  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.
