Local Point Cloud Curvature

• Local point cloud curvature quantifies the bending of a discrete manifold using differential-geometric descriptors like shape operators and principal curvatures.
• Advanced methods, including local PCA with quadratic fitting, normal variation techniques, Laplacian operators, and diffusion measures, robustly estimate curvature despite noise and nonuniform sampling.
• Emergent deep learning techniques enhance curvature estimation by learning adaptive representations that address challenges such as sample complexity and high-dimensional noise.

Local point cloud curvature refers to the quantification of how a manifold—sampled as a discrete, unstructured collection of points in Euclidean space—bends or deviates locally from being flat. Accurate estimation of local curvature from point clouds is essential for geometry processing, 3D vision, manifold learning, biomedical data analysis, and a variety of downstream computational tasks. The problem is technically challenging due to the absence of explicit connectivity, possible high dimensions, nonuniform sampling, and pervasive noise.

1. Differential-Geometric Foundations and Discrete Analogues

For a smooth $d$-dimensional submanifold $\mathcal{M}\subset\mathbb{R}^n$ and a point $x\in\mathcal{M}$, the local curvature is classically encoded by the shape operator $\mathcal{L}_x$, principal curvatures $\{\kappa_j\}_{j=1}^d$, mean curvature $H(x)=\frac1d\sum_j\kappa_j$, and Gaussian curvature $K(x)=\prod_j\kappa_j$ (for $d=2$). These quantities derive from the second fundamental form, the differential of the normal vector field, or the Laplace–Beltrami operator on the embedding.

The central challenge in point clouds is to robustly approximate these objects from finite, typically noisy samples. Prominent discretizations include:

• The discrete Laplacian or umbrella operator for mean curvature, e.g., $$\delta x_i = x_i - \frac1{|\mathcal N(i)|}\sum_{j\in\mathcal N(i)} x_j$$ (Xiu et al., 2022, Ziwen et al., 2019).
• Osculating-circle for curve curvature: $\hat\kappa(p)=1/r(p,p_1,p_2)$ with $r$ as circumradius of three points (Mirzaie, 7 Jun 2025).
• Quadratic surface (jet) fitting in local PCA frames, with $K_G(p)=\det A$ of the least-squares Hessian matrix (Asao et al., 2021).
• The absolute-variation curvature, based on normal variation: $$\Omega_y(x) = \frac{2\sin(\frac12\theta(x,y))}{\|x-y\|}$$ averaged over a neighborhood (Chen et al., 4 Nov 2025).
• Dirichlet energy of the Gauss map for total curvature: $K_i = \sum_{j\in N(i)} w_{ij}\|n_i-n_j\|^2$ (Chen, 2023).

Each approach encodes different trade-offs regarding extrinsic/intrinsic nature, smoothness class, and statistical consistency.

2. Algorithmic Frameworks for Local Curvature Estimation

The state-of-the-art methods for local curvature estimation in point clouds can be grouped as follows:

A. Local PCA and Quadratic Fitting

Algorithms estimate tangent spaces by local Principal Component Analysis (PCA) on an $\varepsilon$-ball or $k$-NN neighborhood. Intrinsic curvature is then extracted via a second-order fit, such as least-squares fitting of a quadratic (height) function and taking the determinant or eigenvalues of the Hessian (Asao et al., 2021, Zhang et al., 6 Feb 2025). For example, Gaussian curvature at $p$ is estimated as $K_G(p)=\det A$, where $A$ is the Hessian in the locally estimated tangent-normal frame.

Adaptive-scale selection is crucial; AdaL-PCA uses the local ratio of explained variance as a function of neighborhood radius to set the scale and further computes principal curvatures from directional samples (Zhang et al., 6 Feb 2025).

B. Variation and Two-Point Normal Curvature

Absolute-variation curvature (Chen et al., 4 Nov 2025) computes for each $x$ an average of the two-point formula over its neighborhood, using normal angles: $$\omega(x) = \lim_{\epsilon\to0}\frac{1}{|\partial B_\epsilon(x)|}\int_{\partial B_\epsilon(x)}\Omega_y(x)d\lambda(y)$$ where $\Omega_y(x)$ uses the angle between estimated normals at $x$ and $y$. This approach admits statistical consistency proofs for vanishing noise and sufficient density, but naive estimators can be heavily biased in high dimensions.

A probabilistic framework based on von Mises–Fisher mixtures for noise modeling and maximum-likelihood decoding corrects for the severe bias of the naive estimator, restoring accuracy up to dimension 12 with relative errors below 5% in realistic noise regimes (Chen et al., 4 Nov 2025).

C. Laplacian and Umbrella Operators

Discrete Laplacians provide a direct proxy for the mean-curvature normal. For example, in 3D, the local Laplacian is proportional to $H_in$, and residual updates of the form $x_i' = x_i + \Delta x_i$ (where $\Delta x_i$ is a learned, possibly anisotropic local Laplacian term) are used both in classic geometry processing and as architectural units in deep learning (Xiu et al., 2022, Ziwen et al., 2019). The Laplacian Unit couples a learnable linear transformation and nonlinearity to produce adaptive smoothing or sharpening, approximating discrete mean curvature flow.

The PLATYPUS system adapts the "umbrella curvature" (unsigned, normal-projected offsets normalized by distance) as a sampling criterion to focus learning and upsampling on high-curvature regions, thus improving feature coverage and downstream accuracy (Kim et al., 1 Nov 2024).

D. Dirichlet Energy and Normal Variation

Per-point total (or bending) curvature can be linked to the Dirichlet energy of the Gauss map, discretized as a sum of squared local normal differences with cotangent or area weights (Chen, 2023). Point clouds are locally triangulated (e.g., via 2D Delaunay in the tangent plane), and curvature is aggregated over the local mesh patch.

E. Varifold and Mean Curvature via Regularized First Variation

Varifold-based methods define a measure $V=\sum_i m_i\delta_{(x_i,P_i)}$ using estimated tangent planes $P_i$. Mean curvature vectors are then computed as regularized first variations of this varifold, involving kernel-smoothed integrals of directional gradients over the point cloud (Buet et al., 2020). This yields local mean curvature that is robust to sampling irregularity and noise.

F. Diffusion and Intrinsic Measures

Diffusion curvature (Bhaskar et al., 2022) defines an intrinsic, scale-adaptive scalar curvature by measuring the "laziness" (return probability) of a random walk defined by the diffusion operator on the point cloud. It has a direct link to Riemannian scalar curvature via the Bishop–Gromov comparison, and can be extended to quadratic-form (Hessian) estimates by combining with neural prediction models.

3. Sample Complexity, Noise, and Robustness

Sample density, noise level, and neighborhood size directly impact the bias and variance of all estimators, and precise tradeoffs are quantified in recent theory (Mirzaie, 7 Jun 2025, Chen et al., 4 Nov 2025). Key results include:

• For $C^3$ curves/surfaces, osculating-circle and "principal-cone" estimators are $O(\epsilon)$-accurate when neighbor distances are bounded by $\epsilon$ (Mirzaie, 7 Jun 2025).
• To guarantee with probability $p_0$ that every point of a curve (resp. surface) has the necessary neighbors, one requires $m=O(\epsilon^{-1}\log(1/(1-p_0)))$ (curve), $m=O(\epsilon^{-2}\log(1/(1-p_0)))$ (surface).
• In high dimensions, classic naive estimators (e.g., angle-averaging) exhibit bias of $O(m/\kappa)$, where $m$ is intrinsic dimension and $\kappa$ the noise concentration, making correction schemes necessary (Chen et al., 4 Nov 2025).
• Probabilistic decoders exploiting explicit noise models (vMF mixture fits) and maximum-likelihood search recover unbiased curvature in high dimensions ($m\leq 12$), with $<5\%$ relative error even at moderate noise levels.

4. Deep Learning Approaches for Local Curvature

Deep neural networks have been adapted to regress local curvature (and often normal vectors) from point cloud patches, circumventing manual tuning of neighborhood scales:

• PCPNet (Guerrero et al., 2017) uses translation and scale-normalized local patches as input to order-invariant, multi-scale PointNet-style architectures, outputting principal curvature estimates $(\kappa_1,\kappa_2)$. The method achieves 2–10× lower RMSE versus jet-fitting across noise conditions.
• CanonNet (Friedmann et al., 3 Apr 2025) introduces canonical ordering and orientation in preprocessing (via Laplacian eigenmaps) and trains compact MLPs solely on analytic quadratic surfaces. Despite 100× fewer parameters and patch sizes of $n=20$ points, CanonNet achieves state-of-the-art accuracy on mean-curvature regression and competitive Gaussian curvature error.
• Progressive local feature encoding architectures for upsampling (e.g., PLATYPUS, (Kim et al., 1 Nov 2024)) utilize curvature-driven subsampling at multiple resolutions to prioritize edge and corner regions.

Learned methods are evaluated using rectified error metrics, e.g., $D_K=|K_{pred}-K_{GT}|/\max(|K_{GT}|,1)$, and demonstrate strong robustness to noise and sampling variance.

5. Applications and Practical Impact

Local curvature features derived from point clouds support a variety of tasks:

Task	Role of Local Curvature	Methods/Notes
Part segmentation/labeling	Boundary amplification, smoothing/sharpening	Laplacian Unit (Xiu et al., 2022)
Feature-aware upsampling	Sampling focus on edges/corners	Umbrella curvature (Kim et al., 1 Nov 2024)
Clustering/representation	Embedding & clustering in curvature-augmented space	PCA/curvature features (Asao et al., 2021)
Visualization/saliency	Smoothing salient features	Curvature smoothing (Ziwen et al., 2019)
Biomedical data embedding	Detect cell-state transitions	AdaL-PCA (Zhang et al., 6 Feb 2025), diffusion (Bhaskar et al., 2022)
Manifold learning/theory	Intrinsic quantity estimation, bias control	Absolute-variation, vMF decoding (Chen et al., 4 Nov 2025)

Empirically, curvature-aware methods increase segmentation accuracy (up to +1.3 mIoU on S3DIS for Laplacian Unit), improve reconstruction sharpness, and enable unsupervised structural discovery and denoising.

6. Limitations, Open Problems, and Future Directions

Several limitations emerge:

• Sample complexity grows rapidly with ambient/intrinsic dimension; many traditional estimators fail for $m\gtrsim 8$ without carefully designed statistics (Chen et al., 4 Nov 2025).
• Nonuniform sampling, missing data, and outliers can undermine PCA, normal-estimation, and neighborhood-based methods unless compensated by robust statistics or topological regularity (Spang, 2023).
• Trade-off between bias (large neighborhoods) and variance (small neighborhoods) remains fundamental, motivating adaptive-scale, de-biasing, and correction frameworks.
• Deep learning approaches require comprehensive synthetic training data or robust transfer pipelines to accommodate real-world sensor and sampling artifacts.

A plausible implication is that future advances will continue to integrate statistical modeling of noise, adaptive and probabilistic estimators, and deep representation learning to further mitigate curse-of-dimensionality effects and adapt to non-IID, real-world data settings.

7. Summary Table: Representative Methods

Paper	Notion Estimated	Dim.	Key Formula/Principle	Robustness/Complexity
(Xiu et al., 2022)	Mean curvature normal (Laplacian Unit)	3	$\Delta x_i\approx H_i n_i$	Adaptive, learns smoothing/sharpening; $k$-NN trade-off
(Asao et al., 2021)	Gaussian curvature (PCA jet-fit)	$n$	$K_G(p)=\det A$ (least-squares Hessian)	LLN-verified, requires $\varepsilon$/$\delta$ sweep
(Spang, 2023)	Principal curvatures (VWME)	$n$	Shape operator via VCM-normal/WME	Strong robustness to noise via VCM
(Chen et al., 4 Nov 2025)	Absolute-variation curvature	$m\leq12$	Angle-averaged, MLE-debiased	vMF calibration removes high-dim bias
(Zhang et al., 6 Feb 2025)	Principal curvatures (AdaL-PCA)	$d$	Weighted directional curvature, adaptive scale	SOTA empirical accuracy, minimal tuning
(Ziwen et al., 2019)	Mean curvature vector (Taubin/plane)	3	$\mathbf{H}(x)=-\frac12\Delta_S\mathbf{x}$	O(NK) per iter, stable to mild noise
(Guerrero et al., 2017)	$(\kappa_1,\kappa_2)$ regression (PCPNet)	3	Deep local patch regression	High robustness, no manual scale tuning
(Bhaskar et al., 2022)	Intrinsic scalar curvature (diffusion)	$n$	$\kappa(x)=\text{mean walk laziness}$	Correlates with Gauss curvature; scalable

These data-driven, statistical, Laplacian-based, and spectral/diffusion-inspired approaches collectively establish a rigorous, versatile toolkit for estimating and utilizing local curvature in point cloud analysis, with explicit quantitative guidelines for sample size, noise robustness, and computational trade-offs.