Neural implicit representations, which encode a surface as the level set of a neural network applied to spatial coordinates, have proven to be remarkably effective for optimizing, compressing, and generating 3D geometry. Although these representations are easy to fit, it is not clear how to best evaluate geometric queries on the shape, such as intersecting against a ray or finding a closest point. The predominant approach is to encourage the network to have a signed distance property. However, this property typically holds only approximately, leading to robustness issues, and holds only at the conclusion of training, inhibiting the use of queries in loss functions. Instead, this work presents a new approach to perform queries directly on general neural implicit functions for a wide range of existing architectures. Our key tool is the application of range analysis to neural networks, using automatic arithmetic rules to bound the output of a network over a region; we conduct a study of range analysis on neural networks, and identify variants of affine arithmetic which are highly effective. We use the resulting bounds to develop geometric queries including ray casting, intersection testing, constructing spatial hierarchies, fast mesh extraction, closest-point evaluation, evaluating bulk properties, and more. Our queries can be efficiently evaluated on GPUs, and offer concrete accuracy guarantees even on randomly-initialized networks, enabling their use in training objectives and beyond. We also show a preliminary application to inverse rendering.

翻译：将表面视为适用于空间坐标的神经网络水平的隐含表层,已被证明在优化、压缩和生成3D几何学方面非常有效。虽然这些表层很容易适应,但不清楚如何最好地评价形状上的几何查询,例如对射线的交叉或寻找最接近的点。主要的方法是鼓励网络拥有一个经签署的距离属性。然而,这种属性通常只是大致存在,导致稳健问题,并且只有在培训结束时才能保持,从而阻止在损失功能中使用查询。相反,这项工作提出了一种新的方法,直接对一系列现有结构的一般神经隐含功能进行查询。我们的关键工具是将范围分析应用于神经网络,使用自动算术规则来约束一个区域网络的输出;我们对神经网络进行范围分析研究,并找出非常有效的近距离算法变量。我们还利用由此而形成的界限来开发几何查询,包括射线、交叉测试、构建空间等级、快速度隐含功能、对现有各种结构进行直接询问。我们的关键工具是将范围分析应用应用应用应用范围应用到更精确性、最接近点的精确度、最精确性、最精确性、最精确性、最精确性、最精确性、最精确性、最精确性地评估。我们在深度查询中可以评估、最精确性、最精确性、最精确性、最精确性能、最精确性能、更精确性能、更精确性能、更精确性能、更精确性能、更精确性能、更精确性能、更精确性、更精确性能、更精确性能评估。