Scheduling the Off-Diagonal Weingarten Loss of Neural SDFs for CAD Models

TL;DR

This work tackles curvature regularization for neural SDFs used in CAD reconstruction by introducing time-varying scheduling of the Off-Diagonal Weingarten (ODW) loss. By starting with a strong regularization and gradually decaying the ODW weight through various schedules (constant, linear, quintic, step, and warm-up), the method stabilizes early optimization while enabling fine-detail refinement. Experiments on the ABC CAD dataset show that all time-varying schedules outperform the fixed-weight baseline, with quintic scheduling often delivering the best Chamfer Distance improvements (up to 35% relative reduction). The results demonstrate that dynamic curvature priors are a simple yet effective extension for robust, high-fidelity CAD reconstruction using neural implicit surfaces, and point toward adaptive weighting for broader shape categories as future work.

Abstract

Neural signed distance functions (SDFs) have become a powerful representation for geometric reconstruction from point clouds, yet they often require both gradient- and curvature-based regularization to suppress spurious warp and preserve structural fidelity. FlatCAD introduced the Off-Diagonal Weingarten (ODW) loss as an efficient second-order prior for CAD surfaces, approximating full-Hessian regularization at roughly half the computational cost. However, FlatCAD applies a fixed ODW weight throughout training, which is suboptimal: strong regularization stabilizes early optimization but suppresses detail recovery in later stages. We present scheduling strategies for the ODW loss that assign a high initial weight to stabilize optimization and progressively decay it to permit fine-scale refinement. We investigate constant, linear, quintic, and step interpolation schedules, as well as an increasing warm-up variant. Experiments on the ABC CAD dataset demonstrate that time-varying schedules consistently outperform fixed weights. Our method achieves up to a 35% improvement in Chamfer Distance over the FlatCAD baseline, establishing scheduling as a simple yet effective extension of curvature regularization for robust CAD reconstruction.

Figures (4)

• Figure 1: Principal curvatures $\kappa_1$ and $\kappa_2$ are the eigenvalues of the Weingarten map $S$. Their product gives the Gaussian curvature $K=\kappa_1 \kappa_2 = \det S$. Because $S$ is self-adjoint with respect to the first fundamental form, its eigenvectors (the principal directions) are orthogonal in the surface metric whenever the eigenvalues are distinct. At an umbilic point (e.g. on a sphere) the entire two-dimensional tangent plane is the eigenspace, so no unique pair of principal directions exists. For a plane the second fundamental form vanishes, i.e., $S=0$.
• Figure 2: Scheduling strategies for the ODW weight $\lambda_{\mathrm{ODW}}(t)$. From left to right: Constant (FlatCAD baseline), Linear (piecewise linear ramp), Quintic (fifth-order polynomial interpolation), and Step (discontinuous jump). All schedules share the same control points ($[(0,10),\,(0.2,10),\,(0.5,0.001),\,(1.0,0)]$, and normalized time $t\!\in\![0,1]$, $n_{\text{iter}}{=}10{,}000$). The apparent kink at $t=0.2$ in the quintic curve arises because the weight is held constant before 0.2 and only begins quintic interpolation afterward.
• Figure 3: Comparison with the original FlatCAD Yin2025FlatCAD. We evaluate our proposed weight scheduling strategies—linear, quintic (fifth-order polynomial), and step interpolation—against the baseline with constant weights. All three schedules achieve comparable or superior reconstruction quality, consistently producing cleaner, more complete, and geometrically faithful surfaces that closely match the ground truth geometry (GT). The improvement arises from curvature-aware regularization that promotes developability, effectively suppressing spurious artifacts. For detailed numerical evaluation, please refer to the quantitative results reported in Table \ref{['tab:interpolation_compare']}.
• Figure 4: Scheduling strategies for the ODW weight $\lambda_{\mathrm{ODW}}(t)$. From left to right: Constant baseline (fixed $w=10$), Decreasing linear (strong--start/decay from $10$ to $0$), and Increasing linear (warm-up from $0$ to $10$). The horizontal axis shows normalized training progress $t \in [0,1]$.