Neural shape representation generally refers to representing 3D geometry using neural networks, e.g., to compute a signed distance or occupancy value at a specific spatial position. Previous methods tend to rely on the auto-decoder paradigm, which often requires densely-sampled and accurate signed distances to be known during training and testing, as well as an additional optimization loop during inference. This introduces a lot of computational overhead, in addition to having to compute signed distances analytically, even during testing. In this paper, we present a novel encoder-decoder neural network for embedding 3D shapes in a single forward pass. Our architecture is based on a multi-scale hybrid system incorporating graph-based and voxel-based components, as well as a continuously differentiable decoder. Furthermore, the network is trained to solve the Eikonal equation and only requires knowledge of the zero-level set for training and inference. Additional volumetric samples can be generated on-the-fly, and incorporated in an unsupervised manner. This means that in contrast to most previous work, our network is able to output valid signed distance fields without explicit prior knowledge of non-zero distance values or shape occupancy. In other words, our network computes approximate solutions to the boundary-valued Eikonal equation. It also requires only a single forward pass during inference, instead of the common latent code optimization. We further propose a modification of the loss function in case that surface normals are not well defined, e.g., in the context of non-watertight surface-meshes and non-manifold geometry. We finally demonstrate the efficacy, generalizability and scalability of our method on datasets consisting of deforming 3D shapes, single class encoding and multiclass encoding, showcasing a wide range of possible applications.

翻译：神经形状表示通常指利用神经网络表示三维几何，例如计算特定空间位置的符号距离或占据值。以往方法倾向于采用自动解码器范式，这通常要求在训练和测试过程中预先知道密集采样且准确的符号距离，并在推理阶段进行额外的优化循环。除了需在测试中解析计算符号距离外，该方法还引入了大量计算开销。本文提出一种新型编码器-解码器神经网络，能够在单次前向传播中嵌入三维形状。我们的架构基于多尺度混合系统，融合了基于图和基于体素的组件，以及连续可微的解码器。此外，网络通过求解Eikonal方程进行训练，仅需零水平集信息即可完成训练与推理。额外的体积样本可在运行时动态生成，并以无监督方式融入训练。这意味着与以往大多数工作不同，我们的网络无需预知非零距离值或形状占据状态，即可直接输出有效的符号距离场。换言之，网络可计算边界值Eikonal方程的近似解，且推理时仅需单次前向传播，而非常见的隐编码优化过程。针对表面法线定义不明确的情况（例如非水密网格与非流形几何），我们进一步提出了损失函数的改进方案。最后，我们在变形三维形状数据集、单类编码及多类编码场景中验证了本方法的有效性、泛化能力与可扩展性，展示了广泛的应用前景。