用有限差分法高效实现神经SDF的曲率正则化，提升训练速度与内存效率。
A Finite Difference Approximation of Second Order Regularization of Neural-SDFs
- 采用泰勒展开的有限差分模板近似二阶导数，避免复杂自动微分。
- 重建精度与自动微分方法相当，显存占用和训练时间减半。
- 适用于稀疏、不完整及非CAD数据，适合实际三维建模应用。
本文提出一种基于有限差分的神经符号距离场（Neural-SDF）曲率正则化框架。现有方法依赖二阶自动微分获取完整海森矩阵以施加曲率先验，虽准确但计算开销大；部分方法虽避免显式海森矩阵组装，仍需高阶微分。本工作用轻量级有限差分模板替代上述操作，通过泰勒展开近似二阶导数，截断误差为O(h²)，可直接替换高斯曲率损失与秩不足损失。实验表明，该方法在重建保真度上与自动微分方法相当，同时将GPU内存占用和训练时间降低至一半以下。在稀疏、不完整及非CAD数据上的测试也验证了其鲁棒性与通用性，为曲率感知的SDF学习提供了一种高效可扩展的替代方案。
原文摘要 · Abstract (English)
We introduce a finite-difference framework for curvature regularization in neural signed distance field (SDF) learning. Existing approaches enforce curvature priors using full Hessian information obtained via second-order automatic differentiation, which is accurate but computationally expensive. Others reduced this overhead by avoiding explicit Hessian assembly, but still required higher-order differentiation. In contrast, our method replaces these operations with lightweight finite-difference stencils that approximate second derivatives using the well known Taylor expansion with a truncation error of O(h^2), and can serve as drop-in replacements for Gaussian curvature and rank-deficiency losses. Experiments demonstrate that our finite-difference variants achieve reconstruction fidelity comparable to their automatic-differentiation counterparts, while reducing GPU memory usage and training time by up to a factor of two. Additional tests on sparse, incomplete, and non-CAD data confirm that the proposed formulation is robust and general, offering an efficient and scalable alternative for curvature-aware SDF learning.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。