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 $x_0$, 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: \mathbb{R}^3 \to \mathbb{R}$0, and analogous expressions for $f: \mathbb{R}^3 \to \mathbb{R}$1 and $f: \mathbb{R}^3 \to \mathbb{R}$2.

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 $f: \mathbb{R}^3 \to \mathbb{R}$3 is approximated as

$f: \mathbb{R}^3 \to \mathbb{R}$4

Near the zero-level set, $f: \mathbb{R}^3 \to \mathbb{R}$5, simplifying $f: \mathbb{R}^3 \to \mathbb{R}$6. The associated loss is $f: \mathbb{R}^3 \to \mathbb{R}$7 or $f: \mathbb{R}^3 \to \mathbb{R}$8.

• FD Rank-Deficiency Loss: The rank-deficiency term is similarly $f: \mathbb{R}^3 \to \mathbb{R}$9, penalized as $x_0$0 or $x_0$1.

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 $x_0$2 and sampling regime:

• Step Size: Empirically, $x_0$3 or smaller captures fine detail, balancing truncation error and numerical noise.
• Sampling Scheme: Shell points $x_0$4 are uniformly sampled in the bounding box (20k per iteration typical), with near-surface projection via $x_0$5. A local tangent frame is constructed at each $x_0$6 using the normal $x_0$7, followed by random tangent directions $x_0$8, $x_0$9. FD stencils require evaluation at eight neighboring positions for mixed derivatives.

Surface-anchored Dirichlet samples $(u,v) \perp n$0 ($(u,v) \perp n$1) 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:

$(u,v) \perp n$2

where:

• $(u,v) \perp n$3 aligns network predictions to observed surface,
• $(u,v) \perp n$4 (Atzmon & Lipman SAL++) penalizes non-manifold solutions,
• $(u,v) \perp n$5 enforces signed-distance constraint,
• $(u,v) \perp n$6 uses $(u,v) \perp n$7 or $(u,v) \perp n$8 from the FD framework.

Hyperparameters typically are $(u,v) \perp n$9, $n = \nabla f / \|\nabla f\|$0, $n = \nabla f / \|\nabla f\|$1. A linear warm-up for $n = \nabla f / \|\nabla f\|$2 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 = \nabla f / \|\nabla f\|$3 surface and $n = \nabla f / \|\nabla f\|$4 off-surface points.
2. Forward Pass: Evaluate $n = \nabla f / \|\nabla f\|$5 and $n = \nabla f / \|\nabla f\|$6 at sampled locations.
3. Curvature Stencil Computation: For each $n = \nabla f / \|\nabla f\|$7, calculate FD stencils using tangent vectors $n = \nabla f / \|\nabla f\|$8, and evaluate $n = \nabla f / \|\nabla f\|$9 at required offsets.
4. Loss Evaluation: Compute $x_0$0, $x_0$1, and curvature loss $x_0$2, 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$3 plus one backward gradient calculation. Memory usage scales as $x_0$4, in contrast to $x_0$5 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 $x_0$6k points, with strong errors only for extremely sparse ($x_0$7k) 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 $x_0$8 values in $x_0$9 as optimal, robust to variation ($$f_{uu} \approx \frac{f(x_0 + h u) - 2 f(x_0) + f(x_0 - h u)}{h^2}$$0 to $$f_{uu} \approx \frac{f(x_0 + h u) - 2 f(x_0) + f(x_0 - h u)}{h^2}$$1).

7. Practical Recommendations and Limitations

Best practices for curvature-aware densification include:

• Selecting $$f_{uu} \approx \frac{f(x_0 + h u) - 2 f(x_0) + f(x_0 - h u)}{h^2}$$2 to match the smallest feature scale (0.1–1% bounding-box diagonal); excessive $$f_{uu} \approx \frac{f(x_0 + h u) - 2 f(x_0) + f(x_0 - h u)}{h^2}$$3 increases truncation error, while overly small $$f_{uu} \approx \frac{f(x_0 + h u) - 2 f(x_0) + f(x_0 - h u)}{h^2}$$4 amplifies noise.
• Ensuring dense off-surface sampling (≥10k per iteration) for stable curvature estimation.
• Implementing a gradual ramp-up of $$f_{uu} \approx \frac{f(x_0 + h u) - 2 f(x_0) + f(x_0 - h u)}{h^2}$$5 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 $$f_{uu} \approx \frac{f(x_0 + h u) - 2 f(x_0) + f(x_0 - h u)}{h^2}$$6 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).