The development of neural networks for point cloud analysis has drawn a lot of interests in recent years [24, 26, 34, 30, 16, 31, 44, 21, 43], and applied to various 3D applications, e.g., shape classification, object detection, semantic scene segmentation, etc. However, the features learned by these networks are not rotation invariant, meaning that they consider a point cloud and an arbitrary rotation of it as two different shapes. Thus, they may extract different features for the same shape, that is merely embedded in different poses in 3D.
To alleviate this fundamental problem, a common approach is to apply rotation augmentation to the training data. However, aggressive rotation augmentation, like arbitrary 3D rotations, often harms the recognition performance, since most existing networks do not have strong capacity to learn effective features from such unstable inputs. Thus, often, only azimuthal rotations (around the gravity axis) are considered. However, such limited augmentation does not generalize well, which could lead to a significant performance drop. See Figure 1(a), e.g., the classification performance of the recent ShellNet  drops from 93.1% down to 19.9%, on rotations with arbitrary axes.
Recently, some works attempt to design neural networks with rotation invariance [23, 27, 9, 42, 5]. One approach employs spherical-related convolutions, and the other employs local rotation-invariant features, e.g., distances and angles, to replace Cartesian coordinates as the network inputs. However, as we shall show, both approaches have limited success. Typically, the former approach has limited capability to embed features and is still sensitive to the rotations, while the latter one encodes mainly local information, which may not be unique, and often, the performance is much lower than using global Cartesian coordinates; see a part segmentation comparison in Figure 1(b) and various quantitative comparison results in Section 3.3, which demonstrate the superiority of our method over the state-of-the-art rotation-invariant methods.
In this work, we revisit the problem of rotation invariance in deep 3D point cloud analysis, and enumerate the considerations for achieving rotation invariance in the aspects of network inputs and network processing.
Accordingly, we then design an effective low-level representation to replace 3D Cartesian coordinates as the network inputs.
Our representation is purely rotation-invariant and encodes both local and global information, as well as being robust to noise and outliers
robust to noise and outliers. Also, we present a deep hierarchical network to embed these low-level representations into high-level features and to extract local relations between points and their neighbors, together with the global shape information. Further, to alleviate global information loss caused by the rotation-invariant representations, we enrich the network features with more global information by introducing a novel region relation convolution in the network to extract both local and non-local information across the hierarchy. Lastly, we evaluate the effectiveness of our method on various point cloud analysis tasks, including shape classification, part segmentation, and shape retrieval. Experimental results confirm that, our method achieves not only consistent results on inputs at any orientation, but also the best performance on all tasks compared with state-of-the-arts.
1.1 Related Works
Deep learning on 3D point sets. The design of robust and effective neural networks to embed point features has been an emerging topic in recent years. The pioneering networks PointNet  and PointNet++  show the potentials of deep networks to directly process 3D point sets. To better capture local neighborhoods, several works [34, 30, 21] suggested to extract point features by considering local graphs. To address the irregular and orderless properties of point sets, some works defined convolutions on non-Euclidean domains, e.g.
, the self-organizing map, -transformation , permutohedral lattice , and parameterized embedding . Some others designed new convolution operators on points, e.g., Monte Carlo convolution , PointConv , ShellConv , and KPConv . Besides supervised methods, some unsupervised and self-supervised networks [39, 45, 6, 13, 28, 14] were designed recently to avoid tedious manual labeling. Besides object recognition, some networks were designed for point set registration [2, 33, 22], upsampling [41, 40, 19], and denoising [15, 46]. Although these networks are translation and permutation invariant, they are not rotation invariant. They embed different features, and likely produce different outputs for the same input given in different orientations.
Rotation-invariant networks for 3D shapes. Since vanilla CNNs only have translation invariance, some works attempted to learn rotationally-equivariant features by designing spherical CNNs [11, 8] and 3D steerable CNNs . These features rotate correspondingly with the input. While these methods generalize well to unseen orientations, their convolutions are defined in a non-spatial domain, thus leading to poorer learning capability than spatial convolutions on regular grids. Also, they can only handle meshes or regular voxel grids.
Recently, some works explored rotation-invariant networks for point clouds. Poulenard et al.  represented points using volume functions, then used spherical harmonics kernels for convolution. The feature embedding capability of such convolution is, however, limited. Rao et al.  adaptively projected points on a discretized sphere and designed a hierarchical feature learning architecture to capture patterns on the sphere. However, the discretized sphere still carries a global orientation and cannot guarantee perfect symmetry, so the learned features are not purely rotation invariant. Hence, a notable performance drop still exists for inputs at arbitrary orientations.
On the other hand, some other methods suggested using low-level rotation-invariant geometric features to replace 3D Cartesian coordinates as the network inputs. Deng et al. 
suggested relative angles between point-wise normal vectors and paired distances. Chenet al.  suggested relative angles between two-point vectors, and vector norm, etc. Zhang et al.  constructed a point’s neighborhood with local triangles, each formed by a reference point, a neighbor point, and the local neighborhood centroid. They then take the triangle side lengths and angles as the rotation-invariant features. Though these representations are rotation invariant, they encode mainly local information, which may not be unique and sufficient; see Section 2.1 for a detailed analysis. In this work, we present a new rotation-invariant representation, capturing both local neighborhood and global shape structures, while being robust to noise and outliers. Also, we formulate a deep network, and introduce a novel region relation convolution to hierarchically process the point regions.
2.1 General Model for Point Feature Extraction
To start, we review a general model for point feature extraction. Denoteas a point cloud of points, where is the 3D Cartesian coordinate of the -th point in . To extract features for a point, say , a general model would include both local and global information, so can be written as
where denotes the global shape information at ; denotes the local shape information at with its -th neighbor point (); is a nonlinear function with learnable parameters ; and is a symmetric aggregation operation, e.g., max or summation, over the neighbor points of .
For general points processing networks without considering rotation invariance, is simply represented by , since 3D coordinates are global. For , different networks have different choices, e.g., PointNet++  uses as , while DGCNN  uses relative position, i.e., , as . Clearly, both and are based on 3D coordinates, so they are not rotation invariant.
To achieve rotation invariance, current attempts [9, 42, 5] proposed different point-wise purely rotation-invariant representations as the network inputs. However, they focus mainly on encoding the local relations between nearby point pairs using, e.g., distances and relative angles, and ignore . Also, most existing methods suffer from the ambiguity of distinguishing between local shapes, meaning that they may produce the same representation for points of different local configurations. More seriously, we could have information loss, where the embedded features are insufficient to describe the underlying shapes.
2.2 Considerations for Rotation Invariance
For a deep points processing network to be rotation invariant, both the network inputs and operations should be rotation invariant. Hence, before we present the design of our network inputs (Section 2.3) and network architecture (Section 2.4), we first discuss the relevant design considerations that we have taken:
Considerations for designing the network inputs.
Denoting as the function to extract rotation-invariant representations (network inputs) from point cloud , a purely rotation-invariant should satisfy
where SO(3)111SO(3) is the space of all 3D rotations in . is an arbitrary rotation. Most geometric quantities are rotation-variant, e.g., Cartesian coordinates and vectors in 3D space. Hence, we build our point-wise representation by carefully choosing rotation-invariant information inside a not-so-small local neighborhood around the point.
Second, simply using distances and relative angles between nearby point pairs may cause a large amount of information loss and easily introduce ambiguity in the representations, as explained earlier in Section 2.1. Hence, we avoid these issues by combining both rotation-invariant global information and rotation-invariant local point representations.
Last, noise is often unavoidable when scanning 3D point clouds. Hence, should be noise tolerant, meaning that the rotation-invariant representations extracted by should not be too sensitive to noise in .
Considerations for designing the network architecture.
A rotation-invariant network should not take point coordinates but only relative geometric information, such as distances and angles, as its inputs. However, without absolute information defined in a global coordinate frame, the network would lack global information. Hence, we should extract more global features, even from the relative geometric inputs, by considering more global relations among points. Existing rotation-invariant methods did not explore the global point relations, as in our work.
Besides, the network should not assume specific order (which may not be rotation invariant) when processing/aggregating point and regional features.
2.3 Our Rotation-Invariant Representations
Before extracting our rotation-invariant representations for and , we first normalize input point cloud to fit it in the origin-centered unit sphere. Then, for each point in (e.g., the red point in Figure 2(a)), we follow PointNet++  to use a query ball of radius to locate neighbor points (including itself) as its local neighborhood (blue points in Figure 2(b)).
Now, we are ready to extract and of . Here, should capture ’s location relative to the whole object and to its local neighborhood, serving like global but rotation-invariant coordinates of in the object. On the other hand, of should capture the local shape around , so we model a local representation for each , serving like local rotation-invariant coordinates of in ’s local neighborhood.
Global representation includes the following five pieces of rotation-invariant information about (see Figure 2(b) for an illustation of these five components).
(i) , simple but global and rotation-invariant information about .
(ii) , the distance from to ’s local neighborhood center (denoted as ). Here, a common choice of is ’s centroid (arithmetic mean), but such is sensitive to outliers and noise, so may not be stable. We propose to use the geometric median, i.e., the point with minimal distance sum to all . Such a choice is more stable, but computationally expensive . So, we resort to a fast but approximate procedure based on the idea of divide and conquer: we first randomly and independently pick subsets of points in , find the centroid of each subset, and cluster the centroids. Then, we take the mean of the centroids in the largest cluster as . Please refer to Section 3.1 for hyper-parameters and , Section 3.6 for a noise tolerance experiment, and our supplementary material for an evaluation of the approximated .
(iii)-(v) we locate (purple point in Figure 2(b)), the intersection between the query ball and line extended from origin to , and form triangle --. Then, we consider , the distance from to , and the cosine of the two angles subtended at and (denoted as and ) as the last three components of ; see Figure 2(b).
In our implementation, the query ball radius increases with the network layer (see Section 2.4), so the underlying structure described by triangle -- will enlarge gradually. To sum up, our global rotation-invariant representation is
Also, note that all distances range (since the input point cloud has been normalized), whereas angles and range . To avoid numerical instability, which hinders the network learning, we use cosine of these two angles in .
Local representation should help uniquely locate relative to in ’s local neighborhood. First, we construct a tetrahedron by joining to triangle --, and consider the three distances , , and from to points , , and , respectively, and the three angles , , and subtended at on three tetrahedron faces; see Figure 2(c). Using these information alone may be ambiguous, since a mirror point of on the opposite side of triangle -- can have the same set of distances and angles; see Figure 2(c). So, we further consider , the angle for rotating plane of triangle -- to plane of triangle -- about line -.
Again, to avoid numerical instability, we take cosine of , , and . As for , since it ranges , we use a nonlinear function , which is monotonic for and also ranges . To sum up, our ambiguity-free local rotation-invariant representation for point relative to is
Please refer to supplementary file for the proof on the ambiguity-free property.
Overall, for each point , we employ Eq. (3) to obtain its and Eq. (4) to obtain its (for each of its neighbor points ). Then, we pack copies of with to form a matrix (see Figure 3) to store the global and local rotation-invariant representations for .
Global relation between points. To supplement and with more global and rotation-invariant information, we further construct , an matrix to encode the global relations between all point pairs in a point cloud (say, of points), where matrix elements and encode the distance between and , and angle between the two vectors from origin to and , respectively. These information are later fed into the region relation convolution in the network to regress point-wise relation weights; see Section 2.4.
2.4 Network Architecture
Guided by the considerations presented in Section 2.2, we design a deep hierarchical network of three layers to embed a rotation-invariant codeword of the input point cloud. Figure 4 illustrates the network architecture, where the green boxes denote 3D point coordinates (e.g., ) sampled from the input point cloud; yellow boxes denote extracted rotation-invariant representations (see Section 2.3); purple boxes denote point indices from farthest sampling; and blue boxes denote embedded features in the network.
Specifically, given a point set of points, like PointNet++ , we first adopt a sample-and-group operator. That is, we use farthest sampling to select a subset of points, then for each sampled point, we use a query ball to find its neighbor points and group an volume of 3D point coordinates; see in Figure 4. We then follow the steps in Section 2.3 to map it into our rotation-invariant representations (yellow box) and compute an global relation matrix (yellow box) from the sampled points. Further, we feed and into the region relation convolution (to be presented later) to obtain the feature map (blue box) of the first layer.
The second layer continues to sample-and-group into a smaller point subset and uses the same set of indices (Idx) to group into . Note that, we set and to allow a progressively enlarging receptive field in the hierarchy. Instead of directly feeding into region relation convolution for feature embedding, we avoid information loss by concatenating and , which are high-level features extracted from low-level representations
via a series of multi-layer perceptron (MLPs); see Figure4. We then feed the concatenated features , together with another global relation matrix from , to another region relation convolution to generate as the output from the second layer.
Further, the third layer samples-and-groups into , and uses the concatenated features for convolution. Now, we only have one single point together with its neighbors (
), so we directly use MLPs followed by max-pooling alongon to produce the global feature vector , which is a rotation-invariant codeword of the input point set.
Next, we can use in various point cloud analysis tasks. For examples, for shape classification, we can follow the common routine of using fully-connected layers to regress the class scores. For part
segmentation, we can adopt the point feature propagation and interpolation to recover the per-point features, then use MLPs to regress per-point scores; please see 
for details. For shape retrieval, we can directly compare the cosine similarity between the codewords of the query and target point clouds.
Region relation convolution. To alleviate the inevitable global information loss in the rotation-invariant representations, we further formulate the region relation convolution (see Figure 5 for its illustration) to regress global region relation weight from the global relation matrix (which is or ) and to refine feature extracted from or . Here, for each reference point and its local neighbors, previous networks [26, 34] commonly apply shared MLPs to the point features and max-pooling along to obtain a feature vector for encoding the local structure around the reference point. The same operation is applied to all points to obtain an feature map . Such operation, however, considers only ’s own local region when extracting point features for , without looking at its relations with other points more globally.
To introduce more global information into the embedded features, compared with conventional convolutions [26, 34], after the shared MLPs and max-pooling, we refine features by regressing rotation-invariant region relation weight from the global relation matrix ; see the top branch in Figure 5. The weights in each row, say , are regressed based on distances and angles of relative to all the other points (see the last paragraph in Section 2.3 for details), so reveals certain global relations between and other points. We then bring such global information into by ), where and mean element-wise addition and multiplication, respectively. Hence, the features of each point encode not only the local structure around the associated point, but also certain non-local relations with other local structures.
3.1 Implementation Details
We implemented our network using TensorFlow
and trained it for 200 epochs in all tasks. Adam optimizer was used with a learning rate of 0.001 and a mini-batch size of six. Also, we set and , and followed  to capture multi-scale local regions with different and in each layer. Besides, we empirically set and to balance the computing time and stability in finding the approximate geometric median. For details on the hyper-parameter settings (i.e., , , and ), please refer to the supplementary material. We shall release our trained models with code upon the publication of this work.
To evaluate the robustness of our network on inputs of arbitrary orientations, besides conventional data augmentation strategies by random scaling and jittering, we followed the settings in recent rotation-invariant methods [42, 5] to train and test our network in three scenarios: (i) z/z (as a reference): train and test with rotation augmentation about azimuthal axis, (ii) z/SO3: train with azimuthal rotations and test with arbitrary rotations, and (iii) SO3/SO3: train and test with arbitrary rotations. Overall, it is expected that an effective rotation-invariant approach should have consistent performance for all scenarios. In the followings, we evaluate the performance of our method against others, both quantitatively and qualitatively, on three tasks: shape classification (Section 3.2), part segmentation (Section 3.3), and shape retrieval (Section 3.4). Then, we show the network component analysis (Section 3.5) and noise tolerance test (Section 3.6).
|Method||z/z (reference)||z/SO3||drop by||SO3/SO3||drop by|
|SubVolSup MO ||89.5||45.5||49.2%||85.0||5.0%|
|PointNet++ (MSG) ||90.7||28.6||68.5%||85.0||6.3%|
3.2 Evaluation: 3D Shape Classification
First, we evaluate our method on the 3D shape classification task by comparing it with both rotation-variant and rotation-invariant methods using the standard ModelNet40 dataset , which has 12,311 CAD models from 40 categories. We adopted the standard split to train our network using 9,843 models and tested it using the remaining 2,468 models. Each input point cloud has 1024 points.
Comparison with rotation-variant methods. Table 1 compares the drop in accuracy (%) for handling inputs at arbitrary rotations. First, existing rotation-variant methods have significant accuracy drops in z/SO3 as compared with z/z, showing that they are not rotation-invariant. Second, for the results in SO3/SO3, their performance still drops considerably, though arbitrary rotations in data augmentation can help improve their performance when testing in SO3. This means their networks cannot learn in SO3/SO3 as effective as in z/z. In contrast, our method has no accuracy drop for z/SO3, which validates the rotation invariance of our method. Also, it outperforms others when testing on SO3, no matter trained in z or in SO3. Note that, the slight drop in accuracy (i.e., 0.1%) of our method in SO3/SO3 is caused by the network re-training.
|Method||z/z (reference)||z/SO3||drop by|
|Spherical CNN ||88.9||78.6||11.6%|
Comparison with rotation-invariant methods. Next, we compare our method with four most recent rotation-invariant methods. From the results shown in Table 2, we can see that the accuracy of the spherical-related methods (Spherical CNN & SFCNN) drops considerably; since their models cannot guarantee perfect symmetry in rotations, their results are still sensitive to rotations. For RI-ShellConv  and ClusterNet , they are formulated with pure rotation invariance, so they have consistent performance. Yet, our method still outperforms them, since our rotation-invariant representations encode both local and global information, and our network can effectively learn high-level features more globally with the help of the region relation convolution and global relations.
|Method (z/SO3)||aero||bag||cap||car||chair||earph.||guitar||knife||lamp||laptop||motor||mug||pistol||rocket||skate||table||avg. mIoU|
|PointNet++ (MSG) ||51.3||66.0||50.8||25.2||66.7||27.7||29.7||65.6||59.7||70.1||17.2||67.3||49.9||23.4||43.8||57.6||48.3|
|Method (SO3/SO3)||aero||bag||cap||car||chair||earph.||guitar||knife||lamp||laptop||motor||mug||pistol||rocket||skate||table||avg. mIoU|
|PointNet++ (MSG) ||79.5||71.6||87.7||70.7||88.8||64.9||88.8||78.1||79.2||94.9||54.3||92.0||76.4||50.3||68.4||81.0||76.7|
3.3 Evaluation: 3D Object Part Segmentation
Next, we evaluate our method on 3D object part segmentation by comparing it with both rotation-variant methods and the recent rotation-invariant method RI-ShellConv  using the ShapeNet dataset . This dataset has 16,881 models from 16 categories, and is annotated with 50 parts. We adopt the per-category averaged intersection over union (mIoU) metric  in the evaluation. Note that we do not compare with ClusterNet , since it is designed for classification.
Table 3 shows the per-category mIoU and averaged mIoU (over all 16 categories) produced by different methods in scenarios z/SO3 and SO3/SO3. Comparing the results shown in top and bottom tables, we can see that the rotation-variant methods yield very different segmentation results for z/SO3 and SO3/SO3, while both RI-ShellConv and our method achieve more consistent performance when tested on inputs at arbitrary rotations. Also, our method outperforms RI-ShellConv and others with the highest averaged mIoU; see the right-most columns in the two tables. Again, due to network re-training, although both RI-ShellConv and our method are rotation invariant, there are slight difference in the results for z/SO3 and SO3/SO3. Further, we show some typical visual comparison results in z/SO3 in Figures 1(b) and 6, where the segmentation results produced by our method are the closest to the ground truths, compared with others. Please see supplementary material for more visual comparisons.
3.4 Evaluation: 3D Shape Retrieval
Besides 3D shape classification and object part segmentation, we further evaluate our method on 3D shape retrieval using the perturbed ShapeNet Core55 dataset . Here, we followed the rules of the SHREC’17 3D shape retrieval contest 
, where each model has been randomly rotated by a uniformly-sampled rotation in SO(3). For a fair comparison, we trained and tested all methods on the provided training/validation/testing sets, and evaluated their performance with the official evaluation metrics,i.e., precision (P@N), recall (R@N), F1-score (F1@N), mean average precision (mAP), and normalized discounted cumulative gain (NDCG). On each shape, 2,048 points are sampled as the network input. To combine the retrieval results of different categories, we followed  to use the macro and micro average strategies on the above five metrics. For a better demonstration, we also compute the average score over all the metrics.
Table 4 reports the evaluation results. Overall, a larger metric value indicates a better retrieval performance. Compared with the contest winner  and also the recent rotation-invariant methods [27, 42], our method achieves the best performance for most evaluation metrics (six out of ten metrics), and also the best average score with a large margin compared with others.
|Furuya  (contest winner)||0.814||0.683||0.706||0.656||0.754||0.607||0.539||0.503||0.476||0.560||0.630|
3.5 Network Component Analysis
|Scenario||Ablation study||Rot.-inv. representation||Network architecture||Full pipeline|
|Case #1||Case #2||RI-ShellConv||PointNet++||DGCNN|
Next, we conduct an ablation study, an analysis on our rotation-invariant representation, and a network architecture analysis to evaluate different aspects of our method using the shape classification task on ModelNet40.
Ablation study. First, we evaluate two major modules in our method:
Case #2. To verify our proposed region relation convolution, we degenerate it into just the shared MLPs followed by max-pooling (see Figure 5).
The leftmost portion of Table 5 shows the results of the two cases. Since both cases are rotation invariant, their classification accuracies are consistent for z/z, z/SO3, and SO3/SO3, so we report only the accuracies under z/SO3. By comparing the result with our full pipeline (rightmost in Table 5), we can see that each module (case) contributes to achieve a better classification performance.
Rotation-invariant representation analysis. To verify the effectiveness of our rotation-invariant representation (both and , as depicted in Figure 3), we replace it with the state-of-the-art rotation-invariant representation proposed in . The resulting classification accuracy is shown in the middle portion of Table 5. Comparing with the full-pipeline result (rightmost in Table 5), we can see that our network achieves better performance with our rotation-invariant representation (89.4%) than with the representation in  (87.8%). However, such performance (87.8%) is still higher than the performance (86.4%) of  (see Table 2). The difference reveals that although both cases use the same rotation-invariant representation, our network with the region relation convolution and global relation information can achieve better performance.
Network architecture analysis. To verify the effectiveness of our network (Figure 4), we replace it with PointNet++  and DGCNN , respectively, while keeping our rotation-invariant representations as the network inputs. The “network architecture” column in Table 5 shows the results. Apparently, our network (full pipeline) achieves higher performance. Also, we explore our network performance with different number of layers; see supplementary material.
3.6 Noise Tolerance Test
Noise is common in the acquisition of 3D point clouds. This motivates us to introduce the geometric median (which is less sensitive to noise) for formulating our rotation-invariant representations. To study our method’s robustness to noise, we test its shape classification performance on ModelNet40 using inputs that are corrupted by Gaussian noise of increasing level (variance). In this test, we consider four cases: (i) our method with geometric median; (ii) our method with arithmetic mean; (iii) ClusterNet ; and (iv) RI-ShellConv .
Figure 7 plots the shape classification accuracy for the four cases over shape inputs of increasing amount of noise. From the results, we can see that using geometric median consistently achieves better performance than using arithmetic mean, while existing rotation-invariant methods are more sensitive to noise.
We presented a rotation-invariant framework for deep 3D point cloud analysis. Given an input cloud at arbitrary orientation, our framework produces consistent, and also the best performance, on multiple point cloud analysis tasks, including shape classification, part segmentation, and shape retrieval, compared with the state-of-the-arts. To achieve this, we introduce a novel low-level purely rotation-invariant representation as the network inputs, which encodes both local and global information, as well as being robust to noise and outliers. Further, we formulate the region relation convolution to enrich the network features with more global information. The extensive experimental results confirm the rotation invariance of our method, and also its superiority over the state-of-the-arts.
Despite the effectiveness of our method (see Figure 7) on handling noisy inputs as compared with others, the performance still drops progressively when the noise becomes larger. In the future, we plan to explore the possibility of designing a noise-resistant network, since LiDAR-scanned real inputs are often contaminated by large amount of noise, particularly for outdoor situations. Besides, we plan also to extend our rotation-invariant framework for the problems of point cloud registration and partial shape matching.
Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., et al.: TensorFlow: A system for large-scale machine learning. In: 12th USENIX Symposium on Operating Systems Design and Implementation. pp. 265–283 (2016)
Aoki, Y., Goforth, H., Srivatsan, R.A., Lucey, S.: PointNetLK: Robust & efficient point cloud registration using PointNet. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 7163–7172 (2019)
-  Chang, A.X., Funkhouser, T., Guibas, L.J., Hanrahan, P., Huang, Q., Li, Z., Savarese, S., Savva, M., Song, S., Su, H., et al.: ShapeNet: An information-rich 3D model repository. arXiv preprint arXiv:1512.03012 (2015)
-  Chen, C., Fragonara, L.Z., Tsourdos, A.: GAPNet: Graph attention based point neural network for exploiting local feature of point cloud. arXiv preprint arXiv:1905.08705 (2019)
Chen, C., Li, G., Xu, R., Chen, T., Wang, M., Lin, L.: ClusterNet: Deep hierarchical cluster network with rigorously rotation-invariant representation for point cloud analysis. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 4994–5002 (2019)
-  Chen, S., Duan, C., Yang, Y., Li, D., Feng, C., Tian, D.: Deep unsupervised learning of 3D point clouds via graph topology inference and filtering. arXiv preprint arXiv:1905.04571 (2019)
Cohen, M.B., Lee, Y.T., Miller, G., Pachocki, J., Sidford, A.: Geometric median in nearly linear time. In: Proceedings of the forty-eighth annual ACM symposium on Theory of Computing. pp. 9–21. ACM (2016)
-  Cohen, T., Geiger, M., Köhler, J., Welling, M.: Spherical CNNs. In: Int. Conf. on Learning Representations (ICLR) (2018)
Deng, H., Birdal, T., Ilic, S.: PPF-FoldNet: Unsupervised learning of rotation invariant 3D local descriptors. In: European Conf. on Computer Vision (ECCV). pp. 602–618 (2018)
-  Duan, Y., Zheng, Y., Lu, J., Zhou, J., Tian, Q.: Structural relational reasoning of point clouds. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 949–958 (2019)
-  Esteves, C., Allen-Blanchette, C., Makadia, A., Daniilidis, K.: Learning SO(3) equivariant representations with spherical CNNs. In: European Conf. on Computer Vision (ECCV). pp. 52–68 (2018)
-  Furuya, T., Ohbuchi, R.: Deep aggregation of local 3D geometric features for 3D model retrieval. In: British Machine Vision Conf. (BMVC). pp. 121.1–121.12 (2016)
-  Han, Z., Wang, X., Liu, Y.S., Zwicker, M.: Multi-angle point cloud-VAE: Unsupervised feature learning for 3D point clouds from multiple angles by joint self-reconstruction and half-to-half prediction. In: IEEE Int. Conf. on Computer Vision (ICCV). pp. 10442–10451 (2019)
-  Hassani, K., Haley, M.: Unsupervised multi-task feature learning on point clouds. In: IEEE Int. Conf. on Computer Vision (ICCV). pp. 8160–8171 (2019)
-  Hermosilla, P., Ritschel, T., Ropinski, T.: Total Denoising: Unsupervised learning of 3D point cloud cleaning. In: IEEE Int. Conf. on Computer Vision (ICCV). pp. 52–60 (2019)
-  Hermosilla, P., Ritschel, T., Vázquez, P.P., Vinacua, À., Ropinski, T.: Monte Carlo convolution for learning on non-uniformly sampled point clouds. ACM Trans. on Graphics (SIGGRAPH Asia) 37(6), 235:1–12 (2018)
-  Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. In: Int. Conf. on Learning Representations (ICLR) (2015)
-  Li, J., Chen, B.M., Hee Lee, G.: SO-Net: Self-organizing network for point cloud analysis. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 9397–9406 (2018)
-  Li, R., Li, X., Fu, C.W., Cohen-Or, D., Heng, P.A.: PU-GAN: A point cloud upsampling adversarial network. In: IEEE Int. Conf. on Computer Vision (ICCV). pp. 7203–7212 (2019)
-  Li, Y., Bu, R., Sun, M., Wu, W., Di, X., Chen, B.: PointCNN: Convolution on -transformed points. In: Conference and Workshop on Neural Information Processing Systems (NeurIPS). pp. 820–830 (2018)
Liu, Y., Fan, B., Xiang, S., Pan, C.: Relation-shape convolutional neural network for point cloud analysis. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 8895–8904 (2019)
-  Lu, W., Wan, G., Zhou, Y., Fu, X., Yuan, P., Song, S.: DeepVCP: An end-to-end deep neural network for point cloud registration. In: IEEE Int. Conf. on Computer Vision (ICCV). pp. 12–21 (2019)
-  Poulenard, A., Rakotosaona, M.J., Ponty, Y., Ovsjanikov, M.: Effective rotation-invariant point CNN with spherical harmonics kernels. In: Int. Conf. on 3D Vision (3DV). pp. 47–56 (2019)
-  Qi, C.R., Su, H., Mo, K., Guibas, L.J.: PointNet: Deep learning on point sets for 3D classification and segmentation. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 652–660 (2017)
-  Qi, C.R., Su, H., Nießner, M., Dai, A., Yan, M., Guibas, L.J.: Volumetric and multi-view CNNs for object classification on 3D data. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 5648–5656 (2016)
-  Qi, C.R., Yi, L., Su, H., Guibas, L.J.: PointNet++: Deep hierarchical feature learning on point sets in a metric space. In: Conference and Workshop on Neural Information Processing Systems (NeurIPS). pp. 5099–5108 (2017)
-  Rao, Y., Lu, J., Zhou, J.: Spherical fractal convolutional neural networks for point cloud recognition. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 452–460 (2019)
-  Sauder, J., Sievers, B.: Self-supervised deep learning on point clouds by reconstructing space. In: Conference and Workshop on Neural Information Processing Systems (NeurIPS) (2019), to appear
-  Savva, M., Yu, F., Su, H., Kanezaki, A., Furuya, T., Ohbuchi, R., Zhou, Z., Yu, R., Bai, S., Bai, X., et al.: SHREC17’ track: Large-scale 3D shape retrieval from ShapeNet Core55. In: Proceedings of the Eurographics Workshop on 3D Object Retrieval (2017)
-  Shen, Y., Feng, C., Yang, Y., Tian, D.: Mining point cloud local structures by kernel correlation and graph pooling. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 4548–4557 (2018)
-  Su, H., Jampani, V., Sun, D., Maji, S., Kalogerakis, E., Yang, M.H., Kautz, J.: SPLATNet: Sparse lattice networks for point cloud processing. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 2530–2539 (2018)
-  Thomas, H., Qi, C.R., Deschaud, J.E., Marcotegui, B., Goulette, F., Guibas, L.J.: KPConv: Flexible and deformable convolution for point clouds. In: IEEE Int. Conf. on Computer Vision (ICCV). pp. 6411–6420 (2019)
-  Wang, Y., Solomon, J.M.: Deep closest point: Learning representations for point cloud registration. In: IEEE Int. Conf. on Computer Vision (ICCV). pp. 3523–3532 (2019)
-  Wang, Y., Sun, Y., Liu, Z., Sarma, S.E., Bronstein, M.M., Solomon, J.M.: Dynamic graph CNN for learning on point clouds. ACM Trans. on Graphics 38(5), 146:1–12 (2019)
-  Weiler, M., Geiger, M., Welling, M., Boomsma, W., Cohen, T.: 3D steerable CNNs: Learning rotationally equivariant features in volumetric data. In: Conference and Workshop on Neural Information Processing Systems (NeurIPS). pp. 10381–10392 (2018)
-  Wu, W., Qi, Z., Fuxin, L.: PointConv: Deep convolutional networks on 3D point clouds. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 9621–9630 (2019)
-  Wu, Z., Song, S., Khosla, A., Yu, F., Zhang, L., Tang, X., Xiao, J.: 3D ShapeNets: A deep representation for volumetric shapes. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 1912–1920 (2015)
-  Xu, Y., Fan, T., Xu, M., Zeng, L., Qiao, Y.: SpiderCNN: Deep learning on point sets with parameterized convolutional filters. In: European Conf. on Computer Vision (ECCV). pp. 87–102 (2018)
-  Yang, Y., Feng, C., Shen, Y., Tian, D.: FoldingNet: Point cloud auto-encoder via deep grid deformation. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 206–215 (2018)
-  Yifan, W., Wu, S., Huang, H., Cohen-Or, D., Sorkine-Hornung, O.: Patch-based progressive 3D point set upsampling. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 5958–5967 (2019)
-  Yu, L., Li, X., Fu, C.W., Cohen-Or, D., Heng, P.A.: PU-Net: Point cloud upsampling network. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 2790–2799 (2018)
-  Zhang, Z., Hua, B.S., Rosen, D.W., Yeung, S.K.: Rotation invariant convolutions for 3D point clouds deep learning. In: Int. Conf. on 3D Vision (3DV). pp. 204–213 (2019)
-  Zhang, Z., Hua, B.S., Yeung, S.K.: ShellNet: Efficient point cloud convolutional neural networks using concentric shells statistics. In: IEEE Int. Conf. on Computer Vision (ICCV). pp. 1607–1616 (2019)
-  Zhao, H., Jiang, L., Fu, C.W., Jia, J.: PointWeb: Enhancing local neighborhood features for point cloud processing. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 5565–5573 (2019)
-  Zhao, Y., Birdal, T., Deng, H., Tombari, F.: 3D point capsule networks. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR). pp. 1009–1018 (2019)
-  Zhou, H., Chen, K., Zhang, W., Fang, H., Zhou, W., Yu, N.: DUP-Net: Denoiser and upsampler network for 3D adversarial point clouds defense. In: IEEE Int. Conf. on Computer Vision (ICCV). pp. 1961–1970 (2019)