# Texture-Guided Graph Transform Optimization for Point Cloud Attribute Compression
## 2 Related Work on Point Cloud Attribute Compression
### Point Cloud Graph Transform Optimization
To mitigate the overhead associated with signaling the graph transform matrix, we exploit the underlying correlation between the attribute and the geometry in the local cluster of a point cloud. As proposed in Section 3.3, our Hilbert-based uniform clustering method can generate a set of point cloud local clusters with preserved spatial correlation, which enhances the geometry-attribute correlation assumption. Therefore, the modified bitrate term \(R_{b}(\mathbf{W})\) in Equation (11) is defined as:  
\[R_{b}(\mathbf{W})=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}aW_{i,j}\big{(}g_{i}- g_{j}\big{)}^{2}/q^{2}, \tag{12}\]  
where \(g_{i}\) and \(g_{j}\) denote the decoded geometry positions for the attribute residuals \(x_{i}\) and \(x_{j}\), respectively. The parameter \(a\) quantifies the correlation between the point cloud geometry and attribute, determined through statistical analysis of the relationship between the decoded geometry and reconstructed attributes from previously decoded point cloud clusters. **Graph sparsity term. ** Recognizing that graph sparsity impacts both coding performance and computational complexity in attribute compression, we introduce the graph sparsity term \(R_{s}(\mathbf{W})\) to enforce graph sparsity constraints during optimization. Following [37], the graph sparsity term \(R_{s}(\mathbf{W})\) is defined as:  
\[R_{s}(\mathbf{W})=-\alpha\mathbf{1}^{\top}\log(W\mathbf{1})+\frac{\beta}{2}\|W \|_{F}^{2}, \tag{13}\]  
where \(\mathbf{1}=[1,\dots,\mathbf{1}]^{\top}\). \(\alpha\) and \(\beta\) are two tradeoff parameters. \(W\mathbf{1}\) describes the connectivity degrees of graph nodes. The first term \(-\mathbf{1}^{\top}\log(W\mathbf{1})\) guarantees a meaningful graph whose each node is connected to at least one edge with another node. The second term \(\frac{1}{2}\|W\|_{F}^{2}\) is the Frobenius norm of the graph adjacency matrix to control the sparsity by penalizing large degrees of node connectivity. **Texture-guided Constrained Graph Transform Optimization. ** To expedite the optimization process while bypassing exhaustive exploration of all conceivable edge connections, we incorporate the graph connectivity mask \(\mathbf{A}\) obtained through intra-cluster attribute analysis as delineated in Section 3.4. This mask serves to impart a texture-guided constraint onto the graph adjacency matrix \(\mathbf{W}\) under optimization. Subsequently, the resultant constrained graph adjacency matrix \(\tilde{\mathbf{W}}\) is formulated as follows:  
\[\tilde{\mathbf{W}}=\mathbf{A}\circ\mathbf{W}, \tag{14}\]  
where \(\circ\) denotes the Hadamard product. Utilizing the constrained adjacency matrix \(\tilde{W}\), the updated attribute bitrate term \(R_{b}(\tilde{\mathbf{W}})\) in Equation (12) and the geometry sparsity term \(R_{s}(\tilde{\mathbf{W}})\) in Equation (13) undergo modification as follows:  
\[R_{b}(\tilde{\mathbf{W}})=\frac{a}{2q^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\tilde{ W}_{i,j}\big{(}g_{i}-g_{j}\big{)}^{2}, \tag{15}\]  
\[R_{s}(\tilde{\mathbf{W}})=-\alpha\mathbf{1}^{\top}\log(\tilde{W}\mathbf{1})+ \frac{\beta}{2}\|\tilde{W}\|_{F}^{2}, \tag{16}\]  
where the parameters \(\alpha\) and \(\beta\) are consolidated into a singular optimization parameter \(\theta=\frac{1}{\sqrt{a\beta}}\) as advocated in [37]. The parameter \(\theta\) is related to the desired graph sparsity \(k_{d}\). We will provide a detailed ablation analysis about the impact of the texture-guided graph sparsity \(k_{d}\) and the sparsity-related parameter \(\theta\) on the point cloud attribute coding performance. Building upon the insights from [37], we introduce a numerical optimization algorithm tailored to tackle the graph transform optimization problem presented in Equation (9).