curvature estimation

This work addresses the limitations of traditional geometric processing methods, which rely on manifold assumptions and struggle with non-manifold geometries featuring singular structures such as sharp features, self-intersections, or branches. The authors propose a novel “tangent blow-up” representation, introducing for the first time the algebraic geometry concept of blow-up into geometric processing. By jointly embedding each spatial point together with its tangent plane into the product space of Euclidean space and a Grassmannian manifold, the method iteratively disambiguates coincident points that differ in tangential or higher-order contact. Within this lifted domain, discrete gradient, divergence, and Laplace operators are rigorously defined. This structured representation enables a natural extension of classical differential operators to singular points, demonstrating effectiveness in tasks including geodesic computation, segmentation, parameterization, and curvature estimation.

This work proposes the first differentiable geometry processing system that seamlessly integrates with modern machine learning frameworks, addressing the longstanding challenge of combining geometric algorithms—typically non-differentiable and reliant on complex control flow—with gradient-based optimization. By unifying the adjoint method with a scatter-gather mesh processing paradigm, the system enables efficient gradient computation for existing geometric algorithms without requiring algorithmic reimplementation. It supports state-of-the-art solvers such as local-global and ADMM schemes and provides native differentiability for classical operations including curvature flows and conformal parameterizations. Evaluated on multiple inverse geometry problems, the approach significantly reduces both memory consumption and computational overhead, outperforming general-purpose differentiable optimization tools in runtime efficiency while dramatically lowering implementation effort.

This work addresses the problem of smooth path tracing under simultaneous upper and lower curvature bounds by proposing a novel curvature-bounded geodesic model. Formulated within the Hamilton-Jacobi-Bellman (HJB) partial differential equation framework, the model introduces bilateral curvature constraints into the HJB formalism for the first time, enabling strong control over the geometric properties of generated paths. An efficient numerical discretization scheme is devised to balance path smoothness, rigidity, and elasticity. Experimental results demonstrate that the method robustly produces high-quality, curvature-constrained optimal paths in applications such as robotic motion planning and image curve structure tracking, significantly extending the capabilities of conventional single-bound constrained models.

This work addresses the challenge of accurately characterizing geometric singularities—such as corners, edges, and self-intersections—in the underlying manifold of point cloud data. We propose a singularity modeling and estimation framework grounded in the graph Laplacian operator. For the first time, we establish explicit functional bounds for the graph Laplacian within singular neighborhoods, thereby bridging spectral methods with local geometric inference. Based on this theoretical foundation, we derive a principled existence test for singularities and develop robust, nonparametric estimators for intrinsic geometric quantities—including manifold dimension and curvature. The method provides rigorous theoretical guarantees while maintaining interpretability. Extensive experiments on both synthetic and real-world point clouds demonstrate high estimation accuracy and strong consistency with theoretical predictions. Our approach establishes a novel paradigm for singularity-aware manifold learning, enabling geometrically faithful analysis of complex, nonsmooth structures in point cloud data.

Traditional calculus and Riemannian geometry struggle to handle non-manifold, noisy real-world data. This work proposes a data-driven framework grounded in diffusion processes that, for the first time, systematically realizes computable formulations of vector calculus and key geometric objects—such as geodesic distances, curvatures, vector field flows, and solutions to partial differential equations—within the paradigm of diffusion geometry. The framework further integrates topological tools from de Rham cohomology and Morse theory. Leveraging efficient numerical linear algebra techniques, it achieves substantial improvements in computational accuracy, noise robustness, and scalability, demonstrating exceptional numerical stability, low computational complexity, and strong robustness across a range of geometric and topological tasks.

This paper addresses the unsupervised extraction of curves and one-dimensional structures in images with known endpoints. The proposed method integrates geometric modeling and variational principles: curves are lifted to a position-orientation space, where sub-Riemannian or Finsler metrics naturally encode curvature-dependent energy; a differentiable energy functional is constructed based on Smirnov’s vector field decomposition theorem, and a bilevel optimization framework is designed to compute globally optimal paths under endpoint constraints. By jointly discretizing the energy and enforcing geometric regularization, the approach significantly improves connectivity robustness in weak-boundary and low signal-to-noise ratio scenarios. Experiments demonstrate superior accuracy and generalizability compared to state-of-the-art variational and learning-based methods—without requiring manual parameter tuning or ground-truth annotations—making it particularly effective for challenging 1D structure extraction tasks such as biological image analysis and vascular segmentation.

This work addresses the susceptibility of existing geodesic active contour models to topological shortcuts when segmenting objects with complex shapes or weak boundaries, primarily due to insufficient constraints on path tangents. To overcome this limitation, the authors propose a unified geodesic framework that integrates spatially varying tangent priors—such as skeleton-based intrinsic shape representations—with curvature regularization in a lifted orientation space. This approach yields, for the first time, a family of Finsler metrics that explicitly enforce hard tangent constraints. The resulting energy minimization problem is efficiently solved via the Hamilton–Jacobi–Bellman equation using an enhanced fast marching method. Experiments demonstrate that the proposed method significantly suppresses topological shortcuts, enhances robustness to weak boundaries, and achieves superior shape fidelity across synthetic, natural, and medical images.

Conventional curvature-based regularization methods struggle to simultaneously preserve sharp edges and maintain isotropic fidelity in surface and image smoothing. Method: This paper proposes a total curvature regularization model incorporating multi-directional normal curvature penalization. We introduce a novel total normal curvature regularizer that adaptively weights principal curvature directions, enabling isotropic modeling for the first time. The resulting high-order nonconvex optimization problem is reformulated as a steady-state partial differential equation (PDE), solved efficiently via operator-splitting time discretization and closed-form subproblem solvers. Contribution/Results: Experiments demonstrate superior robustness across diverse noise types and geometric structures, markedly reduced parameter sensitivity, and significant improvements in both edge preservation accuracy and isotropic fidelity compared to classical curvature regularization approaches.

Accurate boundary detection in high-dimensional data remains a central challenge in unsupervised learning, particularly in the presence of non-linear structures and heterogeneous densities. In this work, we introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning that departs from traditional density-based approaches by explicitly modeling the intrinsic curvature of the data manifold. The method relies on a discrete approximation of the shape operator, estimated from local k-nearest neighbor patches, to compute pointwise mean curvature without requiring explicit manifold parametrization. The key insight of MCBP is to use mean curvature as a principled descriptor of boundary structure: high-curvature regions naturally correspond to transitions between clusters, geometric irregularities, and low-density interfaces. This yields a unified geometric interpretation of boundary, outlier, and transition points. We further introduce an adaptive percentile-based thresholding scheme that enables multiscale boundary extraction without relying on ad hoc density parameters. Beyond detection, we propose a curvature-driven data decomposition that separates samples into smooth (low-curvature) and boundary (high-curvature) subsets, effectively acting as a non-linear geometric filtering mechanism. This representation enhances cluster separability and improves the robustness of downstream unsupervised algorithms. Extensive experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance, particularly in complex and high-dimensional scenarios. These results position MCBP as a concrete contribution to Geometric Machine Learning, highlighting the potential of curvature-aware analysis as a unifying paradigm bridging differential geometry and data-driven modeling.