Point Cloud Attribute Compression via Successive Subspace Graph Transform ^††thanks: 978-1-7281-8068-7/20/$31.00 ©2020 IEEE

The framework overview of the proposed SSGT method is shown in Fig. 1. To transform a point cloud model consists of a large number of points, we first apply the octree decomposition, and the point cloud decomposes into eight octants recursively to form an octree-tree structure with its root being the whole point cloud and its occupied leaf being a point. Those occupied nodes at different levels represent a subspace of the point cloud model with different spatial sizes. We design a weighted graph with self-loop to describe the subspace and define the graph Fourier transform based on the symmetric normalized Laplacian. The attribute signal (e.g., color) can be transformed to the graph spectral domain with the energy compaction property. The first-stage graph transform is applied at the bottom of the octree. Then, multi-stage transforms are processed from all child nodes to their parents stage by stage until the root node is reached. The proposed SSGT framework successively learns the point cloud subspace and offers an efficient and effective way for point cloud compression.

Problem Formulation. We define a point cloud of $N$ points as ${\bf P}=\{{\bf p}_{1},..,{\bf p}_{N}\}$, where ${\bf p}_{n}\in\mathbb{R}^{3}$. Assume ${\bf P}$ occupies the space of size $1\times 1\times 1$, and can be partitioned using the octree decomposition with depth $L$. The octree will contain $2^{3L}$ leaf nodes with $N$ occupied ones and the spatial size of the leaf node is $2^{-L}\times 2^{-L}\times 2^{-L}$.

At the level $l$, the $k$th node represents a subspace of the point cloud of size $2^{-l}\times 2^{-l}\times 2^{-l}$. we can construct a weighted graph with self-loop, $\mathcal{G}_{lk}$, to describe it, where the vertex set is composed of all occupied child nodes. The designed graph model the relationships among the nodes and the corresponding graph Fourier transform, $\Phi_{lk}$, is calculated based on the symmetric normalized graph laplacian. The multi-stage transforms can be applied from level $(L-1)$ to the root level, the represented subspace expands from size $2^{-(L-1)}\times 2^{-(L-1)}\times 2^{-(L-1)}$ to $1\times 1\times 1$ accordingly.

Designed Graph. The graph design is illustrated in Fig. 2. For the $k$th node at the $l$th level, we can construct a weighted graph with self-loop, $\mathcal{G}_{lk}=\{\mathcal{V}_{lk},\mathcal{E}_{lk},W_{lk}\}$, where $\mathcal{V}_{lk}=\{v_{0},...,v_{N_{lk}}\}$ denotes a set of vertices, which contains all occupied child nodes at level $(l+1)$. The set of edges is defined by $\mathcal{E}_{lk}=\{(v_{i},v_{j})|(v_{i},v_{j})\in\mathcal{V}_{lk}^{2}\wedge v_{i}\neq v_{j}\}$. $W_{lk}\in\mathbb{R}^{N_{lk}\times N_{lk}}$ denotes a edge weight matrix and $N_{lk}$ denotes the number of vertices.

The spatial position of the $v_{i}$ is denoted as ${\bf x}_{i}$ and the number of the points in the $v_{i}$ is denoted as $a_{i}$. The weight matrix $W_{lk}$ is defined by

Considering the relationships among vertices, the edge weights are designed in two aspects: 1) the number of points (i.e., $a_{i}$) within each vertex, and 2) the distances between vertex pairs. In the first aspect, we assume each point in the subspace have relations with all other points, representing by the sub-edges to connect every point pairs. In $i$th vertex, there are ${1\over 2}\times a_{i}\times(a_{i}-1)$ sub-edges among points. From $i$th vertex to $j$th vertex, there are $a_{i}\times a_{j}$ sub-edges among points. The number of sub-edges is introduced to the Eq. 1 to represent the impact of different numbers of points in vertices. In the second aspect, distances between vertex pairs are utilized as expressed in Eq. 2, where the parameter $\alpha$ can control the speed of the edge weight decay. And the same $\alpha$ is adopted over all stages of transforms. In the experiments, we simply use the voxelized position of the vertices, and the voxel grid size increases from $2^{-L}$ to $2^{-1}$ when applying the transform from the level $(L-1)$ to the root node recursively.

Graph Fourier transforms. Based on the designed graph, we can compute the degree matrix $D_{lk}=diag(d_{1},...,d_{N_{li}})$, where $d_{i}=a_{i}(a_{i}-1)+\sum_{j\neq i}d_{ij}a_{i}a_{j}$. The symmetric normalized Laplacian matrix [9] is expressed as

The eigenvalue decompoistion of $L^{sym}_{lk}$:

where $\Phi_{lk}=\{{\bf t_{0}},...,{\bf t_{N_{i}}}\}$ is the matrix whose column is the eigenvector of $L^{sym}_{lk}$ and $\Lambda_{lk}=diag(\lambda_{i})$ is the matrix whose diagonal terms are the corresponding eigenvalues and are sorted in an increasing order. The graph signal can be transformed by $\Phi$ into the graph spectral domain.

The reason for using the normalized Laplacian matrix is to introduce the effect of the self-edges in the graph. Based on the properties of the symmetric normalized Laplacian matrix [10], we have $\lambda_{0}=0$ and its eigenvector ${{\bf t_{0}}=D^{1\over 2}\mathbbm{1}}$. We can treat ${\bf t_{0}}$ as the DC transform kernel, and the DC coefficient is the weighted average of the input graph signal. The other eigenvectors as the AC transform kernels to describe the high frequency details. In the proposed SSGT method, only the DC coefficients are propagated to the next stage.

To transform large point clouds, we use the multiple subspace graph transforms in cascade to expand the subspace size gradually. The subspace transform starts at level $(L-1)$ of the octree and applied to their occupied child node in the $L$th level which is a point. The attribute signal defined on the point can be transformed accordingly. The DC coefficients are fed into the upper level as a new signal defined on the nodes at level $(L-1)$. This process can be repeated stage by stage until the root node is reached.

The speed of the subspace expansions can be adjusted in the proposed framework. If we learn the graph transforms at every level of the octree, the size of the subspace represented by the parent nodes is eight times larger than that of their child nodes. We can increase the expansion speed by applying the subspace transform for every two levels, which leads to eight times faster than the previous settings. However, it should be noticed that if the expansion speed is too fast, the represented graph will contain too many vertices, and its computational cost will increase. In contrast, the capability of the small graph is limited to model the subspace well and it would provide the worse compression performance.

The point cloud attributes can be treated as the graph signal defined on points. In a subspace, we can define the signal $F_{lk}$ on the constructed graph $\mathcal{G}_{lk}$ as $f:V_{lk}\rightarrow R,f_{i}=f(v_{i})$. The signal $F_{lk}$ can be transformed to $H_{lk}=\Phi_{lk}^{T}F_{lk},H_{lk}=\{h_{0},...,h_{N_{lk}}\}$. $h_{0}$ is the DC coefficient, and $h_{1},...h_{N_{lk}}$ are all AC coefficients. Only $h_{0}$ will be propagated to the upper level as a new graph signal.

For example, when encoding the color information of point clouds, at level $(L-1)$, $F_{L-1,k}$ can be the color values (e.g., luminance values) at points in the subspace represented by the $k$th node. Then the DC coefficients of $F_{L-1,k}$ would be a new signal defined on the $k$th node and will be further encoded in the next stage. Finally, all AC coefficients and a DC coefficient of the root node are quantized and entropy-coded.

RAHT is a special case of the proposed SSGT scheme. In every level of the octree, the subspace is further partitioned in x,y,z directions, and the created graphs only contain two vertices, denoted as $v_{1}$ and $v_{2}$. The corresponding degree matrix, Laplacian matrix, and the normalized Laplacian matrix are expressed as following:

where $a_{1}$ and $a_{2}$ represent the number of points in $v_{1}$ and $v_{2}$ respectively. Calculate the eigenvalue decomposition of $L^{sym}$, the graph transform matrix $\Phi$ and the eigenvalue matrix is

$\Phi$ is the same as the transform matrix in RAHT. There are two limitations of only adopting two-vertices transforms in RAHT: 1) it treats the x, y, z directions differently, which causes the performance difference when changing the order of the transforms along x, y, and z axes[11], and 2) this simple graph neglects the influence of the pairwise distances among points leading to weak modeling of the point relationships in the subspace.