Curvature-Aware Densification in Neural SDFs

• The paper introduces curvature-aware densification by integrating finite-difference approximations of second-order derivatives into neural SDF learning to achieve developable, feature-preserving surfaces.
• It leverages FD stencils for computing Gaussian and rank-deficiency curvature losses, significantly reducing memory requirements and computational overhead compared to higher-order automatic differentiation.
• Empirical evaluations demonstrate that the approach matches competitive accuracy while offering faster runtimes and robust performance under sparse sampling and incomplete data regimes.

Curvature-aware densification refers to the integration of curvature-sensitive regularization mechanisms into the learning of neural signed distance fields (SDFs), where the explicit modeling of surface curvature is employed to promote the reconstruction of developable, feature-preserving surfaces—even under sparse or incomplete sampling conditions. The finite-difference (FD) framework developed in "A Finite Difference Approximation of Second Order Regularization of Neural-SDFs" (Yin et al., 12 Nov 2025) enables this process through computationally efficient, second-order accurate approximations of differential geometric quantities, replacing costly higher-order automatic differentiation. The approach serves as a scalable and memory-efficient drop-in replacement for existing curvature regularization terms, supporting robust SDF learning across a range of geometric and data regimes.

1. Finite-Difference Stencils and Second-Order Accuracy

The FD framework approximates second derivatives required for curvature regularization through central difference stencils utilizing local Taylor expansions with truncation error $O(h^2)$. For a neural SDF $f: \mathbb{R}^3 \to \mathbb{R}$ evaluated at point %%%%2%%%%, an orthonormal tangent frame $(u,v) \perp n$ is established, where $n = \nabla f / \|\nabla f\|$. The directional second derivatives at $x_0$ are computed as follows:

• $$f_{uu} \approx \frac{f(x_0 + h u) - 2 f(x_0) + f(x_0 - h u)}{h^2}$$
• $$f_{vv} \approx \frac{f(x_0 + h v) - 2 f(x_0) + f(x_0 - h v)}{h^2}$$
• $$f_{uv} \approx \frac{f(x_0 + h u + h v) - f(x_0 + h u - h v) - f(x_0 - h u + h v) + f(x_0 - h u - h v)}{4 h^2}$$

The Taylor expansion confirms second-order accuracy:

$$f(x_0 \pm h u) = f(x_0) \pm h \nabla f \cdot u + \frac{h^2}{2} u^T H_f u \pm \frac{h^3}{6} D^3 f(u,u,u) + O(h^4),$$

yielding $f_{uu} = u^T H_f u + O(h^2)$, and analogous expressions for $f_{vv}$ and $f_{uv}$.

2. Curvature-Aware Regularization Losses

Curvature-aware densification employs FD-derived proxies for surface regularization. The principal mechanisms are:

• FD Gaussian Curvature Loss: The Gaussian curvature at $x_0$ is approximated as

$$K_{FD}(x_0) = \frac{f_{uu} f_{vv} - f_{uv}^2}{\|\nabla f(x_0)\|^4}.$$

Near the zero-level set, $\|\nabla f\| \approx 1$, simplifying $K_{FD} \approx f_{uu} f_{vv} - f_{uv}^2$. The associated loss is $L_G = \mathbb{E}_{x_0}[|K_{FD}(x_0)|]$ or $\mathbb{E}[K_{FD}(x_0)^2]$.

• FD Rank-Deficiency Loss: The rank-deficiency term is similarly $D_{FD}(x_0) = f_{uu} f_{vv} - f_{uv}^2$, penalized as $L_R = \mathbb{E}_{x_0}[|D_{FD}(x_0)|]$ or $\mathbb{E}[D_{FD}(x_0)^2]$.

These losses target zero Gaussian curvature or rank-deficient Hessian matrices to favor developable or singular surfaces as dictated by reconstruction goals.

3. Step-Size Selection and Spatial Sampling

Optimal application of finite-difference regularization depends on careful choice of the spatial FD step-size $h$ and sampling regime:

• Step Size: Empirically, $$h \simeq 10^{-3} \cdot \text{diameter(bounding box)}$$ or smaller captures fine detail, balancing truncation error and numerical noise.
• Sampling Scheme: Shell points $x_0$ are uniformly sampled in the bounding box (20k per iteration typical), with near-surface projection via $f$. A local tangent frame is constructed at each $x_0$ using the normal $n$, followed by random tangent directions $u \perp n$, $v = n \times u$. FD stencils require evaluation at eight neighboring positions for mixed derivatives.

Surface-anchored Dirichlet samples $x_s$ ($|f(x_s)| \rightarrow 0$) anchor known geometry, while off-surface samples stabilize curvature estimates through dense coverage.

4. Integrated Training Objective and Hyperparameters

Curvature-aware densification is realized within a composite loss function:

$$L_{\text{total}} = L_{DM} + \lambda_{DNM} L_{DNM} + \lambda_{\text{eik}} L_{\text{eik}} + \lambda_{fd} L_{fd},$$

where:

• $L_{DM} = \mathbb{E}_{x_s}[|f(x_s)|]$ aligns network predictions to observed surface,
• $L_{DNM}$ (Atzmon & Lipman SAL++) penalizes non-manifold solutions,
• $$L_{\text{eik}} = \mathbb{E}_x[(\|\nabla f(x)\| - 1)^2]$$ enforces signed-distance constraint,
• $L_{fd}$ uses $L_G$ or $L_R$ from the FD framework.

Hyperparameters typically are $\lambda_{DNM} = 0.01$, $\lambda_{\text{eik}} = 0.1$, $\lambda_{fd} \in [0.4,1.0]$. A linear warm-up for $\lambda_{fd}$ over the first few thousand iterations mitigates early training oscillations.

5. Algorithmic Workflow, Complexity, and Memory Profiling

A typical training iteration proceeds as follows:

1. Point Sampling: Sample $N_\text{surf}$ surface and $N_\text{off}$ off-surface points.
2. Forward Pass: Evaluate $f$ and $\nabla f$ at sampled locations.
3. Curvature Stencil Computation: For each $x_0 \in X_{\text{off}}$, calculate FD stencils using tangent vectors $u, v$, and evaluate $f$ at required offsets.
4. Loss Evaluation: Compute $L_{DM}$, $L_{\text{eik}}$, and curvature loss $L_{fd}$, aggregate according to hyperparameters.
5. Backpropagation: Update network parameters using only first-order gradients.

The FD method demands approximately 9 forward passes per $x_0$ plus one backward gradient calculation. Memory usage scales as $$O(\text{batch size} \cdot \text{cost}_{\text{first-order}})$$, in contrast to $O(\text{batch size} \cdot n_{\text{params}})$ for full Hessian autodiff. FD typically halves memory requirements and yields training speeds 1.3–2× faster than second-order differentiation.

6. Empirical Performance and Robustness

Evaluations on ABC subsets (100 shapes, “1 MB” random and “5 MB” curated) establish the FD method’s parity with autodiff proxies:

• Accuracy: FD-NSH and FD-NCR losses match or marginally trail NeurCADRecon/NSH in Chamfer Distance (CD), F1, and Normal Consistency (NC).
• Efficiency: Example metrics for 1 MB set on H100 GPU:

Method	Chamfer D.	Normal Cons.	Time (s)	Mem (GB)
NSH	2.74	93.93%	559	6.1
NSH-FD	2.93	94.96%	363	4.3
NCR	2.65	93.71%	391	6.06
NCR-FD	4.10	93.41%	331	4.03

• Sparse/Incomplete Data: Reconstruction degrades gracefully to $5$k points, with strong errors only for extremely sparse ($1$k) sampling. Incomplete point clouds yield increased CD (+64%), minor reduction in NC (−0.7%), with topology preserved. On non-CAD shapes (Stanford Armadillo), FD reduces runtime by a factor of 1.9, with comparable reconstruction fidelity.

Ablation studies identify $\lambda_{fd}$ values in $[0.6, 1.0]$ as optimal, robust to variation ($0.2$ to $5$).

7. Practical Recommendations and Limitations

Best practices for curvature-aware densification include:

• Selecting $h$ to match the smallest feature scale (0.1–1% bounding-box diagonal); excessive $h$ increases truncation error, while overly small $h$ amplifies noise.
• Ensuring dense off-surface sampling (≥10k per iteration) for stable curvature estimation.
• Implementing a gradual ramp-up of $\lambda_{fd}$ after initial 500–1k iterations to regularize learning dynamics.
• Recognizing benefits: FD decreases GPU memory by approximately 30–40%, reduces wall-clock time by up to 2×, and is compatible as a drop-in regularization replacement.

Limitations entail increased per-iteration forward calls (~8 additional evaluations per sample), with overall faster convergence compared to full second-order approaches. Performance is sensitive to choices of $h$ and off-surface point distribution, necessitating minor hyperparameter tuning.

In summary, finite-difference-based curvature-aware densification constitutes a simple, second-order-accurate, and memory-efficient approach for Gaussian and rank-deficiency regularization in neural-SDF reconstruction, supporting developable, feature-preserving surface synthesis even in regimes of sparse or incomplete geometric input (Yin et al., 12 Nov 2025).