  We present an adaptive deep representation of volumetric fields of 3D shapes
and an efficient approach to learn this deep representation for high-quality 3D
shape reconstruction and auto-encoding. Our method encodes the volumetric field
of a 3D shape with an adaptive feature volume organized by an octree and
applies a compact multilayer perceptron network for mapping the features to the
field value at each 3D position. An encoder-decoder network is designed to
learn the adaptive feature volume based on the graph convolutions over the dual
graph of octree nodes. The core of our network is a new graph convolution
operator defined over a regular grid of features fused from irregular
neighboring octree nodes at different levels, which not only reduces the
computational and memory cost of the convolutions over irregular neighboring
octree nodes, but also improves the performance of feature learning. Our method
effectively encodes shape details, enables fast 3D shape reconstruction, and
exhibits good generality for modeling 3D shapes out of training categories. We
evaluate our method on a set of reconstruction tasks of 3D shapes and scenes
and validate its superiority over other existing approaches. Our code, data,
and trained models are available at https://wang-ps.github.io/dualocnn.
