A Finite Difference Approximation of Second Order Regularization of Neural-SDFs

TL;DR

This work tackles the high cost of curvature priors in neural SDF learning by introducing a finite-difference framework that replaces second-order automatic differentiation with $f_{uu}, f_{vv}, f_{uv}$ Taylor-stencil estimates, achieving $O(h^2)$ accuracy. By deriving FD analogs for Gaussian curvature and the Neural-Singular-Hessian regularizers, the method integrates with standard self-supervised losses and relies only on forward evaluations of $f$ while preserving curvature-aware priors. Empirical results on ABC dataset subsets show reconstruction quality on par with AD-based baselines, with memory and training-time reductions up to $2\times$, and robust performance on sparse, incomplete, and non-CAD data. The approach offers a scalable, geometry-faithful alternative for curvature-aware neural SDF learning with practical impact for large-scale surface reconstruction tasks.

Abstract

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.

Figures (5)

• Figure 1: Comparison with state-of-the-art surface reconstruction. FD and AD proxies yield comparable accuracy. Our FD approach avoids second-order differentiation while maintaining efficiency.
• Figure 2: Qualitative reconstructed results on point clouds with varying levels of sparsity.
• Figure 3: Reconstruction from incomplete input clouds. Missing planar regions are plausibly filled while preserving global continuity.
• Figure 4: Reconstructions with different regularization weights. Results remain stable across a wide range.
• Figure 5: Reconstruction results on Armadillo. FD-NCR and NeurCADRecon give smooth, complete surfaces, NSH show more artifacts.