Learned Nonlinear Predictor for Critically Sampled 3D Point Cloud Attribute Compression

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 $f_{\hat{\theta}}\left(\mathbf{x}_{i}\right)$ 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 ${\mathcal{F}}_{l_{0}}^{(p)} \subseteq \cdots \subseteq {\mathcal{F}}_{L}^{(p)}$, 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 ${\mathcal{F}}_{l}^{(p)}$ represented as low-pass coefficients $F_{l}^{*}$, and $g_{l}^{*}$ is the residual function in an orthogonal subspace ${\mathcal{G}}_{l}^{(p)}$ (where ${\mathcal{G}}_{l}^{(p)} \oplus {\mathcal{F}}_{l}^{(p)}=$ $\left.{\mathcal{F}}_{l+1}^{(p)}\right)$ represented 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 outperforms the MPEG G-PCC predictor by $11 \%-12 \%$ in bit rate.