1. Introduction

2. Related Works

3. Materials and Methods

3.1. Augmented Sampling and Grouping Module

In the field of optical images, the local area of a pixel can usually be represented as a pixel in the vicinity of a fixed image element size. In 3D point cloud operations, the neighborhood of a single point is defined by the metric distance in the 3D coordinate system. Due to the sparse local regions and irregular geometric structure of the point cloud, the sampling and grouping (SG) [16] operation cannot capture well the different 3D geometric structure features among different regions. This indirectly leads to a learning bottleneck in the subsequent nonlinear mapping layer, and a different extractor is required.

Formally, given a set of unordered point clouds $P=\{p_1,p_1,\dots ,p_N\}$ with $p_i\in R^{N\times d}$, where N is the number of points and d is the feature dimension. When each point is represented by 3D coordinates $p_i=(x_i,y_i,z_i)$, then d = 3. For any p_i ∈ P, there are k-nearest points p_j ∈ P, where j ∈ N. This paper uses the k-nearest neighbor (KNN) search scheme based on comparative experiments. But the existing local shape model with KNN is vulnerable to the local density of the point cloud. In other words, points near the centroid are feature-rich, while points far from the centroid are easily ignored. Therefore, we seek an optimized feature extractor. We draw upon the ideas of PCT [16] and PointNorm [39] to design an augmented local neighbor aggregation strategy ASG. The ASG module optimizes the point embedding and augments UFO-Net’s ability of local feature extraction. Therefore, ASG module is the important component of our UFO-Net framework.

The ASG module introduces a geometric affine operator to local feature extraction. First, the input coordinates are projected using two MLPs to increase the dimension to $C_{in}$. In this paper, $C_{in}$ is taken as 64. Then, the specific implementation of the ASG module is divided into three steps: (i) selecting local centroids by using farthest point sampling (FPS); (ii) obtaining grouped local neighbors (GLNs) using the KNN algorithm based on Euclidean distance; (iii) normalizing the GLNs by using the affine function. To obtain features from different local regions, the GLNs are passed through a lightweight geometric affine function. This operation can help overcome the disadvantage of uneven sampling.

The feature process of ASG can be simply described as follows:

$$P_{\mathit{knn}}=\mathit{cat}(P_{\mathit{knn}},{\mathit{xyz}}_{\mathit{knn}})$$

(1)

$$F_C=\mathit{cat}(P_C,{\mathit{xyz}}_C)$$

(2)

$$F_L=\mathsf{\alpha }⨀\frac{F_{\mathit{knn}} -I\left(F_C\right)}{\mathsf{\sigma }+\mathsf{\epsilon }}+\mathsf{\beta }$$

(3)

$$P_{\mathit{ASG}}=\mathit{cat}(F_L,P_C)$$

(4)

The ASG process is shown in Figure 2, where $P_{\mathit{knn}}$ is $t$the k neighbor features found by KNN from the projection coordinates, $P_C$ is the k neighbor features found by FPS from the original coordinates, ${\mathrm{xyz}}_{\mathrm{knn}}$ is the k neighbor points found by KNN from the original coordinates, and ${\mathit{xyz}}_C$ is the centroid computed by the FPS algorithm from the original coordinates. In addition, $\mathsf{\alpha }\in R^d$ and $\mathsf{\beta }\in R^d$ are learnable affine transformation parameters, $⨀$ denotes the Hadamard production of element directions, $\mathsf{\epsilon }$ is a number ${10}^{-5}$ that keeps the value stable, and I is the unsqueeze operation. Very importantly, $\mathsf{\sigma }$ is a scalar describing the deviation of features between all local groups and channels, obtained from the variance. This helps to obtain more useful features. And f (i, j) is denoted as the lightweight geometric affine function $F_L$. It is the ASG that transforms the local features into a normally distributed process that maintains the geometric properties of the original points. Specifically, this method enhances the identification of domain features in the vicinity of each centroid. The sizes of the point cloud are decreased to 512 and 256 points within the two ASG layers.

Herein, different from the sampling and grouping method, ASG continues to consider the projection features sampled at the farthest point. The input feature is a matrix $N_{\mathit{out}}\times (d+2C_{\mathit{in}})$ with $N_{\mathit{out}}$ subsampling points, d-dim coordinates and two $C_{\mathit{in}}$-dim projection features. The output is a feature matrix $N_{\mathit{out}}\times k\times (d+2C_{\mathit{in}})$, where k is the number of points in the nearest domain of the centroid.

Figure 2. The illustration of the proposed augmented sampling and grouping (ASG) module. FPS represents Farthest Point Sampling algorithm. KNN represents k-nearest neighbor algorithm and f (i, j) means a lightweight function.

Figure 2. The illustration of the proposed augmented sampling and grouping (ASG) module. FPS represents Farthest Point Sampling algorithm. KNN represents k-nearest neighbor algorithm and f (i, j) means a lightweight function.

3.2. Effective Local Feature Learning Module

The existing works [10,40,41] usually use symmetric functions such as max/mean/sum pooling to downscale and preserve the main features to solve the disorder of point clouds. The features obtained from the original point clouds processed by the ASG module lack global properties. To solve this problem, we design an effective local feature learning (EFL) module. In order to utilize the feature information collected from ASG, it is necessary to find a reasonable bridging technique between the two feature processing methods, the ASG module and the following UFO layers. Usually, the max-pooling function is applied to k neighbors of each elaborated local graph to obtain feature representations that aggregate local contexts as the center. Here, we denote the EFL module as:

$$F_{\mathit{EFL}}=M\left(A\right({A(P}_{\mathit{ASG}}\left)\right)$$

(5)

For local sampling and grouping regions $P_{\mathit{ASG}}$, we use a shared neural network comprising two cascaded LBRs A and a max-pooling operator M as symmetric functions to aggregate features. The alignment invariance of the point cloud can be fully guaranteed by ELF. By this learning, the output size of ELF changes from the input matrix $N_{\mathit{out}}\times k\times (d+2C_{\mathrm{in}})$ to the feature size $N_{\mathit{out}}\times 2C_{\mathit{in}}$.

Figure 3. The architecture of Effective local feature learning (EFL). After ASG’s learning, EFL gets a high-level feature by a shared neural network comprising two cascaded LBRs and a max-pooing operator.

Figure 3. The architecture of Effective local feature learning (EFL). After ASG’s learning, EFL gets a high-level feature by a shared neural network comprising two cascaded LBRs and a max-pooing operator.

3.3. Stacked UFO Attention Layers

To develop the exposition of the single UFO attention layer, we first revisit the principle of the self-attention (SA) mechanism. The key to SA mechanism is made up of query, key, and value matrices, which are denoted by Q, K, and V, respectively. The Q, K, and V matrices are generated from encoded local features using linear transformations [33]. Here, $d_k$ is the dimension of the key vectors, and the softmax function is applied to the dot product of the query and key matrices. Formally, the traditional SA mechanism is expressed as:

$$\mathit{Attention}(Q,K,V)=\mathit{softmax}(\frac{Q{K}^T}{\sqrt{d_k}})V$$

(6)

The UFO attention mechanism is explained in the forthcoming note. The architecture of the single UFO attention layer is depicted in Figure 4. We use linear transformation and view operation to convert the input features $F_{\mathit{EFL}}$ into three new representations, query, key, and value matrices, respectively. Given an input feature mapping $F_{\mathit{EFL}}\in R^{N{\times d}_a}$, where N is the number of point clouds and $d_a$ is the feature dimension. Formally, the feature dimension, $d_a=h\times d_e$, is given that h is the number of head and d_e is the dimension of each head.

Then the single UFO attention layer is expressed as:

$$U(x)=\mathit{CN}(Q_U\left)·\mathit{CN}\right(K_U^T{V}_U)$$

(7)

$$Q_U=\mathsf{\psi }(F_{\mathit{EFL}})\in R^{N\times h\times d_e}$$

(8)

$$K_U=\mathsf{\Phi }(F_{\mathit{EFL}})\in R^{N\times h\times d_e}$$

(9)

$$V_U=\mathsf{\gamma }(F_{\mathit{EFL}})\in R^{N\times h\times d_e}$$

(10)

where ψ, Φ, γ are linear transformation and view operation. After permutation, $Q_U\in R^{h\times N\times d_e}$, $K_U^T\in R^{h\times d_e\times N}$, $V_U\in R^{h\times N\times d_e}$, note that h = 4 selected by the ablation experiment. We compute the product of K_U^T and V_U to obtain the spatial correlation matrix KV_Attenion for all points. Next, we use CN to normalize KV_Attenion to get KV_Norm. At the same time, we use CN to normalize Q_U to obtain Q_Norm. It is a common L₂-norm, but it is applied along two dimensions: the spatial dimension of K_U^TV_U and the channel dimension of Q_U. Thus, it is called CrossNorm. Then, permutation and view operation are also adopted.

CrossNorm (CN) is computed as follows:

$$\mathit{CN}\left(x\right)=\frac{\mathsf{\lambda }x}{\sqrt{{\sum }_{i=0}^h{x}^2}}$$

(11)

where λ is a learnable parameter initialized as a random matrix and x is the transformed feature. It generates h clusters through linear kernel method. The operation process can be described as:

Let $q_i=\mathit{CN}\left[\left({\left[Q_U\right]}_{i0},{\left[Q_U\right]}_{i1},\dots ,{\left[Q_U\right]}_{\mathrm{ih}}\right)\right],k_i=\mathit{CN}\left[\left({\left[K_U^T{V}_U\right]}_{0i},{\left[K_U^T{V}_U\right]}_{1i},\dots ,{\left[K_U^T{V}_U\right]}_{\mathit{hi}}\right)\right]$, th-en U(x) can be represented as:

$$U\left(x\right)=\left[\begin{array}{ccc}{q}_0·k_0& \cdots & q_0·k_h\\ ⋮& \ddots & ⋮\\ q_N·k_0& \cdots & q_N·k_h\end{array}\right]v$$

(12)

x is replaced by $F_{\mathrm{EFL}}$_.

The computational nature of CN shows that this is a l₂-normalization, acting successively on the feature channel and the feature channel of K_U^TV_U and Q_U, respectively. Similarly, based on the analysis of the graph convolutional network [42] for the Laplace matrix L = D - E in place of the adjacency matrix E, the offset matrix can diminish the effect of noise [16]. This method provides sufficient discriminative feature information. Therefore, we design offset method to efficiently learn the representation of the distinction of the embedded features. Additionally, the output feature is further obtained through a LBR network and an element-wise addition operation with the input feature.

As the output dimension of each layer is kept the same as the input features, the output of the single UFO attention layer is concatenated four times through the feature dimension, followed by a linear transformation, and more features are obtained. This process can be denoted as:

$$F_1={\mathit{UT}}^1\left(F_{\mathit{EFL}}\right)$$

(13)

$$F_i={\mathit{UT}}^i\left(F_{i-1}\right),i=2,3,4,$$

(14)

$$F_o=\mathit{concat}\left(F_1,F_2,F_3,F_4\right)\cdot W_i$$

(15)

where UTⁱ represents the i-th single UFO attention layer, W_i is the weights of linear layer.

We then concatenate the input features and the output of stacked UFO attention layers to fully obtain contextual features.

4. Experiments and Results

4.2. Experiments on ScanObjectNN Dataset

Due to the rapid development of point cloud research, it can no longer fully meet some practical needs. For this reason, we also conducted experiments on the Scanned Object Neural Network dataset (ScanObjectNN) [53], a real-world point cloud dataset based on LiDAR scanning. ScanObjectNN is a more cumbersome set of point cloud category benchmark, dividing about 15k objects in 700 specific scenarios into 15 classes and 2902 different object instances in the real world. The ScanObjectNN dataset has some variables, of which we are considering the most troublesome variable in the evaluation (PB_T50_RS). Each perturbation variable (prefix PB) in this dataset randomly shifts from the box centroid of the bounding box to 50% of the original size along a specific axis. Suffix R and S represents rotation and scaling [53]. PB_T50_RS contains 13,698 real-world point cloud objects from 15 categories. In particular, 11416 objects are used for training and 2282 objects are used for testing. This dataset is especially a huge challenge for existing point cloud classification techniques. In this experiment, each point cloud object sampled 1024 points, and model was trained using only the local (x, y, z) coordinates.

Table 3. Classifification results on ScanObjectNN dataset.

Table 3. Classifification results on ScanObjectNN dataset.

Method	mAcc (%)	OA (%)
3DmFV [9] PointNet [10] PointNet++ [11] SpiderCNN [54] DGCNN [12] PointCNN [30] BGA-DGCNN [53] PCT [16] DRNet [50] GBNet [38] RPNet++ [55] UFO-Net (ours)	58.1 63.4 75.4 69.8 73.6 75.1 75.7 77.3 78.0 77.8 79.9 82.3	63.0 68.2 77.9 73.7 78.1 78.5 79.7 80.0 80.3 80.5 82.0 83.8

For real-world point cloud classification, we use the same network, training strategy, and 1k of 3D coordinates as input. We quantitatively compared our UFO-Net with other state-of-the-art methods on the hardest ScanObjectNN benchmark dataset. In Table 3, we show the results of competing methods for scanning objective network datasets. Our network has an overall accuracy of 83.8% and an average class accuracy of 82.3%, which is a significant improvement on this benchmark. The results show that mAcc is improved by 5%, and OA is increased by 3.8%, compared with the classical PCT. Furthermore, even when measured using the dynamic local geometry capture network RPNet++, we still have a fairly good lifting in mAcc and OA, with increments of 2.4% and 1.8% respectively, which seems to be tailor-made for this dataset. On top of that, we observe that our UFO-Net creates the smallest gap between mAcc and OA. This phenomenon shows that our method has good robustness.

Since the ScanObjectNN dataset has some difficult cases to classify, the presence of feature-independent background points in ScanObjectNN can pose a challenge to the network. To obtain a global representation of the point cloud, we use the ASG module to learn a local fine-grained feature representation. This is because the design of ASG enhances the relationships between points and enriches the information of geometric features distributed on long edges. Furthermore, our approach provides an efficient solution with stacked UFO attention layers aiming to minimize the impact of these points by equally weighting them according to their channel affinity.

5. Ablation Studies

5.1. Impact of Point Density

Robustness of point cloud density. Sampling density has an influence on point clouds as shown in Figure 4. Therefore, we conducted experiments to evaluate the ability of UFO-Net to extract features.

Figure 4. Examples with different point sampling densities on chair and airplane.

Figure 4. Examples with different point sampling densities on chair and airplane.

The accuracy of experimental curves for different point densities is shown in Figure 5. From Figure 5, it can be seen that the use of 1024 points to train the model to extract features is effective for the designed network. In real-world scenarios, however, point clouds are always fragmented and not completely cover the surface of the target object. So, in this section, we also conduct experiments at different point densities to evaluate the performance of our network. Figure 5 shows the overall accuracy at different input points on the ModelNet40 dataset. We trained our network using a random sample of 1024, 768, 512, 256, and 128 input points. The curves in the figure expose the accuracy trend of the classification model test. Compared with other methods, our model has good robustness to point cloud density. Even at low point densities, our network maintains good accuracy. For example, 91.1% can be reached at 128 input points. The results show that this model can be widely used in different point densities. For relatively sparse scenes, the model can still work with contextual features efficiently.

6. Conclusions

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