M×3 , we select a certain axis of XYZ coordinates, and apply sigmoid and normalize function to calculate a probability ρ 1 , of all M points, which will be applied during sampling.

MAE(R)↓ MAE(T)↓ ERM↓
where µ and σ are two introduced weights. ρ 1,i is the probability of P i to be selected during the afterward sampling. P ϵ is a density aware Gaussian noise. Table 10 illustrates the metrics of each method on clean data. It is obvious that PMDNet is still outstanding, reaching MAE(R) ≤ 0.1, MAE(t) ≤ 0.001, 100% Recall. Furthermore, it is interesting that, compared to density-uniform Clean results (Table 8), PMDNet achieves better performance here, while others make little progress. A major factor contributing to this is the higher continuity of the PMD feature. Although the clouds become sparser and noisier with increasing distance, the nearby points remain accurate and dense, producing more information than uniform sampling, which is exactly what was applied in the density-uniform experiment. Table 11 summaries the results on noise data. Be noticed that, PMDNet fails again on three samples. The major reason is discussed in Section 4.3.2. ∠(x c ) produces less accurate representation since the mesh geometry is corrupted by noise. On the other hand, these three samples consist of mostly planes, with little other geometry for learning. Comparing Table10 and Table 11, we can see a larger decline in accuracy, where MAE(R) rises from 0.0622 to 4.6186, and MAE(t) from 0.00041 to 0.03232, than that in density- Figure 8. Qualitative results of the PMDNet on density-uniform clean BIM data. src, re f , and pred clouds are colored green, red, and blue, respectively. Table 9 illustrates the results of each competing method. PMDNet fails on exactly three samples, which leads to high MAE(R) and ERM. Despite the three failed samples, the overall performance of PMDNet is still of top class since the Recall is 90%. Table 9. Results on noise density-uniform BIM scenarios. Bold and underline denote best and second best performance, respectively. PMDNet fails on three scenarios, contributing to large MAE(R) and ERM, but achieves great performance on all the other samples. PMDNet † shows the metrics on successfully predicted scenarios.

Clouds with Varying Density
Our PMDNet achieves great performance on density-uniform scenarios. However, scan in real BIM practice shows a tendency to fade as the distance increases, which leads to sparse points. In this section, a declining density is applied to simulate the real scans, besides which the other settings remain the same as above. Also, we perform experiments on both clean and noisy datasets.
Density Sampling & Noise. For each cloud P ∈ R M×3 , we select a certain axis of XYZ coordinates, and apply sigmoid and normalize function to calculate a probability ρ 1 , of all M points, which will be applied during sampling.
where µ and σ are two introduced weights. ρ 1,i is the probability of P i to be selected during the afterward sampling. P is a density-aware Gaussian noise. Figure 9 visually illustrates the input and output of PMDNet. Table 10 illustrates the metrics of each method on clean data. It is obvious that PMDNet is still outstanding, reaching MAE(R) ≤ 0.1, MAE(t) ≤ 0.001, 100% Recall. Furthermore, it is interesting that, compared to density-uniform Clean results (Table 8), PMDNet achieves better performance here, while others make little progress. A major factor contributing to this is the higher continuity of the PMD feature. Although the clouds become sparser and noisier with increasing distance, the nearby points remain accurate and dense, producing more information than uniform sampling, which is exactly what was applied in the density-uniform experiment. Table 11 summaries the results on noise data. Be noticed that, PMDNet fails again on three samples. The major reason is discussed in Section 4.3.2. ∠(x c ) produces a less accurate representation since the mesh geometry is corrupted by noise. On the other hand, these three samples consist of mostly planes, with little other geometry for learning. Comparing Table 10 and Table 11, we can see a larger decline in accuracy, where MAE(R) rises from 0.0622 to 4.6186, and MAE(t) from 0.00041 to 0.03232, than that in the density-uniform experiment, where MAE(R) rises from 0.1447 to 3.7088, and MAE(t) from 0.00089 to 0.01696.  src, re f , pred clouds are colored in green, red, and blue, respectively.

Time Efficiency
Time efficiency is another important metric regarding registration methods. Thus, we finally test our PMDNet against other learning-based methods in terms of time efficiency. All the methods are evaluated on Windows 11, Intel i5 12400f, NVIDIA RTX 3060TI, 16GB 3200MHz RAM. Table 12 illustrates the results. We perform this experiment on clouds of 512 and 1024 resolution, with each method fixed to 3 iterations. IDAM is definitely the fast of all, followed by DCP v2. Despite slower than IDAM and DCP v2, PMDNet is over 40% faster compared to RPMNet.

Conclusion
In this work, we firstly introduce a novel local reference frame, the PMD feature, to solve the normal ambiguity of boundaries. Moreover, as we suggest feature matching precision of equal importance as spatial accuracy, we introduce feature loss to registration. Qualitative results of the PMDNet on density-decreasing clean BIM data. src, re f , and pred clouds are colored green, red, and blue, respectively.

Time Efficiency
Time efficiency is another important metric regarding registration methods. Thus, we finally test our PMDNet against other learning-based methods in terms of time efficiency. All the methods are evaluated on Windows 11, Intel i5 12400f, NVIDIA RTX 3060TI, 16 GB 3200 MHz RAM. Table 12 illustrates the results. We perform this experiment on clouds of 512 and 1024 resolution, with each method fixed to 3 iterations. IDAM is definitely the fastest of all methods in this study, followed by DCP v2. Despite being slower than IDAM and DCP v2, PMDNet is over 40% faster compared to RPMNet.

Conclusions
In this work, we first introduce a novel local reference frame, the PMD feature, to solve the normal ambiguity of boundaries. Moreover, as we suggest feature matching precision of equal importance as spatial accuracy, we introduce feature loss to registration.
Extensive experiments show our PMDNet achieves state-of-the-art performance. More specifically, PMDNet achieved a 100% Recall in the Unseen scenario with the generic dataset, which is 25% higher than the second-best, RPMNet. Meanwhile, PMDNet also achieved a 100% Recall on density-decreasing BIM scenarios. Last but not least, PMDNet is 40% faster than RPMNet, and 16% slower than DCP v2. However, it is 3× slower than IDAM.
PMD feature is defined to encourage the key points, however, coordinates show little robustness against noise. That's why an obvious decline in accuracy is witnessed on noisy clouds. We are interested in further studies focused on robust detection and feature extraction on key points.