Point cloud denoising is essentially a geometric recovery task that aims to reconstruct the intrinsic structure of a smooth 2D Riemannian manifold embedded in R^3 from noisy, discrete ambient-space samples. Despite the remarkable progress of modern manifold-aware encoders and generative transport models in geometric representation learning, a fundamental objective-geometry mismatch remains underexplored. Theoretically, we identified that this mismatched coupling leads to geometric gradient interference, where conflicting optimization objectives result in structural degradation and point clustering. We introduce Riemannian Orthogonally Decoupled Regularization (RODR) to reformulate the optimization trajectory by disentangling the normal (fitting) and tangential (distribution) components. Guided by a vector-attention and entropy-aware adaptive strategy, RODR effectively preserves high-fidelity geometric details while maintaining sampling uniformity. Experiments demonstrate that RODR reaches performance comparable to state-of-the-art baselines and suggests improved distribution regularity and reduced local aggregation effectively. Our work establishes a generic and interpretable framework for disentangled geometric optimization in point cloud processing.
Point cloud denoising is a fundamental and crucial challenge in real-world point cloud applications. Existing quantitative evaluation metrics for point cloud denoising methods are implemented in a supervised manner, which requires both the denoised point cloud and the corresponding ground-truth clean point cloud to compute a representative geometric distance. This requirement is highly problematic in real-world scenarios, where ground-truth clean point clouds are often unavailable. In this paper, we propose a simple yet effective unsupervised geometric distance (UGD) for real-world noisy point cloud denoising, calculated solely from noisy point clouds. The core idea of UGD is to learn a patch-wise prior model from a set of clean point clouds and then employ this prior model as the ground-truth to quantify the degradation by measuring the geometric variations of the denoised point cloud. To this end, we first learn a pristine Gaussian Mixture Model (GMM) with extracted patch-wise quality-aware features from a set of pristine clean point clouds by a patch-wise feature extraction network, which serves as the ground-truth for the quantitative evaluation. Then, the UGD is defined as the weighted sum of distances between each patch of the denoised point cloud and the learned pristine GMM model in the patch space. To train the employed patch-wise feature extraction network, we propose a self-supervised training framework through multi-task learning, which includes pair-wise quality ranking, distortion classification, and distortion distribution prediction. Quantitative experiments with synthetic noise confirm that the proposed UGD achieves comparable performance to supervised full-reference metrics. Moreover, experimental results on real-world data demonstrate that the proposed UGD enables unsupervised evaluation of point cloud denoising methods based exclusively on noisy point clouds.
High-curvature regions in 3D point clouds encapsulate critical fine-grained geometric semantics yet exhibit a distinct long-tail sparsity in their spatial distribution. The inherent limitations of polynomial volume growth in Euclidean space frequently render these intricate geometric features challenging to adequately resolve within a uniform-scale feature space. Consequently, these regions are frequently overshadowed by smooth global features dominated by low-curvature regions, thereby limiting the discriminative capacity of the network. To address this issue, we propose PointCHR, a curvature-aware hyperbolic rectification (CHR) for point cloud analysis. Utilising the property of exponential volume expansion in the vicinity of hyperbolic manifolds, CHR presents a learnable curvature-guided radial rectification mechanism. By adaptively projecting high-curvature points towards boundary regions endowed with larger effective embedding capacities, PointCHR effectively mitigates the representation crowding problem inherent in Euclidean settings. Extensive experimentation has demonstrated that PointCHR significantly enhances the ability of backbone to capture fine-grained geometric details, achieving state-of-the-art performance across multiple benchmarks.
Point clouds are a fundamental representation for robotic perception tasks such as localization, mapping, and object pose estimation. However, LiDAR-acquired point clouds are inherently sparse and non-uniform, providing incomplete observations of the underlying scene geometry. This makes reliable geometric reasoning challenging and degrades downstream perception performance. Existing approaches attempt to compensate for these limitations by estimating local geometry, but often rely on hand-crafted statistics or end-to-end supervised learning, which can suffer from limited scalability or require large amounts of accurately labeled data. To address these challenges, we explicitly model point cloud geometry under a principled mathematical formulation. We represent local geometry as a statistical manifold induced by a family of Gaussian distributions, where each point is associated with a Gaussian capturing its local geometric structure. Based on this formulation, we introduce Point-to-Ellipsoid (POLI), a deep neural estimator that predicts per-point Gaussian geometry. POLI learns a mapping from point cloud observations to their underlying geometry in a self-supervised manner, removing the need for labeled data while preserving strong geometric inductive biases. The resulting representation integrates seamlessly into existing robotic perception pipelines without architectural modifications. Extensive experiments show that POLI enables accurate and robust geometry estimation and consistently improves performance across diverse robotic perception tasks.