Implicit Neural Representations (INRs) that learn Signed Distance Functions (SDFs) from point cloud data represent the state-of-the-art for geometrically accurate 3D scene reconstruction. However, training these Neural SDFs often requires enforcing the Eikonal equation, an ill-posed equation that also leads to unstable gradient flows. Numerical Eikonal solvers have relied on viscosity approaches for regularization and stability. Motivated by this well-established theory, we introduce ViscoReg, a novel regularizer that provably stabilizes Neural SDF training. Empirically, ViscoReg outperforms state-of-the-art approaches such as SIREN, DiGS, and StEik on ShapeNet, the Surface Reconstruction Benchmark, and 3D scene reconstruction datasets. Additionally, we establish novel generalization error estimates for Neural SDFs in terms of the training error, using the theory of viscosity solutions.

翻译：从点云数据中学习符号距离函数（SDF）的隐式神经表示（INRs）代表了当前几何精确三维场景重建的最先进技术。然而，训练这些神经SDF通常需要强制满足Eikonal方程，这是一个不适定方程，同时也会导致不稳定的梯度流。数值Eikonal求解器长期以来依赖粘性方法进行正则化和稳定性处理。受这一成熟理论的启发，我们提出了ViscoReg，一种新颖的正则化器，可证明地稳定神经SDF的训练。实验表明，在ShapeNet、表面重建基准测试以及三维场景重建数据集上，ViscoReg的性能优于SIREN、DiGS和StEik等最先进方法。此外，基于粘性解理论，我们首次建立了神经SDF的泛化误差估计与训练误差之间的关系。