# Texture-Guided Graph Transform Optimization for Point Cloud Attribute Compression
## 2 Related Work on Point Cloud Attribute Compression
### Point Cloud Graph Transform Optimization  
Graph optimization facilitates the direct apprehension of the latent graph structure underlying point attributes, presuming their smooth distribution on the graph. Employing the acquired graph transform, residual attributes of the point cloud can be effectively decorrelated into a more condensed representation in the frequency domain, which is conducive to subsequent quantization and entropy coding procedures. Pursuing the optimal efficacy of graph transform involves optimizing the lossy coding performance, specifically by minimizing the introduced rate and distortion terms through transform coding. This optimization can be articulated as an optimization problem aimed at minimizing the cost function \(J\) with the Lagrange multiplier \(\lambda\). We formulate the minimization process of the cost function \(J\) as follows:  
\[\min_{\mathbf{W}}J=D(x,\mathbf{W},q)+\lambda\cdot R(x,\mathbf{W},q), \tag{8}\]  
where \(x\) is the attribute residual to be coded, \(\mathbf{W}\) is the graph adjacency matrix, and \(q\) is the uniform quantization step. \(D(x,\mathbf{W},q)\) denotes the attribute distortion introduced by the uniform quantization on transformed attribute residuals. \(R(x,W,q)\) is the total bitrate to encode the transformed attribute residuals after quantization. In our graph optimization approach, we incorporate a graph sparsity constraint to enhance coding performance and reduce computational complexity. Therefore, we propose an optimization-based graph construction scheme employing a joint energy function that integrates three key terms: the distortion term \(D(\mathbf{W})\), which quantifies the attribute distortion between the original attribute residuals and the quantized transformed attribute residuals; the bitrate term \(R_{b}(\mathbf{W})\), representing the bitrate expenditure for coding the quantized transformed attribute residuals; and the graph sparsity constraint term \(R_{s}(\mathbf{W})\), which acts as a regularizer to enforce graph sparsity constraints during the optimization process. We formulate the joint rate-distortion-sparsity graph transform optimization as:  
\[\underset{\mathbf{W}}{\text{min}}\,J=D(\mathbf{W})+\lambda_{b}R_{b}(\mathbf{ W})+\lambda_{s}R_{s}(\mathbf{W}), \tag{9}\]  
where \(\lambda_{b}\) and \(\lambda_{s}\) serve as weights ranging from 0 to 1, determining the tradeoff among the three terms. We provide detailed explanations for each term--the attribute distortion term \(D(\mathbf{W})\), the attribute bitrate term \(R_{b}(\mathbf{W})\), and the graph sparsity term \(R_{s}(\mathbf{W})\)--each pertaining to the optimization objective of the graph adjacency matrix \(\mathbf{W}\). **Attribute distortion term. ** Since the graph transform is a lossless coding tool, it would not introduce distortion on attribute residuals. The attribute distortion is introduced by the uniform quantization on transformed attribute coefficients. Therefore, we define the attribute distortion term using the classic equation in [18] as:  
\[D(\mathbf{W})=\frac{q^{2}n}{12}, \tag{10}\]  
where \(q\) is the uniform quantization step, and \(n\) is the number of attribute coefficients. From the definition, we can see that the distortion is only related to the quantization step. **Attribute bitrate term. ** Inspired by the work in [36], the bitrate expense for transformed attribute coefficients after the quantization can be estimated from the smoothness degree of attribute signals in the graph. Therefore, we define the bitrate term \(R_{b}(\mathbf{W})\) as:  
\[R_{b}(\mathbf{W})=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}W_{i,j}\big{(}x_{i}- x_{j}\big{)}^{2}/q^{2}, \tag{11}\]  
where \(x_{i}\) and \(x_{j}\) are the attribute residuals to be coded. Notably, when the attribute residuals are utilized in the graph optimization process, the optimized graph transform matrix must be transmitted to the decoder.