# Texture-Guided Graph Transform Optimization for Point Cloud Attribute Compression
## 2 Related Work on Point Cloud Attribute Compression
### Problem Formulation  
Building on the insights from [30], it points out that when a signal follows a GMRF model, the eigenvectors of its precision matrix can optimally decorrelate the signal within an underlying graph structure. Once the spatial prediction on those GMRF signals is complete, with the constructed graph structure, the eigenanalysis of the GMRF model's graph Laplacian matrix can serve as the optimal transform on those signals' residual to compact its energy for quantization and entropy coding. Therefore, to obtain the optimal prediction and transform coding performance on compressing point cloud attribute, we propose to formulate the point cloud as a GMRF model and intend to construct the optimal graph within the GMRF model to pursue the most efficient graph transform coding performance. Let \(\mathbf{x}=(x_{1},\ldots,x_{m})^{T}\) represent the point cloud attribute signals within a GMRF model. The density function of \(\mathbf{x}\) with mean \(\mu\) and a precision matrix \(\mathbf{Q}\) can be defined as [30]:  
\[p(x)=(2\pi)^{-\frac{m}{2}}|\mathbf{Q}|^{\frac{1}{2}}exp\left(-\frac{1}{2}(x- \mu)^{T}\mathbf{Q}(x-\mu)\right), \tag{1}\]  
where the matrix \(\mathbf{Q}\) is symmetric and positive definite. \(\mathbf{Q}\) is the inverse of the covariance matrix in a typical multivariate Gaussian distribution. This property ensures that \(\mathbf{Q}\) effectively captures the interdependence among the elements of the attribute signals \(\mathbf{x}\). Certainly, the precision matrix \(\mathbf{Q}\) in Equation (1) can be further partitioned based on the current attribute signals \(\mathbf{x_{c}}\) to be coded and the reference attribute signals \(\mathbf{x_{r}}\) that have already been coded. This partitioning facilitates a more targeted analysis of the interdependencies within the data. Let \(\mathbf{x_{c}}=(x_{c_{1}},\ldots,x_{c_{m}})^{T}\) represent the current attribute signals to be coded, and \(\mathbf{x_{r}}=(x_{r_{1}},\ldots,x_{r_{m}})^{T}\) denote the reference attribute signals that have already been coded. Then, the precision matrix \(\mathbf{Q}\) can be partitioned as [28]:  
\[\mathbf{Q}=\left( \begin{array}{cc}\mathbf{Q}_{c,r}&\mathbf{Q}_{c,r}\\ \mathbf{Q}_{c,r}^{T}&\mathbf{Q}_{r,r}\end{array} \right), \tag{2}\]where \(\mathbf{Q}_{c,c}\in\mathbb{R}^{n\times n}\), \(\mathbf{Q}_{r,r}\in\mathbb{R}^{n\times n}\), and \(\mathbf{Q}_{c,r}\in\mathbb{R}^{n\times n}\) are the sub-matrices. \(\mathbf{Q}_{c,c}\) is the desired precision matrix for the optimal transform of current attributes \(\mathbf{x}_{\mathbf{c}}=(x_{c_{1}},\ldots,x_{c_{n}})^{T}\). \(\mathbf{Q}_{r,r}\) can be obtained with the decoded reference attributes \(\mathbf{x}_{\mathbf{r}}=(x_{r_{1}},\ldots,x_{r_{n}})^{T}\). \(\mathbf{Q}_{c,r}\in\mathbb{R}^{n\times n}\) represents the spatial attribute correlation between \(x_{c}\) and \(x_{r}\). Based on the deviation in [28], the conditional distribution \(x_{c}|x_{r}\) also adheres to a GMRF with mean \(\mu_{x_{c}|x_{r}}\) and a precision matrix \(\mathbf{Q}_{c,r}\in\mathbb{R}^{n\times n}\), which are defined as [28]:  
\[\mu_{x_{c}|x_{r}} =\mu_{x_{c}}-\mathbf{Q}_{c,c}^{-1}\mathbf{Q}_{c,r}(\mathbf{x}_{ r}-\mu_{\mathbf{x}}), \tag{3}\] \[\mathbf{Q}_{x_{c}|x_{r}} =\mathbf{Q}_{c,c},\]  
where \(\mu_{x_{c}}\) and \(\mu_{x_{r}}\) are the mean of the current attributes \(\mathbf{x}_{c}\) and the reference attributes \(\mathbf{x}_{r}\), respectively. Combining Equations (2) and (3), it becomes evident that achieving optimal decorrelation between the current attributes \(\mathbf{x}_{\mathbf{c}}\) and the reference attributes \(\mathbf{x}_{\mathbf{r}}\) involves the following steps: First, subtract the conditional mean \(\mu_{\mathbf{x}|\mathbf{x}}\) as the intra prediction from \(\mathbf{x}_{\mathbf{c}}\). Next, construct a graph to model the spatial relationship \(x_{c}|x_{r}\) among the attributes and derive the precision matrix \(\mathbf{Q}_{c,c}\).