We study 3D point cloud attribute compression via a volumetric approach:
assuming point cloud geometry is known at both encoder and decoder, parameters
$\theta$ of a continuous attribute function $f: \mathbb{R}^3 \mapsto
\mathbb{R}$ are quantized to $\hat{\theta}$ and encoded, so that discrete
samples $f_{\hat{\theta}}(\mathbf{x}_i)$ can be recovered at known 3D points
$\mathbf{x}_i \in \mathbb{R}^3$ at the decoder. Specifically, we consider a
nested sequences of function subspaces $\mathcal{F}^{(p)}_{l_0} \subseteq
\cdots \subseteq \mathcal{F}^{(p)}_L$, where $\mathcal{F}_l^{(p)}$ is a family
of functions spanned by B-spline basis functions of order $p$, $f_l^*$ is the
projection of $f$ on $\mathcal{F}_l^{(p)}$ and encoded as low-pass coefficients
$F_l^*$, and $g_l^*$ is the residual function in orthogonal subspace
$\mathcal{G}_l^{(p)}$ (where $\mathcal{G}_l^{(p)} \oplus \mathcal{F}_l^{(p)} =
\mathcal{F}_{l+1}^{(p)}$) and encoded as high-pass coefficients $G_l^*$. In
this paper, to improve coding performance over [1], we study predicting
$f_{l+1}^*$ at level $l+1$ given $f_l^*$ at level $l$ and encoding of $G_l^*$
for the $p=1$ case (RAHT($1$)). For the prediction, we formalize RAHT(1) linear
prediction in MPEG-PCC in a theoretical framework, and propose a new nonlinear
predictor using a polynomial of bilateral filter. We derive equations to
efficiently compute the critically sampled high-pass coefficients $G_l^*$
amenable to encoding. We optimize parameters in our resulting feed-forward
network on a large training set of point clouds by minimizing a rate-distortion
Lagrangian. Experimental results show that our improved framework outperformed
the MPEG G-PCC predictor by $11$ to $12\%$ in bit rate reduction.

我们通过一种体积方法研究了三维点云属性的压缩：假设点云几何在编码器和解码器都已知，连续属性函数的参数 θ 被量化为 ḣ θ 并进行编码，以便可以在解码器上以已知的三维点 xᵢ 恢复离散样本 fₕₑₐₜ θ (xᵢ)。具体而言，我们考虑一个嵌套函数子空间序列 F⁽ᵖ⁾ₗ₀ ⊆ ⋯ ⊆ F⁽ᵖ⁾ₗ，其中 F⁽ᵖ⁾ₗ是由 p 阶 B - 样条基函数张成的函数族，fₗˡ⁻ ᵢ *  是 f 在 F⁽ᵖ⁾ₗ 上的投影，并编码为低通系数 Fˡ⁻ ᵢ *，gₗˡ⁻ ᵢ * 是正交子空间 G⁽ᵖ⁾ₗ 上的残差函数（其中 G⁽ᵖ⁾ₗ ⊕ F⁽ᵖ⁾ₗ = F⁽ᵖ⁾ₗ₊₁），并编码为高通系数 Gˡ⁻ ᵢ *。本文中，为了在 p=1 情况（RAHT (1)）上提高编码性能，我们研究了在 l 水平上用 l+1 水平的 fˡ⁺ ᵢ * 预测 fₗˡ⁻ ᵢ * 以及 Gˡ⁻ ᵢ * 的编码。在预测方面，我们在 MPEG-PCC 中理论框架下形式化了 RAHT (1) 的线性预测，并提出了一个使用双边滤波器多项式的非线性预测器。我们推导了计算可编码的临界采样高通系数 Gˡ⁻ ᵢ * 的方程。通过最小化速率失真拉格朗日量在大型点云训练集上优化我们的前馈网络的参数。实验结果表明，我们改进的框架比 MPEG G-PCC 预测器在比特率降低方面提高了 11% 到 12%。