SCP: Spherical-Coordinate-Based Learned Point Cloud Compression

Hand-crafted methods based on geometry typically employ tree structures like Octree (Jackins and Tanimoto 1980; Garcia and de Queiroz 2018; Unno et al. 2023) and predictive geometry (ISO/IEC JTC 1/SC 29/WC 7 2019) to organize unstructured point clouds. The Octree structure is widely used in numerous methods, such as (Huang et al. 2008; Kammerl et al. 2012). Predictive geometry, on the other hand, models LiDAR PCs as a predictive tree, where each branch represents a complete point chain scanned by a laser beam. This approach leverages the circular chain structure in LiDAR PCs to predict subsequent points based on the angles and distances of previous points in the chain. Leveraging the circular shape of LiDAR PCs, (Sridhara, Pavez, and Ortega 2021) proposed a transformation from Cartesian-coordinate-based positions to Cylindrical coordinates. This approach, being sensitive to circular shapes, enhances the performance of the G-PCC algorithm in compressing point clouds.

In recent years, learned point cloud compression methods have been emerging. Many of these techniques, including those cited in (Nguyen et al. 2021b; Quach, Valenzise, and Dufaux 2019; Que, Lu, and Xu 2021; Nguyen et al. 2021a; Wang et al. 2022), utilize Octree to represent and compress point clouds.

OctSqueeze (Huang et al. 2020) builds the Octree of the point cloud, predicting voxel occupancy level by level, using information from ancient voxels and known data about the current voxel. Building upon OctSqueeze, methods such as VoxelDNN (Nguyen et al. 2021a), VoxelContext-Net (Que, Lu, and Xu 2021), SparsePCGC (Wang et al. 2022), and OctFormer (Cui et al. 2023) eliminate redundancy by employing the information of neighbor voxels of the parent voxel. Moreover, Surface Prior (Chen et al. 2022) incorporates neighbor voxels which share the same depth as the current coding voxel, into the framework. OctAttention (Fu et al. 2022) further utilizes this kind of voxel, increasing the context size from surrounding 26 voxels to 1024 voxels. This significantly expands the receptive field of the current voxel. Building on OctAttention, EHEM (Song et al. 2023) further exploits the potential of the Transformer framework by integrating the Swin Transformer (Liu et al. 2021) into their method. This effectively increases the context size to 8192 sibling voxels. Simultaneously, it transitions from serial coding to a checkerboard (He et al. 2021a) type, resulting in a substantial reduction in decoding time. In another aspect, (Chen et al. 2022) employs a context with uncle voxels and utilizes the feature of circular shapes in the LiDAR PCs by incorporating a geometry-prior loss for training.

The Octree is a hierarchical data structure that provides an efficient way of representing 3D point clouds and is beneficial for dealing with inherently unstructured and sparse LiDAR PCs. When constructing the Octree, the whole point cloud is taken as a cube, a.k.a a voxel, then divided into 8 equal-sized sub-voxels. This procedure repeats until the side length of a leaf voxel is equal to the quantization step. Then the quantization operation merges points in the same leaf voxel together. In each non-leaf voxel, the sub-voxels with points in them are represented by a bit $1$, and the empty ones are set to value $0$. Therefore, each non-leaf voxel is represented by an occupancy symbol composed of 8 bits $(1-255)$, where each bit indicates the occupancy status of the corresponding sub-voxel.

We introduce different coordinate systems involved in our experiments: 3-D Cartesian coordinates, Cylindrical coordinates, and Spherical coordinates.

The occupancy sequence $\boldsymbol{x}=\{x_{1},...,x_{V}\}$ ($V$ is the number of voxels) of voxels in an Octree is losslessly encoded by an entropy encoder into a bitstream, based on the predicted probability from a neural network. After transmission, the Octree is reconstructed from the entropy-decoded occupancy sequence and converted back into the point cloud. Hence, the objective of point cloud geometry compression is to predict the occupancy of each voxel in the Octree, which can be viewed as a 255-category classification problem. Given all the parameters in our model as $\boldsymbol{w}$ and context information as $\boldsymbol{C}=\{\boldsymbol{C}_{1},...,\boldsymbol{C}_{V}\}$, the optimization objective $\mathcal{L}$ is the cross-entropy between the ground truth occupancy value $t\in[1,255]$ and the corresponding predicted probability of each voxel $\tilde{p}_{t}(x_{i}|\boldsymbol{C}_{i};\boldsymbol{w})$. This can be defined as follows:

The circular shapes and azimuthal angle invariance features in LiDAR PCs lead to a high degree of redundancy. In common-used Cartesian coordinates, redundant points are segmented into discrete voxels, as shown in Fig. 1(a). This segmentation creates an unstable context, in which nearby voxels may not be relevant.

To leverage both features of LiDAR PCs, we propose the Spherical-Coordinate-based learned Point cloud Compression (SCP) method, which converts the points from Cartesian coordinates to Spherical coordinates. When looking up-down, the Spherical-coordinate-based Octree has the same structure as the Cylindrical one, as shown in Fig. 1(a)-left. Points in the same chain are assigned together, making better usage of circular shapes. On the other hand, when looking horizontally from the original point, the Spherical-coordinate-based Octree (Fig. 1(c)) is more likely to assign the points in the same chain to the same voxel compared with the Cylindrical one (Fig. 1(a)-right), harnessing the azimuthal angle invariance feature. Furthermore, the representation of circular structures in Spherical coordinates is simplified from quadratic to linear, making the prediction problem easier to solve, as illustrated in Fig. 3.

When constructing Octree in Spherical coordinates, the quantization procedure differs from that in Cartesian coordinates. The variation range for the $\rho$, $\theta$ and $\varphi$ coordinates in Spherical coordinates are $[0,\rho_{max}]$, $[0,2\pi)$ and $[0,\pi]$, respectively. The $\rho_{max}$ varies in different point clouds. When quantizing the point clouds, we adopt the quantization step $q$ of EHEM (Song et al. 2023) to the $\rho$ coordinate, noted as $q_{\rho}=q$. Therefore, the total number of bins for $\rho$ coordinate is $b=\lceil\rho_{max}/q_{\rho}\rceil$. Subsequently, we divide the ranges of the $\theta$ and $\varphi$ coordinates by $b$ separately to generate their respective quantization steps, as demonstrated in Eq.(LABEL:eq:qs).

where $q$ is the quantization step. By setting these $q_{\theta}$ and $q_{\varphi}$, the bin number of each coordinate are the same.

In the Spherical-coordinate-based Octree, the distance $\rho$ is positively related to the reconstruction error, causing the higher reconstruction error of distant voxels. To maintain a reconstruction error lower or similar to that in Cartesian coordinates, we propose a multi-level Octree for SCP, which allocates additional levels to distant voxels. We initially discuss the relation between distant voxels and reconstruction errors of the quantized point cloud in Cartesian and Spherical coordinates, then illustrate our multi-level Octree method.

For Cartesian coordinates, the upper bound of the error can be expressed as follows:

where $p_{i}$ represents the original position of point $i$, and $\hat{p}_{i}$ is its position after quantization, $q$ is the quantization step size. It is obvious that this error is uniformly distributed throughout the entire space of Cartesian coordinates because the upper bound is only related to the quantization step. The complete derivation can be found in the Appendix.

In the Spherical coordinate system, however, the voxels’ reconstruction error is non-uniform. As depicted in Fig. 2, the further a voxel (with a larger $\rho$ value) is from the origin, the larger its size. This characteristic leads to a higher upper bound of reconstruction error for distant voxels. As shown in Eq.(6), the reconstruction error $\varepsilon_{s}$ linearly correlated with the value of the $\rho$ coordinate.

We put the proof into the Appendix and further discuss the quantization results in the Experimental Results of the Experiments section.

Based on the above findings, if we allocate additional levels (depths) to the distant areas in the Octree, each added level will reduce the quantization step size of the distant area from $q$ to $\frac{1}{2}q$, which in turn halves $\varepsilon_{s}$ according to Eq.(6). Motivated by this, we propose a multi-level Octree method to reduce the reconstruction error of distant voxels by adding additional levels. Specifically, we divide the point cloud into $N$ parts to assign different numbers of additional levels. Each part has $n\in\{0,1,...,N-1\}$ additional levels with $\rho_{n}\in[t_{n}\rho_{max},t_{n+1}\rho_{max})$, where $0\leq t_{n}<t_{n+1}<1$, $t_{0}=0$, $t_{N}=1$. The assigned extra levels can be represented by the reduction of the quantization $q_{n}=\frac{q}{2^{n}}$, making the reconstruction error upper bound of each part $n$ lower or similar to the upper bound of $\varepsilon_{c}$, which can be expressed as

where we assign $n$ additional levels for the part $n$. In our experiments, we describe how to pick the optimal $t_{n}$ for each part to keep $\varepsilon_{n}\leq\varepsilon_{c}$.

We designate SCP with a multi-level Octree as the default method, referred to as “SCP”, while the SCP method without a multi-level Octree is called “SCP w/o multi-level” or “SCP w/o”. The SCP methods with EHEM (Song et al. 2023) and OctAttention (Fu et al. 2022) as the backbone are represented as “SCP-EHEM” and “SCP-OctAttn”, respectively.

To validate the effectiveness of SCP and the multi-level Octree method, we conducted ablation studies on them, as shown in Fig. 6. For the selection of metric, we find the curve tendencies of 3 metrics in Fig. 4 are similar, so we follow the ablation settings of EHEM, using D1 PSNR as the metric.

To verify the good performance of SCP, we experiment on all three coordinate systems, Spherical, Cylindrical, and Cartesian, with both backbone models EHEM and OctAttention. Note that the Cartesian-coordinate-based methods are the original EHEM/OctAttention. These experiments are conducted without the multi-level Octree method because it is designed for angle-based coordinate systems and is not applicable to Cartesian-coordinate-based Octree. As shown in Fig. 6, Spherical-coordinate-based methods (Spherical ones) are better than both the Cartesian-coordinate-based ones (Cartesian methods) and the Cylindrical-coordinate-based ones (Cylindrical methods). As mentioned in Fig. 1, the Octree under Spherical coordinates has better relevant context for each point in the point cloud than that under Cylindrical/Cartesian coordinates. The improvement is introduced by the azimuthal coordinate and polar ($\rho$, $\theta$) coordinates in the Spherical coordinate system. For the ablation study on the multi-level Octree method, our SCP methods outperform the version without our multi-level Octree method (SCP-$*$ w/o), proving the effectiveness of our method.

We illustrate the derivation of Eq.(5) and Eq.(6) in this section. We use the $L2$-norm to evaluate the reconstruction error. This $L2$-norm is used in all three metrics in our experiments.

For the Octree in Cartesian coordinates, the leaf voxels are cubes whose side length is quantization step $q$. The position of a voxel is the center of it, marked as $(x_{c},y_{c},z_{c})$. The upper bound of reconstruction error can be reached at the corners of the cube, one of which can be written as $(x_{c}+\frac{q}{2},y_{c}+\frac{q}{2},z_{c}+\frac{q}{2})$. Therefore, the upper bound of reconstruction error in Cartesian coordinates is

To compare with former methods, we need to convert Spherical coordinates back to Cartesian coordinates for reconstruction error calculation. In Spherical coordinates, the size of voxels is only relevant to coordinate $\rho$, therefore, we analyze the one voxel beside coordinate $x$, as the green box shown in Fig. 7, which is easier for derivation. The voxel center can be marked as $(\rho_{c}\cos\frac{q_{\theta}}{2},\rho_{c}\sin\frac{q_{\theta}}{2},\rho_{c}\sin\frac{q_{\theta}}{4})$. We use $q_{\theta}$ to replace $q_{\varphi}$ according to Eq.(LABEL:eq:qs). In the most situation, $q$ is much smaller that $\rho_{c}$, so we can set $\cos\frac{q_{\theta}}{2}=1$, $\sin\frac{q_{\theta}}{2}=\sin\frac{q_{\theta}}{4}=1$. Then the position can be rewritten as: $(\rho,\rho\frac{q_{\theta}}{2},\rho\frac{q_{\theta}}{4})$. On the other hand, one of the furthest points in the voxel is indicated in Fig. 7. The position of it is $(\rho_{c}+\frac{q}{2},0,0)$, which can be approximated as $(\rho_{c},0,0)$ since $q$ is much smaller than $\rho_{c}$. In conclusion, the upper bound of the reconstruction error can be derived by calculating the distance between these two points, as shown in Eq.(9).