We propose DualConvMesh-Nets (DCM-Net) a family of deep hierarchical
convolutional networks over 3D geometric data that combines two types of
convolutions. The first type, geodesic convolutions, defines the kernel weights
over mesh surfaces or graphs. That is, the convolutional kernel weights are
mapped to the local surface of a given mesh. The second type, Euclidean
convolutions, is independent of any underlying mesh structure. The
convolutional kernel is applied on a neighborhood obtained from a local
affinity representation based on the Euclidean distance between 3D points.
Intuitively, geodesic convolutions can easily separate objects that are
spatially close but have disconnected surfaces, while Euclidean convolutions
can represent interactions between nearby objects better, as they are oblivious
to object surfaces. To realize a multi-resolution architecture, we borrow
well-established mesh simplification methods from the geometry processing
domain and adapt them to define mesh-preserving pooling and unpooling
operations. We experimentally show that combining both types of convolutions in
our architecture leads to significant performance gains for 3D semantic
segmentation, and we report competitive results on three scene segmentation
benchmarks. Our models and code are publicly available.

提出了 DualConvMesh-Nets (DCM-Net) 模型，它是一种基于 3D 几何数据的深层分层卷积神经网络，结合了两种卷积类型：测地线卷积和欧几里得卷积，并且还采用了几何处理领域中的网格简化方法来定义网格池化和反池化运算。实验结果表明，我们提出的模型在 3D 语义分割任务上取得了显著的性能提升并在三个场景分割基准测试中获得了竞争性的结果。

DualConvMesh-Net: 3D 网格上的测地和欧几里得卷积联合

DualConvMesh-Net: Joint Geodesic and Euclidean Convolutions on 3D Meshes

Generative models for 3D geometric data arise in many important applications
in 3D computer vision and graphics. In this paper, we focus on 3D deformable
shapes that share a common topological structure, such as human faces and
bodies. Morphable Models and their variants, despite their linear formulation,
have been widely used for shape representation, while most of the recently
proposed nonlinear approaches resort to intermediate representations, such as
3D voxel grids or 2D views. In this work, we introduce a novel graph
convolutional operator, acting directly on the 3D mesh, that explicitly models
the inductive bias of the fixed underlying graph. This is achieved by enforcing
consistent local orderings of the vertices of the graph, through the spiral
operator, thus breaking the permutation invariance property that is adopted by
all the prior work on Graph Neural Networks. Our operator comes by construction
with desirable properties (anisotropic, topology-aware, lightweight,
easy-to-optimise), and by using it as a building block for traditional deep
generative architectures, we demonstrate state-of-the-art results on a variety
of 3D shape datasets compared to the linear Morphable Model and other graph
convolutional operators.

本文提出了一种新颖的图卷积算子，通过对网格的本地顺序强制执行连续性，实现对固定底层图形的归纳偏差建模，并将其作为传统深度生成建筑的构件，以多种 3D 形状数据集为例，与线性 Morphable Model 和其他图卷积算子相比，展示出最先进的结果。