Geodesic Self-Attention for 3D Point Clouds
Anonymous Author(s)
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Abstract
Due to the outstanding competence in capturing long-range relationships, Selfattention mechanism has achieved remarkable progress in point cloud tasks.
Nevertheless, point cloud object often has complex non-Euclidean spatial structures, withthe behavior changing dynamically and unpredictably.
Most current self-attentionmodules highly rely on the dot product multiplication in Euclidean space, whichcannot capture internal non-Euclidean structures of point cloud objects, especiallythe long-range relationships along the curve of the manifold surface of point cloudobject.
To address this problem, in this paper, we introduce a novel metric on the
Riemannian manifold to capture the long-range geometrical dependencies of pointcloud objects to replace traditional self-attention modules, namely, the Geodesic
Self-attention (GSA) module.
Our approach achieves state-of-the-art performanceon object classification, few-shot learning and part segmentation benchmarks
Introduction
Motivated by the rapid development of applications in robotics, autonomous vehicles, and so on,research in 3D information processing with deep neural networks (DNNs) is likewise undergoing arevolution.
As one of the most readily available and vital data in the 3D vision field, point cloud hasgarnered considerable research attentions.
The essence of a point cloud is discrete 3D coordinates
[Figure]
sampled from a continuous 3D shape.
Therefore, it isalso typically considered to represent a geometric surface or a 2D manifold embedded in Euclidean space.3D point cloud data contains only geometric coordinate information, whereas word vector and imagepixel matrix contain rich semantic features.
Thus thegeometric features should be the first concern whenanalyzing point cloud.
Although there have been alot of impressive works on point cloud geometricanalysis methods in the past [26, 36, 15, 13], most ofthem are based on the Euclidean or local geometry,which is not sufficient to represent the non-Euclideangeometric semantics of point clouds accurately.
Oneof the most typical geometric-semantic confusion scenarios is illustrated in Fig.1 to further elaborate onthis notion.
To address this problem, we hope to introduce an accurate metric that can capture both thelocal and global features of the point cloud.
Associating the powerful capability of capturing the long-range dependencies of Transformer [34], we think itcould be the critical bridge to introduce such a metric.
Nowadays, Transformer is another popular research topic.
Relying on the powerful global informationcapturing and long-range dependencies aggregation capabilities of self-attention mechanism, various types of networks with Transformer as the backbone have dominated in areas such as natural languageprocessing [7, 28, 3] and computer vision [9, 2, 23, 14, 11].
Although existing dot-product selfattention [9] is effective in image processing, in 3D tasks, especially in point cloud tasks, we inferthat the dot product multiplication in Euclidean space may not be the optimal attention solution, forits limitation in capturing internal non-Euclidean structures of point cloud objects.
Since existingworks [13, 10, 42, 24] have achieved impressive results, we just want to explore whether there issome better and geometrically relevant metric for the self-attention mechanism on point cloud tasks.
Inspired by Riemannian geometry [17], we introduce a metric on Riemannian manifold into pointcloud analysis.
For a Riemannian manifold, the geodesic represents the local shortest path betweentwo points and dictates the geometric relationship.
For the implicit surface points represented bypoint cloud feature vectors, the geodesic can obtain more accurate measurement between two pointsthan the dot product multiplication in Euclidean space.
Simply, the closer the geodesic distancebetween two data points has, the more closely related they are.
Following this idea, we compute theattention map by point-wise geodesic distance to capture the accurate global dependencies betweenpoint cloud data points.
In summary, in this paper, we propose a novel non-parametric geodesic self-attention (GSA) modulefor transformers on point cloud tasks and introduced it into a standard transformer structure, namely,
Point Geodesic Transformer (Point-GT).
Our main contributions are as follows:
• To the best of our knowledge, we are the first to propose that the Riemannian metric
should
be more appropriate for computing an attention map for the point cloud transformers and
the first to use geodesic attention to measure the topological relationships of the underlying
geometric structure represented by the point cloud.
• To verify the feasibility of our idea, we propose two feasible methods of the geodesic
distance computation for the point cloud features.
We apply the geodesic self-attention to
the standard transformer structure, finally achieving better convergence and state-of-the-art
performance with fewer
parameters and faster speed.
Related Work
Geometry Analysis of Point Cloud Point clouds are one of the most readily available 3D datain our daily lives.
They can be sampled from various 3D models discretely or captured by 3Dscanners (Lidar, RGB-D, et al.).
Due to the irregular, non-grid characteristics of point clouds, itwas common in the early days to project the original point cloud onto a multi-view map [41, 20] orvoxel [6, 29, 30] to utilize well-explored 2D convolution for processing.
PointNet [25] pioneered amethod of embedding point cloud coordinates directly using MLP, which inspired many subsequentpoint cloud analysis efforts.
PointNet ++ [26] introduced a hierarchical aggregation paradigm forpoint cloud local geometric information based on PointNet and became a common means of pointcloud embedding in many exciting point cloud work in the future.
DGCNN [36] proposed a graphbased means of point cloud processing by constructing local dynamic graphs to characterize edgerelationships between base points and neighboring nodes and using EdgeConv to learn implicitpoint cloud topological geometric features.
Inspired by transformers, PCT [13] first combined theself-attention mechanism with edge features and proposed the Offset-Attention.
Although the aboveworks have all made significant contributions to the geometric analysis of point cloud data, most ofthe geometric explorations of point clouds in the past have only considered the Euclidean metric orthe local geometry of point clouds.
For the learning ability of the self-attention mechanism for globalinformation and the geometric representation ability of geodesic, we combine both of them to explorethe global geometric features of point clouds.
Geodesic Distance Computation In previous work, the geodesic distance between 3D data isoften computed by path wandering on the mesh surface [4, 5, 31] or vertex shortest path estimation[8, 12, 16].
On the one hand, these works have an unbearable time complexity (often above O ( n 2 )for one graph).
On the other hand, most of the point cloud data in real-world scenarios do notcontain meshes, making it even more challenging to obtain geodesic distances.
In order to reduce thecomputational cost of geodesic distance during network training, GeoNet [15] approaches a Geodesic
Matching (GM) layer to learn potential geodesic features.
Before training the GM layer, the geodesicdistance of sampled points on the mesh surface has been manually computed as ground truth.
After training the GM layer, GeoNet utilizes the trained model to learn the potential geodesic features of theextracted point cloud features.
Although the GM layer alleviates the difficulty of acquiring geodesicfeatures during network training, we believe that pursuing the acquisition of explicit geodesic featurestends to increase the inductive bias and reduce the learnability and robustness of the network.
It is alsonot in line with nowadays mainstream network design thinking.
Whiteley et al.
[37] propose a graphbased method for recovering the true geodesic distance of data features from a high-dimensional
Hilbert space and prove the feasibility theoretically.
Qi et al.
[27] study few-shot learning with the
Riemannian metric by mapping the features extracted from Euclidean space on oblique manifold(OM).
The OM ( n, m ) is considered as a Riemannian submanifold of the embedding Euclidean space
R n × m with unit Euclidean norm, which is an intrinsic property similar to the l 2 - norm in Euclideanspace.
Due to these properties, we consider Qi’s mapping method can preserve the geometric featuresof the data.
Inspired by these works, we try to propose two methods to learn dynamic topologicalfeatures of point cloud data and obtain geodesic attention maps from the graph and data manifoldtheory, respectively.
Point Cloud Transformers Recently, Transformers have achieved great success in NLP.
Its powerful sequential data processing and global features aggregation capability play a significant role.
ViT[9] has successfully introduced the Transformers structure to computer vision tasks.
3D point clouds,a prevalent topic in computer vision, represent the underlying surface of 3D objects.
Therefore,the geometric properties are critical to point cloud analysis.
Since 3D point clouds can also beconsidered a special kind of sequence data, we believe Transformers’ structure is beneficial forgeometric analysis.
In addition, to the possible future applications of multimodal work with pointclouds and images, we think the Transformer backbone is well suited for point cloud tasks.
PCT[13] is one of the earliest works to use the Transformer structure for point cloud tasks, which designsa modified self-attention mechanism with the core idea of Edgeconv [36].
PoinTr [42] utilizes amodified DGCNN [36] as a feature embedding layer to capture geometric information.
Receivinginspiration from PoinTr, Point-BERT [43] designs a discrete Variational AutoEncoder (dVAE) forgenerating point tokens, successfully introducing BERT-style [7] pre-training to point cloud tasks.
Point-MAE [24] further explores pre-training methods for point cloud Transformers.
Inspired by
Masked Autoencoders (MAE) [14] self-supervised learning method, it implements a point cloud taskpre-training pipeline with entirely standard Transformer structures.
The above works build a solidfoundation for future work on point cloud tasks with Transformers.
However, these works are basedon dot-product self-attention of coordinate feature vectors.
We believe that such self-attention mapcomputation does not accurately represent the geometric structure of the point cloud.
To demonstrateour idea and solve the problem, we propose geodesic self-attention, which aggregates the globalgeometric features, for point cloud Transformers.
Point Geodesic Transformer
This section will detail our approach of introducing geodesic self-attention into the point cloud
Transformers structure.
First, we will describe how the Transformers structure can be applied to thepoint cloud task.
Then, we will describe how geodesic self-attention works in our network structure.
The overall structure of the Point Geodesic Transformer is illustrated in Fig.2.
3.1 Transformer Backbone
In order to compare with dot-product self-attention directly, inspired by Point-Bert [43], our mainexperiments are performed on the standard Transformer [34] structure, except for the modification ofthe self-attention function.
Patches Generation Unlike the classical visual tasks with regular grids, point clouds consist ofunordered points describing 3D coordinates.
It is challenging to find a suitable method to embedthe point cloud feature.
Following the previous point cloud Transformers [13, 10, 42, 24], we adoptan embedding strategy that encodes point clouds as irregular local point patches.
To be specific,given an input point cloud P ∈ R N × 3 ( P = { p 1 , p 2 , p 3 , · · · , p N } ) , we first perform Farthest Point
Sampling (FPS) to sample g center points from the raw point cloud data.
Then, the k -nearest neighbor( k NN) algorithm captures the g centers’ k nearest neighbor points, and a normalization operation isperformed to make the local neighbors unbiased.
F
PS · · ·
Patches
Positional Embedding
Embedding Layer
Transformer Blocks Transformer Blocks
Transformer Blocks Transformer Blocks · · ·
CLS CLS
Classification
W
GDS
SoftMax
MatMul
X
Geodesic Self-Attention
Points Embedding In order to maintain the permutation invariance of the point cloud [25], we donot choose the patches embedding method like ViT [9], which simply flattens the input patches andembeds them with a trainable linear projection.
For each input point patches, several MLP layersand aggregation operations which can be considered as a lightweight PointNet[25] are utilized toproject the point patches to point embedded features.
Given the center point x and its neighbors
X = { x 1 , x 2 , x 3 , · · · , x k } , the embedding operations can be described as:
[Equation]
where □ denotes an aggregation function and f Θ ( · ) can be considered as an embedding layer.
In thisstep, there are many optional aggregation functions [36], such as max - pooling and mean - pooling ,and the aggregation function captures the local geometric information of the input point patches.
Tobe consistent with the previous work [13, 10, 42, 24], we choose max - pooling as the aggregationfunction here.
As for the positional embedding, a naive thought is that this operation may be superfluous for the pointcloud data containing position information.
However, since the normalization operation mentionedin patches generation stage decouples the relative coordinate relationships, we need to map thelocation features of the corresponding patch centers to the patch features by a simple MLP.
Since theembedded feature can be processed directly by the standard Transformer, the rest structure of ournetwork is the same as the standard Transformer[34].
3.2 Geodesic Self-Attention
The original definition of attention function can be described as mapping a query and a set ofkey-value pairs to an output.
For the self-attention function, the query and the key are the inputvector itself.
Following the exact definition, the geodesic self-attention (GSA) score map canbe described as the point-wise geodesic distance scores of the input vectors.
Although the 3D pointcloud data represents the underlying shape of 3D objects, it does not contain an explicit manifoldsurface.
Therefore, how to define the geodesic of the points and how to obtain the geodesic distancescore are crucial.
Inspired by previous work [37, 27], we define two ways to compute the geodesicdistance score, based on graph and data manifold, respectively.
3.2.1 Geodesic Distance Score
Graph-based Geodesic Whiteley et al.
[37] recover the true distances and true coordinates betweenfeatures through matrix decomposition and dimensionality reduction.
Isomap [32] constructs the graph structure between feature points and calculates the geodesic distances between nodes by k NNand shortest path algorithms.
Since the previous studies [36, 19] have supported the analysis of pointclouds as a graph structure, we combine the thoughts from the above works, by firstly recovering thetrue distances of the input features of each Transformer block, then constructing a dynamic weightedadjacency graph for these features through k NN.
For adjacent points, we set the weights of theadjacency graph to Euclidean distance, and for non-adjacent points, we set the weights to infinity.
Webelieve that dynamically updating the adjacency graph can not only learn explicit geodesic featuresbut also recover the latent topological features of point clouds, which enrich the representation powerof point clouds [36].
After constructing the dynamic adjacency graph, we apply Floyd’s shortest pathalgorithm [12] to estimate the geodesic distance of all pairs of the points, and the output matrix is thegeodesic distance score matrix.
Algorithm
Input: X = { x 1 , x 2 , x 3 , · x g }
Output: GDS
Since Floyd’s algorithm is nonlinear and cannot be computed in parallel, we use Numba [18] toimplement a CUDA operator for computational acceleration.
Each computation of the adjacencygraph is implemented by a single CUDA kernel and achieves more than a hundred times fastercomputation.
To solve the back-propagation problem of Floyd’s algorithm, we record a table ofupdates of the adjacency graph and pass the gradient to the updated vectors, following the backpropagation pattern of max - pooling .
Data Manifold-based Geodesic Riemannian geometry is the study of smooth manifolds by introducing Riemannian metrics based on differential geometry, defining geometric concepts suchas angles, length of curves, etc.
The oblique manifold (OM) is a submanifold of the embedding
Euclidean space with unit Euclidean norm columns and is always utilized for independent componentanalysis [1].
Formally, OM ( n, g ) is defined as:
[Equation]
where diag ( · ) denotes the diagonal matrix.
Previous work [27] models few-shot learning on OM.
The authors project the data features extractedin the Euclidean space on OM and then design an oblique distance-based classifier to achieve theclassification task for the projected features.
Since the OM offers an unit l 2 - norm intrinsic property,we consider the above mapping preserves the relative geometric structure of the features, whichsatisfies the need for the attention mechanism to capture relative dependencies.
Following this notion,we consider constraining the point cloud features from Euclidean space to oblique manifold to extractthe underlying geometric structure.
Since the positional embedding recovers the relative geometricdistance of the point cloud patches, we directly map the input patches embedded in Euclidean spaceto OM to compute the implied geodesic relations between patches.
Considering the patch matrix
P = { p 1 , p 2 , · · · , p g } does not naturally satisfy the Eq.2, in other words, P is not a member of OM.
We apply a projection Proj ( · ) to get the manifold-valued point cloud feature matrix:
[Equation]
where ∥ · ∥ denotes the square Frobenius norm in the ambient space.
After the projection, the geodesic distance of an input pair of points { K, V } on OM can be calculatedas:
[Equation]
Note that computing the geodesic distance score in this way is even faster and more space-efficientthan the original multi-head dot-product way for its non-parametric characteristic and paralleledimplementation.
More details are presented in Section4.4.
After solving the dilemma of obtaining the geodesic, we need to consider how the geodesic distancescore guides our attention mechanism.
Based on the principle that the more distant the geometricrelationship is, the less the dependencies between the point cloud sampling points, we simply makethe geodesic distance score negative and apply a softmax function to obtain the attention mapweights.
In addition, we have also tried to take the inverse of the geodesic distance score or normalizeit before softmax .
However, through the experiments, we found that the most naive way achievedthe best result instead, and we inferred that the redundant operation might impair the representationability of GSA for the distance of points itself is relatively small after the scaling of data augmentation.
The experiment results are presented in Section4.4.
Therefore, given the input X , the computationpipeline of GSA is illustrated in Fig.2.
Experiments
In this section, we comprehensively evaluate the performance of Point-GT on several benchmarks,including the classification task, part segmentation and few-shot learning.
In the ablation studies, wequalitatively and quantitatively assess the effectiveness of GSA.
4.1 Object Classification Tasks
We report the accuracy (%) and the number of sampled points in the input.
[Table] [Table]
PointCNN [21] 1k 92.
DGCNN [36] 1k 92.
RS-CNN [22] 1k 92.
[T] PCT [13] 1k 93.
[T] NPCT [13] 1k 91.
[T] PCT-GSA (OURS) 1k 93.
[ST] Transformer [43] 1k 91.
[ST] Point Transformer [10] 1k 92.
[ST] Point-GT (OURS) 1k 93.
[STP] Point-BERT [43] 1k 93.
[STP] Point-MAE [24] 1k 93.
[STP] Point-GT-MAE (Ours) 1k 93.
[STP] Point-BERT [43] 8k 93.
[STP] Point-MAE [24] 8k 94.
[STP] Point-GT-MAE (Ours) 8k 94.
Object Classification on ModelNet40 ModelNet40 [38], one of the most popular 3D objectclassification datasets, contains 12,311 synthesized CAD models from 40 categories.
We splitthe dataset into 9,843 training and 2,468 validation instances following the previous standardpractice.
During the training, standard randomscaling and random translation are applied fordata augmentation.
More experiment details areprovided in supplementary materials.
The experiment results are presented in Table1, and we denote our methods as Point-GTand Point-GT-MAE.
The former is trained fromscratch, and the latter is pre-trained with MAE[24] strategy 1 .
Since the graph-based GSA requires significant computational overhead evenwhen accelerated with CUDA arithmetic, wetested it only on the ModelNet40 dataset with1024 input points, and all the results of our
Point-GT represented below are implementedwith the data manifold-based GSA.
When training from scratch, our graph-based Point-GTachieves 92.6% accuracy without voting, whichis better than the base Transformer (91.4%) [43].
The result demonstrates the feasibility and superiority of the graph-based GSA.
For fair comparison, our methods utilize the standard voting method[22] during testing, and thedifferent number of input points is also presented.
Yu et al.
[43] observe that increasing the numberof input points cannot improve the performance of a model without pre-training.
Therefore, we justtest Point-GT with 1,024 input points.
In addition to the comparison of classic point cloud models,several Transformer-based models with dot-product self-attention are illustrated, where [T] denotesthe Transformer model with some special designs of Transformer blocks or FeedForward Network[34], [ST] denotes a standard Transformer architecture, and [STP] represents a [ST] model withpre-training.
Since our GSA focus on the self-mechanism of point cloud Transformers, we also testthe GSA on PCT [13] by modifying its self-attention function.
Since dot-product self-attention playsa less important role in PCT, a minor improvement is also valuable.
We also design an experiment onthe resistance to rotation, and the results are presented in supplementary materials.
Compared with the point cloud Transformers with dot-product self-attention, our Point-GT achievesan excellent result (93.3%) which is even better than a model, Point-BERT (93.2%), with pre-trainingwhen the input points are 1,024.
Increasing the number of input points to 8,192, we can also find that
Point-GT with MAE pre-training achieves state-of-the-art performance (94.1%) and is better than
Point-MAE (94.0%), which is the best Transformer-based point cloud work up to date.
Consideringdot-product attention plays a relatively less important role in PCT, our GSA does not bring as muchof a boost as it does in vanilla Transformer.
Object Classification on ScanObjectNN Since ModelNet40 is a noise-free artificial dataset andhas been extensively studied, it has gradually failed to meet current research needs, so we alsoevaluate our approach on the ScanObjectNN [33] benchmark.
ScanObjectNN contains about 15,000objects sampled from 2,902 real-world instances with 15 categories.
Since the real-world dataset willinevitably be affected by noise or occlusions, it is a much more challenging dataset for point cloudanalysis methods.
We follow previous works [43, 24] to evaluate our approach on three variants:
OBJ-BG, OBJ-ONLY, and PB-T50-RS.
In comparison, our Point-GT largely improves the baseline
Transformers by 7.7%, 7.0% and 4.5% on three variants without voting and our Point-GT-MAE alsooutperforms the pre-trained point cloud Transformers.
[Table]
4.2 Few-Shot Learning
Following previous works [43, 24], we conduct few-shot learning experiments on ModelNet40.
Adopting a “ K -way N -shot” setting, where K classes are randomly selected from the dataset and Ndenotes the number of instances randomly sampled for each class, we train the model on K × Nsamples (support set) and another 20 × K instances (query set) are randomly sampled from K classes for evaluation.
The results with the setting of K ∈ { 5 , 10 } and N ∈ { 10 , 20 } are presented in Tab.3.
We conduct ten independent experiments for each setting and report the mean accuracy as well as thestandard deviation over ten runs.
According to the results, our approach achieves SOTA performance.
[Table]
4.3 Part Segmentation
Object part segmentation is a challenging task with a high requirement of model representationcapability.
It aims to predict a more fine-grained class label for every point.
We evaluate our Point-GTon ShapeNetPart [40] dataset, containing 16,881 instances of 16 categories.
Following previousworks [43, 24], we sample 2,048 points as input for each instance and design a simple segmentationhead without any propagating operation or DGCNN [36].
Specifically, we concatenate the learnedfeatures from the 4 th , 8 th and 12 th layer of Transformer blocks and then apply mean - pooling andmax - pooling separately to obtain two global features.
Then we up-sample the concatenated featuresto 2,048, which is the size of input points to obtain the predicting features.
The mean IoU across all categories and the mean IoU across all instances are illustrated in Table 4.
Our training from scratch model, Point-GT, outperforms the pre-training model Point-BERT whichdemonstrates the superior fine-grained representation capability of the GSA.
[Table]
[Table]
[Table]
4.4 Ablation Study
In this subsection, we conduct several controlled experiments to explore how to design our geodesicself-attention and compare the performance of different attention types.
The experiments are basedon the classification task on ModelNet40 and the number of input points is 1,024.
All the resultspresented are evaluated without voting.
Attention Score Regularization
In section3.2, we design two methods to compute the GDS.
Before using softmax function to obtain the attention map, we need to convert the GDS intoattention score.
Following the dot-product self-attention [34], we try to scale the attention score.
Weattempt three scaling settings, including "Norm.", "Inv."
and "Neg."
denote making the attention scorenormalization, inverse and negative, respectively.
The results are illustrated in Fig. ?? .
The accuracyresults of the three settings are 92.5%, 92.3% and 92.8%, which show that the redundant operationmight impair the representation ability of GSA.
Attention Types We compare several types of self-attention mechanism in Transformer block.
The results are illustrated in Table 5.
Four conditions are evaluated.
"Dot." denotes the standarddot product multiplication attention used in previous works.
Given the input X , the operation candescribed as:
[Equation]
"Dot.-less" denotes a non-parametric dot product multiplication attention without redundant lineartransformations which can described as:
[Equation]
"Non." is a condition that we only apply a simple point-wise MLP to map the input X withoutattention mechanism and the operation can described as:
[Equation]
"GSA" denotes our geodesic self-attention.
The results fully demonstrate the superiority of ourapproach and the assumption that dot product multiplication in Euclidean space cannot capture theinternal non-Euclidean structure of point cloud objects.
Further analyzing the results, we argue thatthe general linear transformation causes scale ambiguity and destroys the geometric structure of thefeatures, which is fatal for point cloud analysis.
Meanwhile, the redundant linear transformationsmap the input features to different high-dimensional spaces, aggravating scale ambiguity.
To keep thegeometric structure of the input features in the original space, Fei et al. [11] replace the conventionallinear transformation with an orthogonal transformation and get a better result in 2D vision tasks,which confirms the above conclusions.
[Figure]
Regularization The validation accuracy curves and loss curves of three settings are presented.
[Figure]
[Table]
Conclusion
In this paper, we first throw out an assumption that dot production multiplication in Euclidean spaceis not the optimal self-attention mechanism for Transformers aiming at point cloud tasks and thenpropose a novel non-parametric geodesic self-attention mechanism to solve this problem.
The keyinsight behind the GSA is that the geodesic can obtain more accurate measurement of non-Euclideanstructure of point clouds.
Since the geodesic of point cloud is difficult to define and obtain, wepropose two feasible methods to compute it.
To verify our approach, we introduce the GSA into astandard Transformer structure, Point-GT, and it outperforms related works on different benchmarks.
We hope that our work can provide some new thoughts on the self-attention mechanism of pointcloud Transformers.
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