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

TL;DR

This paper presents a finite-difference approach for curvature regularization in neural SDFs that reduces computational costs while maintaining high reconstruction quality, making it more scalable and efficient.

Contribution

It introduces a lightweight finite-difference framework for curvature regularization in neural SDFs, replacing costly second-order differentiation with efficient approximations.

Findings

Achieves comparable reconstruction fidelity to automatic differentiation methods.

Reduces GPU memory usage and training time by up to 50%.

Demonstrates robustness on sparse, incomplete, and non-CAD data.

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.