Finite-Difference Curvature Regularization

• The paper introduces finite-difference approximations to curvature regularization, eliminating the need for full Hessian evaluations.
• Key methods include central and monotone stencils that offer reliable second derivative estimates with reduced computational cost and proven convergence.
• Applications in neural SDF learning, image reconstruction, and PDE solvers demonstrate up to 2× speed improvements while maintaining geometric accuracy.

A finite-difference framework for curvature regularization refers to the suite of discretization techniques and workflows that approximate geometric curvature quantities—such as mean, Gaussian, or affine curvature—using central or monotone finite-difference stencils, and leverage these within variational regularization or PDE-based optimization schemes. Such methods, spanning applications from neural signed-distance fields and image processing to the numerically robust approximation of PDEs for surfaces with prescribed curvature, avoid explicit computation of Hessians or high-order derivatives where possible, reducing memory and computational footprints while retaining geometric accuracy and theoretical convergence guarantees.

1. Foundations: Geometric Curvature and Motivation

Curvature regularization leverages differential geometric invariants to impose priors or constraints on the solution of variational and PDE-driven problems. In the context of a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ defining a surface $S=\{x | f(x) = 0\}$ or the image-level set $u : \Omega \subset \mathbb{R}^2 \to \mathbb{R}$, the mean and Gaussian curvature are expressed via the eigenstructure of the projected Hessian (Weingarten operator) or directly from surface differential geometry.

Traditional implementation in neural or variational settings requires full Hessian evaluation—typically via second-order automatic differentiation—leading to significant computational overhead. Central finite-difference approximations for second directional derivatives ($f_{uu}, f_{vv}, f_{uv}$ in local tangent frames) offer second-order Taylor-accurate proxies for these curvature components (Yin et al., 12 Nov 2025). The resulting stencils act as efficient, architecture-agnostic surrogates for curvature-based regularization throughout modern geometric learning and PDE-simulation pipelines.

2. Finite-Difference Stencils for Second Derivatives

Central-difference formulas approximate second derivatives with high accuracy—$O(h^2)$ truncation error for scalar fields $f(x,y,z)$—enabling practical computation of curvature quantities:

• Along axis $x$:

$$\frac{\partial^2 f}{\partial x^2} \approx \frac{f(x+h, y, z) - 2f(x, y, z) + f(x-h, y, z)}{h^2}$$

• Mixed partials:

$$\frac{\partial^2 f}{\partial x \partial y} \approx \frac{f(x+h, y+h, z) - f(x+h, y-h, z) - f(x-h, y+h, z) + f(x-h, y-h, z)}{4 h^2}$$

For SDFs, the tangent-plane basis $(u, v)$ at each sample is constructed orthogonal to $\nabla f$, and stencils such as

$$f_{uu} \approx \frac{f(x_0 + h u) - 2f(x_0) + f(x_0 - h u)}{h^2}$$

are evaluated at off-surface points $x_0$ (Yin et al., 12 Nov 2025), with analogous expressions for $f_{vv}$ and $f_{uv}$.

In specialized proxies such as those in FlatCAD (Yin et al., 19 Jun 2025), only the mixed term $u^\top H_f v$ is needed. This is efficiently computed via the four-point formula:

$$D_{uv}(x_\Omega) = \frac{f(x_\Omega + h u + h v) - f(x_\Omega + h u) - f(x_\Omega + h v) + f(x_\Omega)}{h^2}$$

where convergence is $O(h)$, and efficiency is maximal as no second-order graphs are built.

3. Curvature Regularization Strategies via Finite Differences

Finite-difference approximations of curvature enable various regularization terms in neural or variational reconstruction. Principal strategies include:

• Gaussian curvature regularization:

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

promoted as either a penalty $|K_{FD}|$ or squared-form in the objective (Yin et al., 12 Nov 2025).

• Mean curvature regularization:

$H_{FD} = \frac{f_{uu} + f_{vv}}{\|\nabla f\|}$

• Rank-deficiency loss (Neural-Singular-Hessian)—enforcing surface developability:

$$L_{\mathrm{rankFD}}(x_0) = |f_{uu} f_{vv} - f_{uv}^2|$$

This is directly linked to developable CAD-style behaviors and can serve as a convex proxy for the determinant minimization with robust convergence properties (Yin et al., 12 Nov 2025).

• Image domain discrete curvature: In grid-based image settings, mean and Gaussian curvatures are estimated from 3×3 neighborhoods by constructing tangent planes and evaluating signed distances and arclengths, e.g.,

$\kappa_\ell \approx \frac{2 d_\ell}{s_\ell^2}$

The principal curvatures at each pixel are then defined by the extremal values over directions, with aggregate regularizers minimized across the grid (Zhong et al., 2019).

4. Viscosity Solutions and Monotone Schemes

The stability and convergence of nonlinear PDEs involving curvature (including Monge–Ampère-type and affine curvature flows) are critically dependent on monotonicity, consistency, and ellipticity in the design of finite-difference quotients (Oberman et al., 2016, Froese, 2016). Explicitly:

• Monotone/Elliptic discretizations: Use of wide-stencil medians or one-sided (upwind) differences ensures degenerate ellipticity, a prerequisite for comparison and maximum principles in the viscosity-solution framework.
• Consistency: Taylor expansion is used to verify local approximation order ($O(h^2)$ for central differences; $O(h)$ for certain mixed proxies).
• Lipschitz regularizations: Operators like $(p^2 q)^{1/3}$ (arising in affine curvature flow) are regularized to ensure global Lipschitz bounds essential for explicit time-stepping.
• Filtered schemes: Blending high-accuracy (non-monotone) and monotone (robust) stencils achieves formal second-order accuracy in smooth regions while guaranteeing convergence in singular or degenerate areas.

The Barles–Souganidis theory then guarantees that any stable, consistent, and monotone scheme converges locally uniformly to the viscosity solution (Oberman et al., 2016).

5. Algorithmic and Computational Aspects

Integrating finite-difference curvature proxies into learning or optimization frameworks offers major computational advantages:

Proxy Variant	SDF Calls	Backward Passes	GPU Memory	Error Order	Typical Speedup
Full Hessian (AD)	-	~7/sample	2–3× baseline	0 (exact)	Baseline
FD Proxy (e.g., FlatCAD)	4/sample	1/sample	$\sim$Eikonal only	$O(h)$	$\sim 2\times$ faster
FD Curvature Regularizer (Yin et al., 12 Nov 2025)	6–8/sample	1/sample	$\sim$Eikonal only	$O(h^2)$	$\sim$2× faster
AutoDiff Proxy	-	2/sample	1×	$O(1)$	Comparable to FD

• Training loop structure: Forward SDF evaluations at offset points, assembly of tangent basis, and stencil application per sample. The loss combines Eikonal, Dirichlet/matching, and curvature regularization, with Adam or similar optimizers (Yin et al., 12 Nov 2025).
• Step size $h$ selection: Recommended $h \in [10^{-3}, 10^{-2}]$ times the domain scale; empirically, convergence and stability are robust within this window.
• Implementation drop-in capability: Framework-agnostic proxies—no need to modify autodiff engine internals; only standard network forward and backward passes are used.

6. Application Domains and Empirical Validation

Finite-difference curvature regularization is validated in several computational domains:

• Neural SDF learning (CAD and non-CAD): Experiments on ABC and Armadillo datasets report normal consistency, Chamfer distance, F1 scores, and resource footprint. FD approaches achieve normal consistency and Chamfer within $\sim$10% (and frequently tighter) of AD baselines, with up to 2$\times$ memory/runtime reduction. On sparse and incomplete data, FD variants preserve global topology and yield F1 $\approx 0.99$ where baselines require nearly twice the wall-clock time (Yin et al., 12 Nov 2025).
• Image reconstruction: Discrete curvature regularized variational TV models solved via ADMM demonstrate strong edge connectivity and detail preservation, with per-iteration complexity $O(m^2 \log m^2)$—significantly less than prior elastica-based methods (Zhong et al., 2019).
• Nonlinear PDE solvers: Monotone wide-stencil finite-difference methods guarantee convergence to viscosity solutions of Monge–Ampère equations and affine/mean curvature flows. Rigorous L∞ or L¹ error tables confirm interior convergence across smooth, singular, and boundary-layer regimes (Oberman et al., 2016, Froese, 2016).

7. Benefits, Limitations, and Extensions

The principal benefit is computational efficiency—empirically, a factor of $\sim$2 improvement in memory and time—without sacrificing geometric fidelity in regularization targets (Yin et al., 19 Jun 2025, Yin et al., 12 Nov 2025). Advantages include framework-agnosticity, drop-in applicability, and avoidance of explicit second-order computation.

Limitations are rooted in the discretization error ($O(h^2)$ or $O(h)$), the need for multiple network evaluations per sample, and noise sensitivity for too small $h$. The accuracy-memory trade-off can be tuned by $h$ and (in theory) by deploying higher-order stencils ($O(h^4)$) or adaptive $h$ selection. Potential extensions cover application to other geometric operators, integration with adaptive/multiresolution grid sampling, and GPU-accelerated evaluation kernels (Yin et al., 12 Nov 2025).

A plausible implication is that such finite-difference frameworks make curvature-driven geometric regularization tractable for large-scale neural and numerical PDE applications where full Hessian evaluation would otherwise be prohibitive.