# Texture-Guided Graph Transform Optimization for Point Cloud Attribute Compression
## 2 Related Work on Point Cloud Attribute Compression
### Point Cloud Reorganization and Clustering  
The requirement for point cloud reorganization stems from the arbitrary arrangement of points in the point cloud, posing challenges for efficient attribute redundancy removal during compression. Thus, before compressing point cloud attributes, meaningful reorganization is essential. Proper reordering is crucial for preserving attribute spatial correlation, which significantly influences the efficiency and quality of the compression codec. Based on the requirement for point cloud reorganization, we select the Hilbert curve-based clustering method based on its advantages in computational efficiency and clustering quality. In terms of computational complexity, constructing a Hilbert curve involves recursively subdividing the 3D space into smaller cells and ordering them along the curve. Once constructed, assigning points to clusters based on their proximity along the curve can be relatively efficient. In contrast, the computational complexity of the k-means clustering method depends on the number of clusters (k) and the size of the dataset, often requiring multiple iterations until convergence. Regarding the clustering quality, the Hilbert curve's ability to preserve spatial locality is crucial for maintaining attribute correlation within the point cloud. Studies such as [25] have shown that the Hilbert curve consistently outperforms other space-filling curves, like the Morton code, in preserving attribute correlation across point clouds with varying sparsity levels. While k-means clustering aims to minimize within-cluster variance, its performance can be sensitive to initialization and the choice of k, leading to inconsistent clustering quality. The Hilbert curve-based clustering method offers advantages in preserving spatial locality and potentially reducing computational complexity for point cloud reorganization. These advantages align well  
Figure 1: The pipeline of the proposed texture-guided graph transform optimization scheme for point cloud attribute compression. with the goals of our proposed point cloud attribute codec, hence our decision to employ the Hilbert order to reorganize the points in the point cloud. Leveraging the geometry spatial correlation, the Hilbert-based point cloud reorganization aims to preserve attribute spatial correlation within a one-dimensional (1D) sequence of points. The Hilbert order of a point is derived from its 3D positions within the point cloud, represented as an ordering number on the Hilbert space-filling curve. Table 1 illustrates the mapping rules from 3D locations in a space cube to the Hilbert code. Utilizing these mapping rules, all 3D points in the point cloud can be organized into a 1D sequence of points, akin to a fitted Hilbert filling curve. The Hilbert-based point reorganization process is demonstrated in Figure 2. Figure 2a showcases an example of the fitted Hilbert filling curve within a \(2\times 2\times 2\) 3D cube, with the maximum geometry displacement along the curve limited to 1 unit. In Figure 2b, it is evident that compared to the popular Morton code-based order, the Hilbert curve yields a more compact reordered point sequence with smaller geometry displacements. This compactness is advantageous for preserving geometry-associated attribute spatial correlation. It is posited that maintaining nearby points in close proximity in 3D space during the point cloud reorganization process is desirable, as it enhances the contribution of attribute prediction and transform coding. Therefore, it is desired to keep nearby points in 3D space close in the point cloud reorganization process. By arranging points into a 1D sequence using the Hilbert space-filling curve, we facilitate the creation of more efficient and compact blocks over the sorted points in the point cloud. Following Hilbert-based point cloud reorganization, we apply uniform clustering to the reordered 1D point sequence to generate point cloud clusters, which serve as attribute