  We study 3D point cloud attribute compression using a volumetric approach:
given a target volumetric attribute function $f : \mathbb{R}^3 \rightarrow
\mathbb{R}$, we quantize and encode parameter vector $\theta$ that
characterizes $f$ at the encoder, for reconstruction
$f_{\hat{\theta}}(\mathbf{x})$ at known 3D points $\mathbf{x}$'s at the
decoder. Extending a previous work Region Adaptive Hierarchical Transform
(RAHT) that employs piecewise constant functions to span a nested sequence of
function spaces, we propose a feedforward linear network that implements
higher-order B-spline bases spanning function spaces without
eigen-decomposition. Feedforward network architecture means that the system is
amenable to end-to-end neural learning. The key to our network is space-varying
convolution, similar to a graph operator, whose weights are computed from the
known 3D geometry for normalization. We show that the number of layers in the
normalization at the encoder is equivalent to the number of terms in a matrix
inverse Taylor series. Experimental results on real-world 3D point clouds show
up to 2-3 dB gain over RAHT in energy compaction and 20-30% bitrate reduction.
