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.
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.
In point clouds, noise directly perturbs point coordinates that encode both spatial location and geometry, making one-to-one correspondence construction more challenging than in images. Existing methods impose statistical mappings across noisy variants via noise or optimal transport, but suffer from correspondence ambiguity. In this work, we propose Self-Induced Mirror-Point Consistency (SIMPC) to learn deterministic correspondences between points and the underlying surface in an unsupervised manner. For each noisy point, SIMPC generates a mirror-point on the opposite side of the underlying surface, guided by geometric priors during the denoising process. By encouraging consistency between the denoising targets of the original point and its mirror counterpart, SIMPC effectively localizes the position of underlying surface. Extensive experiments on synthetic and real-world datasets demonstrate that SIMPC significantly outperforms state-of-the-art unsupervised methods and surpasses several strong supervised counterparts.
In recent years, neural networks have become the dominant models in most point cloud upsampling methods. Although these approaches are achieving good results, they do have drawbacks, such as a lack of interpretability and data dependency. Moreover, they have to be trained on a dataset that is similar to the test data in order to perform well. To avoid these disadvantages, we propose Point Cloud Upsampling through Patch-based Frequency Superposition (PUtPFS), an optimization-based approach that selects subsets of points and estimates the surface of this set through superpositioning spatial frequencies. Then, new points are placed on this surface. By successively selecting points in the least dense regions of the point cloud, a uniform upsampling can be reached. With this method, we surpass the current best upsampling results in the commonly considered point-to-surface distance. Furthermore, we achieve the best Chamfer and Hausdorff distance among the optimization-based approaches. As an additional advantage, our method does not need any training data and is mathematically interpretable.