In stark contrast to the case of images, finding a concise, learnable discrete representation of 3D surfaces remains a challenge. In particular, while polygon meshes are arguably the most common surface representation used in geometry processing, their irregular and combinatorial structure often make them unsuitable for learning-based applications. In this work, we present VoroMesh, a novel and differentiable Voronoi-based representation of watertight 3D shape surfaces. From a set of 3D points (called generators) and their associated occupancy, we define our boundary representation through the Voronoi diagram of the generators as the subset of Voronoi faces whose two associated (equidistant) generators are of opposite occupancy: the resulting polygon mesh forms a watertight approximation of the target shape's boundary. To learn the position of the generators, we propose a novel loss function, dubbed VoroLoss, that minimizes the distance from ground truth surface samples to the closest faces of the Voronoi diagram which does not require an explicit construction of the entire Voronoi diagram. A direct optimization of the Voroloss to obtain generators on the Thingi32 dataset demonstrates the geometric efficiency of our representation compared to axiomatic meshing algorithms and recent learning-based mesh representations. We further use VoroMesh in a learning-based mesh prediction task from input SDF grids on the ABC dataset, and show comparable performance to state-of-the-art methods while guaranteeing closed output surfaces free of self-intersections.

翻译：与图像情况形成鲜明对比的是，寻找简洁且可学习的3D表面离散表示仍是一项挑战。特别是，虽然多边形网格可能是几何处理中最常用的表面表示，但其不规则的组合结构往往使其不适合基于学习的应用。本文提出VoroMesh，一种新颖且可微的基于Voronoi图的水密3D形状表面表示方法。通过一组3D点（称为生成器）及其关联的占据状态，我们利用生成器的Voronoi图定义边界表示：Voronoi面中两个关联（等距）生成器具有相反占据状态的子集，生成的 polygon mesh 构成对目标形状边界的水密近似。为学习生成器位置，我们提出一种新型损失函数（VoroLoss），通过最小化真实表面样本到Voronoi图最近面的距离（无需显式构建整个Voronoi图）实现。在Thingi32数据集上直接优化VoroLoss以获得生成器，证明我们的表示相较于公理网格划分算法和近期基于学习的网格表示具有几何高效性。进一步在ABC数据集上，我们将VoroMesh用于基于输入SDF网格的学习性网格预测任务，在保证输出封闭表面无自交的前提下，展现出与最新方法相当的性能。