coordinate transformations

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.

To address the challenge of establishing stable inter-subject local correspondences in anatomical shape statistics—traditionally reliant on explicit registration—we propose a registration-free, boundary- and interior-consistent geometric modeling framework. Our method models target deformations as diffeomorphic transformations of an ellipsoid, embedded within a globally optimized skeleton-driven fitting scheme that simultaneously constructs a consistent coordinate system on both the object’s boundary and interior. We further introduce an evolutionary s-rep representation, the first to encode intrinsic geometric features directly in the fitted coordinate space, enabling robust point-wise correspondence across subjects without mesh alignment. The approach integrates differential-geometric deformation modeling, skeleton-guided fitting, and boundary-driven intrinsic coordinate generation. In hippocampal disease classification, it significantly outperforms two state-of-the-art methods, demonstrating superior discriminative power and statistical stability of the learned features.

This study addresses the generalization bottleneck in point cloud geometric representation by proposing transferable Geometric Neural Operators (GNOs) as foundational models. Methodologically, it introduces the first pretraining framework for unordered, unstructured point clouds: leveraging mesh-free, coordinate-agnostic functional mappings; embedding differential-geometric priors—such as covariant derivatives and curvature constraints; and employing self-supervised geometric losses to enable robust representation learning across shapes, topologies, and noise levels. Contributions include: (1) unified support for curvature estimation, geometric PDE solving on manifolds, and curvature-driven deformation modeling; (2) state-of-the-art performance across multiple benchmarks, significantly outperforming existing methods; and (3) open-sourced code and pretrained weights enabling plug-and-play integration.

This study addresses the challenge of constructing rotation-invariant vector representations for planar shapes by proposing a method that strictly encodes star-shaped normalized contours into Euclidean vectors. The resulting representation guarantees that Euclidean distances between vectors faithfully reflect shape dissimilarities while enabling efficient shape analysis. The approach is the first to simultaneously achieve strict invariance under rotation (and controllable reflection), injectivity, and robustness to small perturbations. By discretizing functions defined on the unit circle and employing an offset-based parameterization, the method constructs an ε-approximate vector in O((1/ε) log(1/ε)) time, yielding an O(1/ε)-dimensional embedding amenable to efficient nearest-neighbor search and clustering. Experimental results confirm that the representation maintains high accuracy and computational efficiency without compromising invariance properties.

Three-dimensional (3D) mappings are fundamental in computational mechanics (CAE), computer graphics, and medical imaging; however, conventional vertex-coordinate-based representations struggle to simultaneously ensure geometric fidelity and intuitive, controllable editing. To address this, we propose the first theoretically rigorous and computationally tractable 3D quasiconformal representation—extending the Beltrami coefficient to three dimensions—to characterize local scaling distortion in a mathematically sound manner. We further design an invertible reconstruction algorithm that stably and accurately recovers the original mapping from its distortion representation. Our approach integrates 3D quasiconformal theory, partial differential equation (PDE)-based modeling, and numerical optimization. Experiments demonstrate that our method significantly outperforms state-of-the-art alternatives in 3D mapping reconstruction, keyframe interpolation, and compression—achieving superior accuracy, robustness, and editability while preserving theoretical guarantees.

Existing 3D representation learning methods often rely on extrinsic geometry or high-level semantics, making it difficult to capture the intrinsic structure and manifold topology of shapes. This work proposes PRISM, a novel pretraining paradigm that, for the first time, leverages geodesic distance recovery as a self-supervised signal to learn intrinsic geometry through isometric embeddings. To address the inherent imbalance in geodesic distance distributions, the approach introduces a topology-preserving latent space constraint and a two-stage training strategy. The method demonstrates high accuracy, robustness, and efficiency in geodesic distance prediction and achieves state-of-the-art performance on downstream tasks including shape recognition, surface parameterization, and non-rigid correspondence.

This work addresses the limitations of traditional 3D reconstruction methods, which predict point maps in camera-centered coordinates, struggle to incorporate scene structural priors, and suffer from high rotational degrees of freedom across views, leading to inconsistent reconstructions. To overcome these issues, the authors propose predicting point maps in a gravity-aligned upright coordinate system, thereby reducing inter-view rotational ambiguity through a shared vertical axis. They introduce the Gravity Grounded Geometry Transformer (G3T) model and the G3T-Long incremental reconstruction framework, which for the first time integrate gravity-aligned coordinates into point map prediction by combining a Transformer architecture, gravity-aware pose estimation, and a submap stitching strategy. Experiments demonstrate that this approach significantly improves reconstruction accuracy and robustness, outperforming existing methods in incremental 3D reconstruction and validating the effectiveness of gravity-aligned representations.

This work addresses the insufficient integration of learning-based methods and geometric constraints in camera pose and scene structure estimation by proposing a modular framework. The approach first employs a learning model (VGGT) to generate initial hypotheses for depth and relative pose, which are subsequently refined and validated using classical geometric algorithms such as point-to-plane RGB-D ICP. Crucially, the framework explicitly distinguishes the roles of learning as a “proposer” and geometry as a “referee,” emphasizing that the geometric module serves not merely as post-processing but as an essential mechanism for verifying and integrating learned outputs. Experiments on the TUM RGB-D dataset demonstrate that, in moderately challenging rigid scenes, the system significantly outperforms both purely learning-based and purely geometric baselines when the learned depth aligns geometrically with the camera intrinsics and undergoes optimization by the geometric backend.

This work addresses the limited robustness and discriminability of local features under arbitrary 3D rotations in point cloud registration by proposing the first strictly rotation-equivariant registration framework that operates without a local reference frame. Built upon SO(3) representation theory, the method employs spherical harmonics to construct a rotation-equivariant neural network that jointly learns rotation-invariant descriptors and equivariant geometric features. This design enables each putative correspondence to directly model the underlying rigid transformation, substantially reducing reliance on extensive RANSAC sampling. Experiments on the 3DMatch, 3DLoMatch, and KITTI benchmarks demonstrate that the proposed approach achieves significantly higher registration accuracy than existing methods under large rotational perturbations.

This work proposes the first projective-geometric framework for measuring representational drift, grounded in the Fubini–Study metric, which treats high-dimensional representations as points in projective space and quantifies their geometric evolution along trajectories. Conventional metrics—such as Euclidean or cosine distance—are prone to conflating genuine structural changes in data with artifacts induced by parametrization ambiguities, such as global scaling or sign flips. In contrast, the proposed approach is gauge-invariant, effectively disentangling intrinsic representational dynamics from spurious perturbations due to parameterization freedom. It further introduces a computable, monotonic quantity to rigorously quantify representational churn. Experiments on real high-dimensional data demonstrate that this framework avoids the systematic overestimation of drift inherent in traditional measures, yielding stable and interpretable diagnostics suitable for general-purpose empirical analysis pipelines.