1 Introduction
3D deep learning has achieved rapid progress recently, there are many different methods for 3D deep learning, some convey 3D shapes to voxel[27][13] while some project 3D objects to different image planes[20][12]. However, the former represent the 3D data in a sparse way which is memory space consuming and ineffective, the latter relies on heavy computation and needs additional view rendering.
Point cloud learning has made a lot of progress and methods consuming point cloud directly as input have formed a new research field, which has shown advantages in accuracy and speed compared to other methods. The first point cloud dedicated network is PointNet [5]
, which uses a transform net to align 3D shapes, and max pooling as global symmetric function to handle unordered point cloud. One weakness for PointNet is that it treats each point individually which leading to the loss of local structure information, Methods like PointNet++
[16] and DGCNN [25] then try to solve this problem by taking neighbor points into account to improve the performance, but as the neighbor points in each patch are still treated individually, they fail to learn correlation of points within a patch.Besides, since the coordinates of the points depends on the coordinate system, these methods require the orientation to be similar between training set and test set to achieve high accuracy in recognition tasks. 3D rotations have been a nuisance for 3D shape recognition tasks, most of point cloud based methods can only tackle it with data augmentation. Until recently some methods tries to solve the problem by projecting 3D points to a unit sphere and perform spherical CNN to learn equivalence 3D features [7][28]. However these methods need preprocessing and are time consuming which limits the versatility of these methods.
We propose a transforminvariant 3D deep net taking a point cloud as direct input which we called 3DTINet, we designed a transforminvariant feature encoder and a deep net based on GCN(graph convolutional network) to learn transforminvariant 3D descriptor, Translation invariant and be easily achieved by recentering the point cloud to the it’s centroid, however, rotation invariant remains a challenge in 3D deep learning, Rotation invariant features are important in tasks like classification, segmentation or highlevel task like 3D retrieval in practical application, for the reason that the point cloud collected by depth sensors may not aligned with the training data.
3DTINet can tackle rotation invariant problems without data augmentation and is more computationally efficient. Our model needs no pretraining and could consume point clouds directly without complex preprocessing.
The key contributions of our work are as follows: (1) As far as we know, we are the first one to propose a graphbased transforminvariant 3D geometry feature for 3D point cloud learning, together with theoretical analysis on the transforminvariant property and the spatial meaning.
(2) We propose a novel, general GCNbased deep net to learn 3D geometry feature represented by graph signals more effectively.
(3) We propose a fast and effective transforminvariant feature based pooling method and a dynamic graph update strategy to achieve multiresolution learning on graphs.
The rest of this paper is organized as follows: Related work is shown in section II. Problem statement of our work is given in section III. Details of our propose method is shown IV. Experiment results are detailed in section V. Finally, we conclude this paper in section VI.
2 Related Work
3D Geometry Feature: 3D feature descriptors could be split into two classes, handcrafted features and learned features. Various handcrafted 3D features descriptors haver been proposed for segmentation, matching and classification, including spin images[11], shape context[3] and histogram based approaches[17][24][18], most of which could be found in the Point Cloud Library[1]. However, these methods have reached a bottleneck in handling noisy and lowresolution data, and those methods were designed for specific application which means they are hard to generalize for different situations.
Datadriven methods provide a better way with the success of CNN in 2D vision tasks. learned 3D features has achieved rapid progress since the introduction of 3D deep learning.
Su H et al. [15] learns 3D shapes by a collection of the rendered views on 2D images and achieves dominating performance in 3D recongnition tasks. Zhou Y et al. [13] propose VoxelNet which vonverts a point cloud into 3D voxels and transforms a group of points into a unified feature representation. However, both multiview rending and voxelization lead to high time and space cost, besides, voxelization may cause an unnecessary information loss. Both methods above tried transforms point cloud to more regular data since point clouds are not in a regular format.
Deep Learning on Point clouds: Charles R Q et al. [5] firstly design a novel deep network, named PointNet, for unordered point clouds. PointNet provides a unified architecture for various 3D based applications, such as object classification, part segmentation and scene semantic parsing. In order to better integrate local structure information, the authors propose another network called PointNet++ [16]. PointNet++ applies PointNet recursively on a nested partitioning of the input point set. Inspired by PoineNet, more researchers have propose different deep learning methods on point clouds [25].
Graph Signal processing: Graph signal processing are emerging field mean to combine signal processing and spectral graph theory, which aims at generalize fundamental signal operations from regular to irregular structures represented by graphs. we need new mathematical definitions to extend standard operations such as convolution, filtering, dilation and down sampling to graphs.
More research on graph signal processing tries to process data on irregular graph domains, readers could refer to [14] for an overview, which summarizes recent development of basic graph signal processing tools, including methods for sampling, filtering and graph learning. Chen S et al. [6] propose a resampling framework for 3D point clouds and a featureextraction operator based on graph filtering.
Graph convolution neural network:
[4]defined convolution over graphs in the spectral domain, however the computation complexity is high due to the eigendecomposition of the graph Laplacian matrix in order to get the eigenvector matrix. Defferrard M
et al. [8]then tries to impove GCN through fast localized convolutions, where Chebyshev expansion is deployed to approximate graph Fourier transform.
Graph convolution networks have achieved great performance on learning data lying on irregular nonEuclidean domains like point clouds and social networks[22].
Rotation Invariant 3D features Learning: Convolutional neural networks have translational invariance. However, 3D rotation equivariance in classical convolutional neural networks still remains unsolved. [9][7] propose a novel spherical convolutional network that implements exact convolution on the sphere. In order to achieve rotation equivariance, authors of [23]
introduced a tensor field neural network that inputs and outputs tensor fields: scalars, vectors and higherorder tensors.
[26] provided an example of a CNN with linear equivariance to 3D rotations and 3D translations of voxelized data.3 Problem Statement
In order to propose a general network for point cloud learning, two challenges exits, the first one is to learn inner 3D geometry features in a transforminvariant way, the second one is to learn point cloud more generally and effectively with a deep net. When given a 3D object represented by point cloud , where is the number of points and is number of feature channels, we need to design a transforminvariant feature encoder (abbreviated as TIencoder) as the frontend to describe the shape of the 3D object, the encoded features should be able to express the inner geometric features. Let be the TIencoder, where represents the dimension of the feature space after encoding, the encoded feature can be written as , Let be a random rotation matrix, then should be satisfied to achieve a transforminvariant expression. When given a point cloud with points and each point with attributes, in classification experiments we use 3D coordinates or the encoded transforminvariant features to show that we can learn the geometry of a point cloud effectively, there are many attributes like colors, normal vectors or textures to further describe the characteristics of a point cloud, normals are used as additional attributes in segmentation experiments to better compare with previous methods. We need to design a feature extractor as the backend to learn a feature descriptor, where is the length of the output feature descriptor. should be able to extract features of high degree of division, high computational efficiency and be robust to noise and point cloud intensity.
4 Method
4.1 Porposed Archetecture
Our proposed model is shown in Figure 1, the raw input of the model is point cloud with points and each with three coordinates, In section 4.2, the point cloud coordinates will be encoded as high dimensional transforminvariant feature through transforminvariant feature encoder(TIlayer), MultiGCN layers are used to extract more abstract features, and in order to reduce the amount of parameters and better integrate local features, we perform pooling after the first MultiGCN layer to achieve multiresolution graph learning, the number of nodes is hence reduced and the information carried by each node becomes more abundant. Max pooling is applied to extract global feature descriptors, followed with three fully connected layers, which outputs the final predicted labels.
4.2 3D Geometry Representation
The coordinates of a point cloud depends on the choice of the 3D coordinate system. Therefore, to achieve rotation invariant geometry feature learning, we need to convert the point cloud coordinates to another transform invariant expression , should be able to express inner 3D geometry characteristic carried by the original 3D shape represented by the point cloud.
The geometric features are implicitly expressed by the relationship between different points. When the coordinate system where the point cloud located is fixed, a simple relationship between points can be represented as the difference in coordinate values , where are two points in point cloud , representing the relative position in the coordinate system, from which we can get the distance information between two points: . Relative direction information in the determined coordinates , and can be completely determined by multiplying the length by a direction vector.
However, if the coordinate system is not fixed, the relative direction will change with the coordinate system while distance between points remain unchanged, functions based on distance are called RBFs(radial basis functions) which are widely used in handcrafted 3D rotation invariant features, one important property for RBFs is that they are rotation invariant. At the same time, if the distance between two points is known, the relationship between two points can be reconstructed when given a fixed coordinate system, and the geometric information carried by the two points can then be completely expressed. For a point cloud, if the pairwise distance between points are known, the inner geometric features carried by points can be completely reconstructed except for an additional mirror symmetry uncertainty.
Therefore, the geometric features can be implicitly represented by the connection relationship between points, the edge connecting each points are weighted by pairwise distance. this expression depends only on the distance between points, hence is transforminvariant. The graph structure can be used to describe the relationship between points. All the points in point cloud serve as nodes. The weighted pointtopoint connection relationship is the edge. There are many ways to express a graph. such as adjacency matrix , Laplacian matrix and other descriptions.
Graph signals: We use the coordinates of the point as the origin signal, i.e. the signal of node is , and means the whole signal consisting of all the nodes on the graph.
Graph construction: Graph can be regarded as a discrete approximation of the original surface. we use the distance in the original 3D space or in the feature space to construct a graph, it is also feasible to take additional feature into account. We could then define an undirected graphs , where is a finite set of verticles, is a set of edges and is a weighted adjacency matrix encoding the connection weight between verticles. First we calculate pairwise distances of the point cloud, then find its neighbors according to the Euclidean distance of each point in 3D coordinate space to get the nearest neighbor adjacency distance matrix , represents the Euclidean distance between the th and th point in the point, is the neighbor points of th point, the weight between the center point and the neighbor point can then be computed based on the distance matrix:
(1) 
where is the selfadaptive normalization parameter, and indicates the connection strength between two nodes, we can construct the adjacency matrix according to the weight. The symmetric normalized Laplacian matrix can then be calculated where is the degree matrix of the adjacency matrix . We also use randomwalk normalized Laplacian matrix as the input of our transforminvariant feature encode layer to achieve rotation invariance.
4.3 Rotation Invariant Geometry Feature on Graphs
We have constructed the graph structure in previous section, which is represented by the Laplacian matrix , we also defined the signal on the graph. Graph filtering is a method that takes a graph signal as raw input and outputs a processed graph signal. Let be a graph shift operator, there are many common choices for in graph signal processing theory, such as adjacency matrix and Laplacian matrix . When the graph shift operator acts on a node on the graph, the graph signal value of the central node is replaced by a weighted average of its neighbors, Each linear translation invariant graph filter kernel can be written as a polynomial of the graph shift operator [19]:
(2) 
Where represents the filter coefficient, K is the length of filter, and the output of the filtered signal is . let , the filter kernel acts on the graph signal , and the filtered output is .
Theorem1: Let be the coordinates of a point cloud recentered to it’s centroid, define features on graph as follows: , Where represents norm of each row, then the is rotational and translation invariant.
Proof: Translation invariant can be easily achieved by the recentering operation, To prove rotation invariance, let be a random rotation matrix, and the rotated point cloud coordinates is . Then the feature:
(3) 
TI (TransformInvariant) Layer:
Given a graph represented by Random Walk Laplacian matrix together with graph signal , the filtered graph signal is , where is the number of feature map, the polynomial order is , then the th output channel of TI layer can be write as:
where are the bias of channel of the feature map, are K trainable filter coefficients which can be regarded as the convolution kernel.
Transform invariant feature visualization:
The visualization result of point cloud features after transform invariant encoding. is shown in Figure 2, It can be seen that the value is associated with contours of 3D object.
4.4 Spatial interpretation of Rotation Invariant feature
As mentioned in section 4.2, when applying the graph shift operator on a point, the operation on the spatial domain is calculating the weighted average of the neighbor points around the center point. For Random Walk Laplacian matrix , the sum of each row equals to , so can be be decomposed into two parts: calculates the weighted average of the neighbor points around the central point. the weight is calculated by the distance between neighbor points and the center point. calculate the the relative coordinates of center point to the weighted average point. Therefore represents the Euclidean distance from the center point to the weighted average point, it’s a more comprehensive understanding of its rotation invariance property since the Euclidean distance is an invariant variable in
. We call this distance contour variance. The rotation invariantce can be achieved by calculating the norm of the relative coordinates, but the direction information is ignored. let
be a relative coordinate, and the coordinate value of any point is . the direction vector of a each point can be defined as which is also associated with the choice of coordinate system. The direction of a single node is also meaningless, the geometric information still hidden in the orientation relationship between points. Thus we can treat the direction vector as a new graph signal and perform feature extraction operation in the same way, that is, we apply a graph filtering to the new graph signal to obtain a description of the direction variance between each points. The feature is also rotation invariant and can be easily proved in the same way, we call this new feature direction variance, we then concatenate contour variance and direction variance to get the final output of TI layer.4.5 Rotation Invariant convolution on graphs
In section 4.2, the point coordinates , which will change with rotation, are converted into a highdimensional rotationinvariant feature by the frontend. that is, The absolute 3D coordinate value can then be transformed to relative rotation invariant feature through the relationship between points. we still need to design a network to extract deep rotation invariant feature.
GCN is used to extract feature descriptor of the point cloud. the convolution operator on a graph is originally defined in the spectral domain [4], however, two limitations exits: (1) it is not localized in space, (2) due to the eigendecomposition of the graph Laplacian, it has high computational complexity. Defferrard et al. propose to use Chebyshev polynomials to approximate the spectral filtering [8]. The Klocalized filtering operation can be defined as:
(4) 
is the th Chebyshev coefficient. is the Chebyshev polynomial of order . It can recurrently calculated by , where . Then the complexity is reduced from to .
Klocalized GCN is used as our convolution layer. The output of the TI layer is transform invariant: where are arbitrary rotation and translation matrices. So the entire convolution process is rotation invariant:
(5) 
We then use a relu activation function to achieve the nonlinearity of the network:
(6) 
where are trainable weights, are the bias, is the final output.
4.6 Multiresolution learning on graphs
Fast transforminvariant pooling on graphs: The pooling operation consists of two steps. first we need to down sample the point cloud and keep those points that contributes more on final classification result after coarsening, they are similar to the ’critical points’ in PointNet which are usually the contour points of an object. Then we need to cluster points that are most similar to the sampled points. Coarsening on point cloud is a challenging task, some methods simply treat each point equally and use uniform down sampling to handle this problem, but uniform sampling suffers uncertainty and the sampled points does not contributes more on the final result, others [10] use furthest points sampling but still have the same problem and is far more time consuming. pooling is frequently used in deep net so pooling on point cloud must be computation effective too. The TI feature we propose is related to the contour variance of the point, i.e. the difference between the center point and it’s neighbor points and we have explained the spatial meaning of it in section, Just like the edges in 2D images, we think the contours of a point clouds are points that are different from their neighbor points, so we just sort the TI feature and select top points that vary most from their neighbors. the comparison of our coarsening strategy with uniform sampling is show in 3 After coarsening, we search nearlist neighbors of each sampled point, followed with a local maxpooling to integrate local information, is set to 8 in our experiment.
Dynamic Graph Updating: The dynamic graph has three meanings, first, the number of neighbors used to construct the graph is dynamic in order to adjust the receptive fields of different layers. Second, in classification task, the number of nodes of the graph is dynamic since the amount of nodes is reduced after pooling. Third, the connection relationship and the weight of the edge are determined by the distance in the feature space in segmentation task.
4.7 3DTINet Baseline Model
To show that 3DTINet is effective in learning inner 3D geometry features, a simple network architecture is designed as our baseline model. The baseline mainly consists of 3 parts, the translation invariant geometry feature encoder layer, feature extraction layer based on graph CNN, and a global feature extraction layer. The point cloud coordinates can be encoded as high dimensional transforminvariant feature through rotation invariant feature encoder layer, then pass through feature extract layer which is composed of 2 GCN layers to learn feature that represents structure information, a golbal max pooling is then applied to extract global rotation invariant 3D geometry feature descriptor.
5 Experiment
5.1 Transform Invariance Test
The transform invariance property of our network are tested through 3D shape classification task and 3D retrieval task. For classification task, when given a set of 3D objects with label, we need to train a function to predict the label of a new object. We choose ModelNet40 as our dataset which is widely used in 3D object classification tasks. ModelNet40 contains 12311 CAD models from 40 categories, for each model, 1,024 points are uniformly sampled from the mesh faces and normalized to the unit sphere. We only use the coordinates of the sampled points in classification task. We train the model with the origin oriention and test the data with random rotation to see whether our model could achieve translation invariant in classification task (z/SO(3) mode).
Implementation Details
There exits two strategies to avoid overfitting, we add dropout after fully connected layer and add regularization term to total loss, we also use a weighted gradient descent to prevent the model from over fitting to classes with majority samples. The models are trained using an NVIDIA TITAN GPU, with a batch size of 16 and epoch number of 400.
Method  SO3/SO3  z/SO3  inp. size  Para 
PointNet++ [16]  85.0  28.6  1.7M  
SubVolSup MO [15]  85.0  45.5  17M  
MVCNN 12x [20]  77.6  70.1  99M  
MVCNN 80x [20]  86.0    99M  
RotationNet 20x [12]  80.0  20.2  58.9M  
SpericalCNN [9]  86.9  78.6  0.5M  
Ours baseline    81.6  0.4M  
Ours  87.1  83.9  4.8M^{1}^{1}1The model size of mode z/SO(3) is 2.6M 
We compared our methods with stateofthearts, our proposed method outperforms previous stateoftheart SpericalCNN by a gap 5.3%. Previous coordinate based methods [12, 16] could achieve good performance when the orientation of the training data and the test data are aligned, but there is a sharp drop in accuracy when the test data are randomly rotated. MultiView based methods could be regard as a brute force approach of equivariance, they could generalizes to unseen orientations[9], but still, our method outperform them with a magnitude fewer parameters and no pretraining. Previous point cloud based methods have tried to handle rotation invariance by data argumentation, which require higher model capacity. As shown in Section 5.2, the backend of 3DTINet is also capable of learning the coordinates directly to achieve higher performance in recognition tasks. We trained an additional model with data argumentation i.e. using the rotated training set, and then we concatenate the descriptors learned by two models and trained a voting network to output the final result, we achieve stateoftheart prediction accuracy when the training set and test set are all randomly rotated (SO(3)/SO(3) mode).
5.1.1 3D Retrieval
We perform retrieval experiment on ShapeNet Core55, which is composed of more than 50 thousands models over 55 common categories in total for training and evaluating, In our experiment each query and retrieval results are treated equally across categories, and therefore the results are averaged without reweighting based on category size. We use the same model architecture shown in Figure 3, we follow the setting of [9], the network are trained for classification on 55 core classes with an additional loss to minimize the distance between matching categories textbfwhile pulling apart nonmatching categories at the same time, we use the distance between descriptors for retrieval. the elements whose distance are below the threshold are returned. As shown in 2, our method achieve the best tradeoff between the model complexity, input size and retrieval performance. The result also proves that 3DTINet are effective in learning inner features of 3D objects, and the learned transform invariant features can be widely used in matching and recognition tasks.
mothod  P@N  R@N  mAP  input size  params 

Furuya [10]  0.814  0.683  0.656  8.4M  
Tatsuma [21]  0.705  0.769  0.696  3M  
Zhou [2]  0.660  0.650  0.567  36M  
SpericalCNN [9]  0.717  0.737  0.685  0.5M  
ours  0.749  0.756  0.698  2.6M 
5.2 Backend Capability Test
Our model is effective in learning 3D transform invariant feature but it’s not limited to this condition, in this section we will show that our model is a general framework in learning 3D geometry feature represented by graph signal, we train and test the model without rotation just like most previous point cloud based models did, we consider two prototypical tasks in this field: classification and segmentation. We perform classification experiment on Modelnet40 and segmentation experiment on ShapeNet to test the capability of our model.
5.2.1 Classification
Architecture The network architecture is illustrated in Figure 1. we follow previous point cloud based methods [5, 16, 25] train and test without rotation in ModelNet40, the result is shown in Table 3.
Method  overall acc  fwd time(MS)  model size 

PointNet [5]  89.2  25.3  3.5M 
PointNet++ [16]  90.7  163  1.7M 
RotationNet 20x [12]    92  58.9M 
DGCNN[25]  92.2  94.6  2.1M 
Ours baseline  89.5  10.8  0.4M 
Ours  91.7  60  2.6M 
Though there is still a gap between our method and multiview based methods, we achieve competitive performance with previous stateofthearts which comsume point cloud directly, while our outperform them by fewer parameters and more effective in computation.
Pressure test: We jitter the coordinates of the point cloud with Gauss noise , we set the mean of
to zero, then we increase standard deviation
from 0.01 to 0.1 to see how the prediction accuracy changes, the result is shown in Figure 6, the result shows that our model is robust to noise.We train and test using point cloud with different intensity, we test the prediction accuracy and compare our result with PointNet, as shown in Figure 6, our model is more stable then PointNet and keeps accuracy even with only 256 points.
5.2.2 Segmentation
Point segmentation is classifying the label of each point in a 3D point set, which is more challenging than point set classification. We present experiment on ShapeNet, which contains 16881 shapes from 16 categories and annotated with 50 labels. In the experiments, we first extract 2048 points from each model with random sampling, most sampled point sets have less than six labeled parts. We then feed the coordinates and normal as raw input to our model.
As shown in Figure 5, the input is a point cloud with coordinates and normal as attributes, we then construct a graph represented with Laplacian matrix, we set the nearest neighbor to 200 to better capture local geometry feature, we pass the graph signal through a GCN layers, then we dynamicly update the graph using the distance in feature space and raise the nearest neighbor number search to 1000, in last MultiGCN layer, we use fully connected graph to better integrate global information, finally we concatanate the output feature of three MultiGCN layers and use a fully connected layer to predict the final output segmentation scores.
We assume that a category label for each shape is already known and use the mIoU (mean intersection over union) metric to eveluate the result. We compare our method with stateoftheart approaches, as shown in Table 4, our method outperform PointNet and achieves competitive performance with the stateofthearts.
We also performed segmentation experiment in with transfrominvariant feature, we achieve decent performance while other methods suffer sharp drops in mIOU, that’s mainly because previous methods relies heavily on the coordinates instead of the inner 3D geometry feature implicitly represented by the correlation between points.
6 Conclusion
In this work, we propose, a novel general deep neural network 3DTINet for point cloud learning, which could not only learn shape represented by coordinates but also learn inner geometry features represented by correlation between points. We propose a learned 3D transform invariant geometry feature based on graph, we show its performance on various tasks, the success of our model suggests that intrinsic features are also very important, for a point cloud, the intrinsic geometry lies on the correlation between points, which can be easily represented by a graph. And the geometry represented by graph can be learned using GCN, the features we learn are spatially localized since the graph filters with length combine information strictly from Khop neighbors, which are encoded in the constructed graph. 3DTINet consume point cloud directly with no preprocessing, pretraining which makes the method suitable for various application scenarios. The model has a small amount of parameters and is computational efficiency. These characteristics make the method suitable for realtime applications, such as semantic slam.
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