GQE-Net: A Graph-based Quality Enhancement Network for Point Cloud Color Attribute

Recently, graph-based neural networks have achieved great success in 3D point cloud processing. Notably, the EdgeConv module, a prominent graph convolution technique, was introduced by Wang et al. [18] to facilitate high-level feature extraction from point clouds. This module uses a dynamic graph across network layers, significantly enhancing the capacity to capture intricate features. Additionally, the incorporation of graph-attention mechanisms plays an important role to focus on key points within the point cloud. Chen et al. [19] introduced a graph attention-based point network layer (GAPLayer) to learn local geometric representations by multiple single head graph attention mechanisms. Wang et al. [20] proposed a graph attention convolution (GAC) that adapts its kernels to the structure of an object to capture structured features for fine-grained segmentation and avoid feature contamination between objects.

Many advancements in point cloud analysis have been achieved through the synergistic fusion of graph neural networks with classical models or mechanisms. This collaborative approach enhances the capability of these networks to tackle complex tasks effectively. Shi et al. [21] proposed a graph neural network for object detection. The network uses an auto-registration mechanism to reduce translation variance and a box merging and scoring operation to combine detections from multiple vertices accurately. Feng et al. [23] proposed a deep auto-encoder with graph topology inference and filtering to achieve compact representations of unorganized 3D point clouds in an unsupervised manner. Liang et al. [24] proposed a hierarchical depth-wise graph convolutional neural network for point cloud semantic segmentation. A customized block called depthwise graph convolution (DGConv) was designed for local feature extraction.

While these methods effectively extract correlation between points using the graph structure, they do not fully exploit the available information. The distances between points and the normals play a crucial role in determining the strength of correlation between two points. Moreover, as different points have varying importance, it is important to focus on key points and design more effective attention modules. In the field of quality enhancement, it is particularly important to capture points with high distortion or those located in key positions. Therefore, we design modules (PSGA and FR) that adaptively extract features and further exploit correlations between points.

3D point cloud geometry denoising is crucial for various applications such as autonomous driving. The goal of point cloud denoising is to remove outliers or fill in holes to enable further processing or downstream applications. Within this context, the integration of graph Laplacian regularization (GLR) has proven effective in enhancing denoising capabilities [25][26]. Dinesh et al. [25] proposed a signal-dependent feature graph Laplacian regularizer which can be used as a signal prior. Zeng et al. [26] proposed a discrete patch distance measure to quantify the similarity between two surface patches of the same size for graph construction. Given the huge data volume in point cloud processing, manifold-based methods hold promise for features projection and dimensionality reduction. Hu et al. [27] proposed a method based on spatial-temporal graph representation that uses the temporal consistency between surface patches that correspond to the same underlying manifold. Xu et al. [28] proposed a transformer-based end-to-end network with an encoder-decoder architecture. The encoder uses a transformer, while the decoder learns the latent manifold of each sampled point. To achieve optimal denoising, multi-stage or stepwise strategies are often applied. Chen et al. [29] proposed a task-specific point cloud denoising network, RePCDNet, which consists of four key modules based on recurrent neural networks (RNNs) for feature extraction and enhancement. Jia et al. [30] proposed a two-step method for removing geometry artifacts and improving the compression efficiency of V-PCC. The first step is learning-based pseudo-motion compensation, aiming at denoising the artifacts. The second step takes advantage of the strong correlation between the near and far depth fields decomposed from geometry to further improve the results. Furthermore, within the specific context of V-PCC, the work in [31] introduced an adaptive denoising method, which applies the Wiener filter on the 2D geometry images.

The works reviewed in this section suggest that extracting effective geometry features is crucial for improving the quality of the point cloud. Techniques such as domain transformation, surface fitting, normal approximation, and distance measurement can be used to detect and eliminate noise in the point cloud. In addition, geometry features can provide important information to enhance color quality.

Most of the previous work in the field of point cloud quality enhancement is based on traditional algorithms. For example, Irfan and Magli [32, 33] make full use of graph transform or optimization to reduce color distortion. In [32], a point cloud quality enhancement method was proposed, which uses a spectral graph wavelet transform to jointly exploit both geometry and color in the graph spectral domain. The authors [33] later proposed a graph-based optimization method that aims to remove both geometry and color distortion simultaneously. The method assumes that the color and location of a point are related to each other. Similarly based on the graph, Yamamoto et al. [34] proposed a deblurring method for point cloud attributes, which is inspired by multi-Wiener Stein’s unbiased risk estimate-linear expansion of thresholding deconvolution of images. The blurred textures are considered as graph signals that are decomposed into sub-bands after Wiener-like filtering, and then all filtered signals are added together with the coefficients obtained from the linear equation. Another work using Wiener filter for color quality enhancement is [35], which is specifically designed for G-PCC. Optimal coefficients of the filter for each color component are calculated in the encoder and selectively written into the bit stream according to the rate-distortion cost. For V-PCC, the only color enhancement method we are aware of is the low-complexity color smoothing method [36] included as an optional mode in the V-PCC test model. In this method, the decoded points are split into 3D grids, and the color centroid of each grid is calculated and then smoothed with tri-linear filtering.

The methods discussed above often offer straightforward yet effective ways to enhance the color quality of point clouds. It becomes evident that the relationship between geometry and color is crucial for driving performance improvements. Nonetheless, these methods do have certain limitations. For instance, their objectives can sometimes be confined to specific tasks, such as G-PCC or V-PCC, resulting in a rather narrow scope. This has motivated the exploration of learning-based methods. Sheng et al. [37] proposed a multi-scale graph attention network for removing attribute compression artifacts for G-PCC. This method uses Chebyshev graph convolutions to extract features of point cloud attributes and a multi-scale scheme to capture short and long-range correlations between the current point and its neighbors. However, the applicability of the method is restricted to G-PCC geometry lossless coding. Moreover, the need for separate models trained at various bit rates imposes limitations.

In summary, while current point cloud quality enhancement methods have shown promising results, there is still room for improvement, particularly within a point cloud coding system. Enhancing quality not only improves the reconstruction quality but also increases the efficiency of the coding system. As color plays a significant role in the overall subjective quality of point clouds and has an impact on subsequent tasks [38], we propose GQE-Net as a post-processing filter for a point cloud compression system to reduce the compression artifacts.

To avoid color distortions caused by geometry losses, we assume that the compression of the geometry information is lossless, i.e., the coordinates of each point in the reconstructed point cloud remain unchanged after compression. We first convert the color space from RGB to YCbCr. We denote the original point cloud by $\bf{P}\in{{\mathbb{R}}^{\emph{N}\times\emph{6}}}$, where $N$ is the number of points, and $6$ corresponds to the 3D geometry coordinates and three color components. After decoding, the original point cloud is denoted by $\bf{P^{\prime}}\in{{\mathbb{R}}^{\emph{N}\times\emph{6}}}$. By applying the proposed method to $\bf{P^{\prime}}$, we obtain a quality-enhanced point cloud $\bf{\hat{P}}\in{{\mathbb{R}}^{\emph{N}\times\emph{6}}}$, as shown in Fig. 3. The aim of the proposed method is to minimize the difference between $\bf{\hat{P}}$ and $\bf{P}$.

Due to limitations in GPU memory capacity, it is challenging to input the entire point cloud directly into the neural network. As an alternative, we partition the point cloud into smaller 3D patches of fixed size $n$. These points are obtained as follows. Let $r$ be a parameter used to control the overlap between the 3D patches and let

Using farthest point sampling (FPS), we select $m$ points from the point cloud. We call these $m$ points key points. Next, the k Nearest Neighbor (k-NN) algorithm [39] is applied to each key point to identify its $n-1$ closest neighbors. Each key point and its $n-1$ neighbors are grouped into a 3D patch represented as $\{\bm{\textbf{S}},\bm{\textbf{C}}\}$, where $\bm{\textbf{S}}\in{{\mathbb{R}}^{n\times 3}}$ denotes the geometry coordinates and $\bm{\textbf{C}}=\{\bm{y},\bm{u},\bm{v}\}\in{\mathbb{R}}^{n\times 3}$ denotes the three color components of the $n$ points in the 3D patch. Thus, for each color component, the 3D patches have the same geometry but different color information. The architecture of GQE-Net for these three sets is identical, but with different parameters. For example, GQE-Net for the Luma component (denoted as Y) can be written as a function:

where $\Psi(\cdot)$ represents GQE-Net, $\Theta_{\bm{\textbf{Y}}}$ represents the learnable parameters for the Y component, $\bm{y^{\prime}}$ represents the initial reconstructed Y component, and $\bm{\hat{\bm{y}}}$ represents the final, quality-enhanced Y component. The final color $\hat{\bm{\textbf{C}}}=\{\hat{\bm{y}},\hat{\bm{u}},\hat{\bm{v}}\}$ of the patch is generated by combining the three quality-enhanced color components.

Using S, we merge the 3D patches to create a high-quality point cloud, $\hat{\bm{\textbf{P}}}$. However, in this process, it is likely that some points will be selected multiple times across different patches, while others may not be selected at all. To address this problem, we use the following approach. For points that are processed in different patches, we take the average as the restored value. For points that are not included in any patch, we keep the same color.

GQE-Net is a graph-based neural network, with most of its operations carried out on the graph topology. Given $n$ points with $L$-dimensional color features $\bm{\textbf{F}}=\{\bm{f}_{1},\bm{f}_{2},…,\bm{f}_{n}\}\in{{\mathbb{R}}^{n\times L}}$ and 3D coordinates $\bm{\textbf{S}}=\{\bm{s}_{1},\bm{s}_{2},…,\bm{s}_{n}\}\in{{\mathbb{R}}^{n\times 3}}$, at the beginning, the $L$-dimensional feature is simply the corresponding color, i.e., $L=1$. We construct a directed and self-loop graph $\bf{G}=(\bf{V},\bf{E})$, where $\bf{V}$ represents the nodes and $\bf{E}$ represents the edges. To exploit the relationship between features of a node and its neighbors, while also simplifying the graph, each node is only connected to its $k-1$ closest neighboring nodes, as well as itself. These nodes are called the $k$ neighbors of the current node in the rest of the paper. The subsequent operations are carried out on the edge features $\bm{e}_{ij}=(\bm{f}_{i},\bm{f}_{ij}-\bm{f}_{i})\in{{\mathbb{R}}^{n\times 2L}}$, where $\bm{f}_{i}$ is the feature of the current node, $\bm{f}_{ij}$ is the feature of the $j^{th}$ connected node ($j=1,2,…,k$), and $\bm{f}_{i1}=\bm{f}_{i}$. By combining the edge features of all nodes, we obtain the final features $\bm{\textbf{P}}^{\prime}\in{{\mathbb{R}}^{n\times k\times 2L}}$. Thus, for each node we calculate a $k\times 2L$ dimensional feature based on its $k$ nearest neighbors. This graph construction is shown in Fig. 4 and can be written as

The graph convolution block (GCB) is a pivotal element within GQE-Net, facilitating the efficient extraction of features from the graph (Fig. 5). By integrating a maxpooling layer, the GCB aids in the selection of the most pertinent features from various channels. The input edge feature, $\bm{\textbf{F}}_{g\_in}$, has a size of $n\times k\times L_{1}$, where $L_{1}$ is the number of feature channels. The input features are processed by three single blocks in series. Each single block contains a 2D convolution layer with a kernel size of $1\times 1$, a batch normalization layer, and a Leaky ReLU for activation. After this process, the feature $\textbf{F}_{g\_mix}$, whose number of feature channels is $L_{2}$, can be obtained. At this point, the feature $\bm{\textbf{F}}_{g\_mix}$ is still an edge feature. To capture more abstract features from $\bm{\textbf{F}}_{g\_mix}$, a max pooling layer is used. It is worth noting that the max pooling layer selects the most important feature from the $k$ neighboring nodes at each channel. As a result, we can obtain the output feature $\textbf{F}_{o}$ with a size of $n\times L_{2}$. In other words, the neighboring information has been embedded into the $L_{2}$-dimensional feature, $\textbf{F}_{o}$.

Points located in different areas may experience different levels of distortion after compression. It is important to focus on points with significant artifacts or those located in areas that significantly affect the subjective quality, such as high-frequency edges. To achieve this, we propose a PSGA module based on a multi-head mechanism [19] [40], as shown in Fig. 6 (a). Unlike existing methods, it adopts a parallel-serial structure. In the first layer, two single head attention (SHA) blocks ($\rm{SHA}_{1}$ and $\rm{SHA}_{2}$) are used in parallel to extract attention features separately. The output attention features from these two blocks are then combined and sent to a third SHA block ($\rm{SHA}_{3}$) for feature integration and reinforcement. This design is beneficial for exploring feature correlation and further integrating attention features.

The input to PSGA is F with $L$ feature channels. The graph construction is done before each SHA block. The outputs of the parallel SHA blocks include an attention feature $(\textbf{F}_{ma}=\{\textbf{F}_{pa1},\ \textbf{F}_{pa2}\in{{\mathbb{R}}^{n\times 2\times L}}\})$ obtained through graph-based self-attention on the input features, and a graph feature $(\textbf{F}_{mg}=\{\textbf{F}_{pg1},\ \textbf{F}_{pg2}\in{{\mathbb{R}}^{n\times k\times 2\times L}}\})$ which contains the neighboring features of each point. For the two parallel heads, the graph features and the attention features are concatenated, respectively. A third SHA block $(\rm{SHA}_{3})$ takes the concatenated attention features as input for fusion. Finally, $\textbf{F}_{ma}$ and the output attention feature of $\rm{SHA}_{3}$ are concatenated to get the final features $\textbf{F}_{a}$, while $\textbf{F}_{mg}$ and the output of $\rm{SHA}_{3}$ are also concatenated to get the updated graph features $\textbf{F}_{g}$.

The structure of a single head attention is illustrated in Fig. 6 (b). We project the feature $\textbf{F}^{\prime}$ into a higher-dimensional space, with a dimension of $L^{\prime}$, using two branches. The output of one branch is $\textbf{F}_{pg}$. Specifically, we apply a graph convolution on $\textbf{F}^{\prime}$ to obtain $\textbf{F}_{pg}$. In PSGA, the graph convolution layer uses a 2D convolution with a kernel size of $1\times 1$, followed by batch normalization and a Leaky ReLU activation function. Next, the graph features from both branches are compressed into the $n\times k\times 1$ dimension using a graph convolution layer without an activation function to obtain the most distinctive features of each neighboring point. We then add the features of the two branches and apply a Leaky ReLU activation function. The output, $\textbf{A}\in{{\mathbb{R}}^{n\times k}}$, is then normalized to weights ranging from $0$ to $1$. We use $softmax$ function to each row of A:

where $\textbf{A}_{i}$ is the $i^{th}$ row of A, and $\textbf{A}_{ij}$ is the $j^{th}$ feature value of $\textbf{A}_{i}$. The final attention feature, $\textbf{F}_{pa}$, is obtained by multiplying the graph feature $\textbf{F}_{pg}$ with the normalized attention weights. $\textbf{F}_{pa}$ provides information about the degree of distortion and importance of each point by taking into account the local information.

The FR module is designed to effectively analyze and use the geometric information in the vicinity of each point, allowing for efficient leverage of the inter-correlation among points. The structure of the FR module is illustrated in Fig. 7. Specifically, the normal vector of each point is calculated and integrated with the input features. After constructing the graph, a weight matrix is also computed based on the distance between each point and its neighboring points, which is then multiplied by the graph feature.

The normal gives a measure of the angle of the tangent plane on which a point is located. The normals of points can reflect changes in the curvature of the surface. Both the coordinates and the normal contribute to the correlation between points. For example, if the normal discrepancy between two neighboring points is large, their color may differ greatly because they are not on the same plane and do not have similar local features. Therefore, we incorporate the normal of each point into the features, increasing the dimension of the features from $L_{in}$ to $L_{out}=L_{in}+3$.

The computation of the normal vector is achieved through principal component analysis (PCA) [41], a method that uses the k nearest neighbors of a given point to approximate the underlying plane. By extracting a specific direction from this plane, the distribution of all points projected onto this direction becomes the most concentrated, effectively revealing the least significant vector in PCA. This calculated normal vector subsequently enriches the feature set, enabling a more comprehensive characterization of each point.

Then, we also extract local correlation based on the distances between points. Since points that are farther away from the current point may have less structural similarity, we aim to reduce the influence of features from distant neighboring points. To do this, we first calculate the distance $d_{ij}$, $i\in[1,n]$, $j\in[1,k]$ between the $i^{th}$ node and each of its $k$ neighboring nodes in the graph. This defines a weight matrix $\textbf{W}=(w_{ij})\in{\mathbb{R}}^{n\times k}$, where

and

Note that $w_{ij}$ is close to $1$ when $d_{ij}$ is close to $0$, while $w_{ij}$ is close to $0$ when $d_{ij}$ is large. Since the number of channels in the current graph features is $L^{\prime}$, we replicate matrix W $L^{\prime}$ times to generate a tensor $\textbf{W}^{\prime}$ with dimension $n\times k\times L^{\prime}$ to deal with each channel of the graph features. Finally, we calculate an element-wise product between $\textbf{W}^{\prime}$ and the graph features to generate the output features.

Fig. 2 shows the architecture of GQE-Net. The input of the network is divided into two parts: distorted color ${\textbf{C}_{i}}^{\prime}$ and geometry coordinates S. The PSGA module is applied to the input color to focus on important points or points with large distortions. The output of the PSGA module is graph features $\textbf{F}_{g1}$ and attention features $\textbf{F}_{a1}$:

The number of feature channels of $\textbf{F}_{a1}$ and $\textbf{F}_{g1}$ is $64$. In the FR module, the attention features $\textbf{F}_{a1}$ are fused with the input color ${\textbf{C}_{i}}^{\prime}$. The normals and distance are taken into account, and the neighbors are weighted by the geometry distance. The output of the FR module is the refined features

where $cat(\cdot)$ represents concatenation. To effectively fuse and extract features, the proposed GCB is applied to the graph feature $\textbf{F}_{r1}$:

where $GCB(\cdot)$ denotes the graph convolution block. This operation changes the number of feature channels to $64$. To further extract features point-by-point, another PSGA module is applied:

where the number of feature channels for $\textbf{F}_{a2}$ and $\textbf{F}_{g2}$ is $256$. The normals and neighbors are also combined for further feature refinement:

where $\textbf{F}_{r2}$ aggregates two features ($\textbf{F}_{1}$ and $\textbf{F}_{a2}$) and incorporates distance and normal information, resulting in $643$ channels. Then, another GCB is applied for feature fusion and to select the most important neighbor in each feature channel.

This results in $256$ feature channels for $\textbf{F}_{2}$. Graph features are crucial for obtaining local information of each point. Therefore, we use the graph features generated by the PSGAs and process them through GCBs to generate $\textbf{F}_{3}$ and $\textbf{F}_{4}$:

The features obtained through this process have incorporated information from the neighborhood through graph convolution and also emphasized important details. Moreover, the features are fused through skip connections to enhance the representation capabilities of the network. In this way, the features from different layers can be fully exploited:

Finally, by using an $1\times 1$ convolution layer without batch normalization and activation, the mixed features are restored, and residual learning is also used to speed up convergence and ensure the performance of the proposed network:

where $Conv(\cdot)$ is a convolution layer with kernel size of $1\times 1$.

The loss function needs to reflect the distance between the quality-enhanced patch and the ground truth for each color component. Therefore, we use the mean squared error (MSE) as the loss function of GQE-Net:

where $l_{i}$ is the loss of the $i^{th}$ color component, C and $\hat{\textbf{C}}$ represent the true color and the restored color, respectively. Note that the loss is calculated in the YCbCr color space.

We evaluated the performance of our method on point clouds encoded with G-PCC Test Model Category 13 version 14.0 (TMC13v14.0) using a computer with an Intel i7-7820X CPU and 64GB of memory. The encoding process was conducted under the Common Test Condition (CTC) C1 [42], which involves lossless geometry compression and lossy attribute compression. We used the lifting transform [6] in the coding procedure.

We trained two GQE-Net models, one for dense point clouds and one for sparse point clouds. To train each model, we used point clouds compressed at various bit rates. Training was carried out on an NVIDIA GeForce RTX3090 GPU, using PyTorch v1.10. The experimental details, results, and analysis for each case are presented in the following sections.

To evaluate both the objective quality and coding efficiency, we used the commonly used BD-PSNR and BD-rate [45] metrics. The BD-rate measures the average bit rate reduction in bits per input point (bpip) at the same PSNR. A negative BD-rate indicates better performance compared to the test model. In addition to calculating the PSNR for each color component, we used YCbCr-PSNR [46] to evaluate the overall color quality gains brought by our method to the point cloud. The results for Cat_A and Cat_B are shown in Table I and Table II, respectively. More results can be found in the supplemental materials.

From Table I, we can see that for Cat_A point clouds GQE-Net achieved an average improvement of $0.43$ dB, $0.25$ dB and $0.36$ dB for the Y, Cb, and Cr components, respectively, corresponding to -$14.0\%$, -$9.3\%$, -$14.5\%$ BD-rates. The largest PSNR gains were $1.36$ dB (Luma component of Andrew at R05), $0.83$ dB (Cb component of soldier at R06), and $0.93$ dB (Cr component of soldier at R06). We observed that the performance gains were limited at the high bitrates. This is due to two reasons. First, since the quality of the point clouds at high bitrates is already high, the room for improvement is smaller than at low bitrates. Second, since we trained GQE-Net with point clouds compressed at various bitrates, the training process is likely to have geared the network towards minimizing large distortions. Such large distortions typically correspond to low bitrates. Fig. 8 compares the rate-PSNR curves for Cat_A point clouds. It can be seen that the coding efficiency can be greatly improved by incorporating the proposed method into the coding system, particularly at medium and high bit rates.

Table II shows the performance of our second GQE-Net model. The average BD-rates for the Y, Cb, Cr components were -$1.3\%$, -$4.0\%$ and -$2.8\%$, respectively. This is because the relationship between the points in these sparse point clouds is more challenging to capture. Moreover, the correlation between points is also relatively low, making it difficult to improve the quality of the current point using its neighbors. Furthermore, the inherent noise present in Cat_B point clouds may also have a negative impact on performance. If we consider the low bitrates only (i.e., R01, R02, R03, R04), the BD-rates can improve, achieving -$3.1\%$, -$4.6\%$ and -$4.9\%$, for Y, Cb, Cr, respectively.

Fig. 9 compares the original point clouds (ground truth), the point clouds reconstructed without any enhancement, and the point clouds processed by GQE-Net. The texture of the point clouds processed by GQE-Net is clearer and the color transitions are smoother, resulting in a better overall visual experience.

To further demonstrate the effectiveness of GQE-Net, we applied the trained GQE-Net models directly to the point clouds in Cat_A encoded under the G-PCC lossless geometry, lossy attribute and RAHT configuration. The BD-PSNR and BD-rate results are presented in Table III. More results are available in the supplemental materials.

It can be seen that competitive results can be achieved at all bit rates. Compared with G-PCC TMC13v14.0, GQE-Net achieved $0.28$ dB, $0.26$ dB, and $0.36$ dB BD-PSNR, corresponding to -$9.5\%$, -$11.0\%$, and -$13.9\%$ BD-rates. Moreover, the PSNR gain was up to $0.84$ dB (Cr component of redandblack at R04). We believe that the performance could be improved if GQE-Net is fine-tuned or retrained based on point clouds compressed with RAHT.

To further demonstrate the effectiveness of GQE-Net, we used it to improve the quality of point clouds in Cat_A, which were compressed with V-PCC TCM2v18.0 under the lossy geometry and lossy attribute configuration. The quantization parameters for the geometry coding were set to $32$, $28$, $24$, $20$, and $16$. Since the coding structure of V-PCC is different from that of G-PCC, we retrained GQE-Net using the WPCSD dataset. The results, shown in Table IV, indicate that an average BD-PSNR of $0.18$ dB, $0.23$ dB, and $0.24$ dB can be achieved for the Y, Cb, and Cr components, respectively, corresponding to $6.5\%$, $12.4\%$ and $11.5\%$ BD-rate savings. The performance of GQE-Net for V-PCC was slightly reduced compared to G-PCC. This is because the geometry of the reconstructed point cloud is lossy, making the FR module unable to extract the correct distance and normal information. A comprehensive comparison of rate-distortion and rate-PSNR performance with V-PCC is provided in the supplemental materials.

Table V compares GQE-Net to G-PCC TMC13 for a lossy geometry, lossy attribute condition. More results are provided in the supplemental materials. The results show that GQE-Net remains capable of enhancing performance, although to a lesser extent compared to the lossless geometry, lossy attribute condition.

In this section, we evaluate the effectiveness of the PSGA module and FR module. We also study the effect of patch overlap on the performance of GQE-Net.

In this section, we first compare GQE-Net to the multi-scale graph attention network (MSGAT) [37], which was the first neural network used for enhancing the color quality of point clouds. Note that MSGAT trained $12$ separate models for the Y, Cb, and Cr components with bit rates ranging from R01 to R04. This means that MSGAT is only suitable for point clouds at specific bit rates, limiting its applicability. For a comprehensive comparison, we considered two versions of GQE-Net. The first one, which we call GQE-Net_1, consists of a single model for all bit rates, which was trained on the WPCSD training set for the corresponding color channel. The second one, which we call GQE-Net_2, consists of several models, where each model was trained on the training set in [37] for one target bit rate and color channel. In both cases, the training and test point clouds were compressed using G-PCC TMC13v12.0 with the lifting transform according to the configuration adopted in [37]. Table VIII gives the BD-rates of MSGAT, GQE-Net_1, and GQE-Net_2 on the test set in [37]. The results confirm the superiority of our approach.

Next, we compare the proposed method and a Wiener filter-based point cloud color quality enhancement method [35], in which the optimal coefficients of a Wiener filter are calculated in the encoder and transmitted to the decoder. We used G-PCC TMC13v14.0 with the lifting transform for compression. Table IX shows the BD-rates and the time complexity of the encoding and decoding for Cat_A and Cat_B point clouds. We can observe that the proposed method performed better no matter which category of point clouds was used.

In Table IX, we provide the processing time of G-PCC, the additional processing time of Wiener filtering, and the additional processing time of the proposed method in different stages (patch generation, GQE-Net, and patch fusion). We can see that the time complexity of the proposed method was higher than that of G-PCC and the Wiener filter-based method. It is a common problem that neural network-based methods consume more computational resources.

Finally, we compared the performance of GQE-Net to that of V-PCC TMC2v18.0’s low-complexity color smoothing method [36]. Table X compares the BD-rates. As can be seen, our method was more stable and provided clear BD-rate gains over the V-PCC test model color smoothing method.