# Texture-Guided Graph Transform Optimization for Point Cloud Attribute Compression
## 2 Related Work on Point Cloud Attribute Compression
### Attribute Inter-Cluster Prediction and Intra-Cluster Analysis  
With the generated point cloud clusters from the previous stage, we compress the attributes in those clusters in sequential order. The optimal predictive transform for the current attributes essentially considers the correlation between points given a reference set, which is the conditional probability distribution problem. As formulated in Equations (2) and (3) of Section 3.1, achieving the optimal decorrelation of current attributes \(\mathbf{x_{c}}\) with reference attributes \(\mathbf{x_{r}}\) involves a multi-step process: Firstly, the intra prediction involves subtracting the conditional mean \(\mu_{\mathbf{x}|\mathbf{x}}\) from \(\mathbf{x_{c}}\). Subsequently, a graph is constructed to model the spatial relationship \(x_{c}|x_{r}\) among the attributes, facilitating the derivation of the precision matrix \(\mathbf{Q}_{\mathbf{c},\mathbf{c}}\). Leveraging the eigenvector matrix of \(\mathbf{Q}_{\mathbf{c},\mathbf{c}}\), the residuals of \(\mathbf{x_{c}}\) are decorrelated into the frequency domain, achieving a more compact representation. Building upon this foundation, we engage in inter-cluster attribute prediction to diminish spatial redundancy within the current cluster, leveraging previously decoded clusters as references. In our inter-cluster attribute prediction scheme, the mean of the decoded attributes in the reference point cloud cluster, denoted as \(\mu_{\mathbf{x}\mathbf{r}}\), serves as a reference for the mean of the current attributes \(\mathbf{x}\mathbf{c}\) (as per Equation (3)). The attributes in the current cluster are then subtracted from the mean of the reference attributes, yielding attribute residuals. These residuals have effectively mitigated inter-cluster spatial redundancy, priming them for further compression via graph transform coding tools. To better guide the design of graph transform coding tools, we propose an intra-cluster attribute analysis scheme aimed at extracting texture priors before the graph optimization process. Graph construction operates under the assumption that signals on the graph exhibit smoothness concerning an underlying sparse representation in the graph spectral domain. Traditional graph optimization algorithms [31; 33] often consider all possible edges between points in a point cloud cluster, resulting in high computational costs per iteration. However, fully connected graphs do not always yield optimal compression performance [20]. Oftentimes, appropriate sparsity parameters are necessary to achieve optimal compression performance and processing efficiency. To address this, we propose an intra-cluster attribute analysis to estimate the desired graph sparsity, i.e., the desired average number of connecting neighbors per point before the graph optimization process. The specifics of our intra-cluster attribute analysis are as follows:  
Let \(k_{d}\) represent the desired graph sparsity to be estimated within the point cloud cluster. To mitigate the complexity associated with fully considering all possible edges for deriving \(k_{d}\), we constrain \(k_{d}\) within a limited range \((1,\cdots,k_{0})\). Subsequently, we employ the k-NN method to construct a graph with edges connecting points to their k nearest neighbors, wherein various k values are tested within the range \((1,\cdots,k_{0})\). We evaluate all \(k_{0}\) constructed graphs using our texture complexity evaluation. For each graph, we compute the texture complexity variation \(\Delta E\) between the attributes of all graph nodes and the average attributes within the graph, employing the classic CIEDE2000 Color-Difference Formula [35].