Certified Point-Cloud

A geometrically valid placement can still be difficult to execute because the selected grasp changes the required end-effector pose, collision geometry, and transport motion. Placement is formulated as a pre-execution ranking problem in which supplied grasp-placement candidates are scored before planning. The model combines a typed target-conditioned point cloud with three pose descriptors and hierarchical heads for planning success and execution success conditioned on planning. On a 30-object, 1,235-scene dataset with scene-group-held-out splits, three-seed top-1 success on covered test groups reaches 85.63 +/- 1.08% for joint selection and 79.84 +/- 0.16% for fixed-target ranking. For the designated frozen seed-42 checkpoint, top-1 success improves from 72.84% to 85.78% over full-pool cuMotion for joint ranking and from 59.65% to 79.67% for fixed-target ranking. Frozen transfer to xArm7/MoveIt requires no xArm-specific retraining. Across 27 locked cases, 13 complete end to end (48.15%). Of the 16 cases that pass Top-5 preflight and begin execution, 13 succeed (81.25%). Candidate-level deployment-feasibility prediction reaches 81.25% recall, 85.20% specificity, and 83.23% balanced accuracy.

Pre-trained 3D vision models have substantially advanced point cloud analysis, yet adapting them to downstream tasks via full fine-tuning is computationally expensive and storage-intensive. Parameter-Efficient Fine-Tuning (PEFT) offers a promising alternative by reducing both adaptation cost and storage burden. However, existing prompting-based approaches ignore the intrinsic geometric structures of point clouds, thereby limiting their adaptation capability. This limitation stems from their inability to encode both fine-grained geometric cues and coarse-grained structural semantics, as well as failing to propagate such information effectively through the model hierarchy. To address these challenges, we propose GAPrompt++, a multi-granular geometry-aware prompting method that provides richer geometric guidance for efficient 3D task adaptation. Specifically, we introduce a Point Shift Prompter that extracts multi-granular geometric features across different scales, enabling instance-specific geometric adjustments during adaptation. Next, a Keypoint Prompter adaptively generates point-level prompts to highlight local geometric saliency and fine-grained structural details. Furthermore, a Prompt Propagation mechanism injects these multi-granular geometric cues throughout the feature extraction hierarchy, strengthening the ability to capture essential geometric characteristics. Extensive experiments show that GAPrompt++ achieves state-of-the-art performance among prompting-based PEFT methods and even surpasses full fine-tuning across diverse benchmarks, while requiring less than 2% trainable parameters. In addition, to address the saturation of existing evaluation datasets, we construct two more challenging benchmarks derived from 3D Gaussian Splatting and Multi-View Stereo reconstruction, offering diverse and realistic point cloud scenarios to promote future research.

Cranial implant generation is an important task in medical imaging. Recent point cloud based generative methods, particularly flow matching, offer strong reconstruction quality and efficient sampling, but still require multiple neural function evaluations during inference. This limits rapid generation of multiple plausible implant candidates. We propose Teacher-guided Endpoint Distillation (TED), a simple one-step distillation framework for conditional cranial implant generation on point clouds. TED trains a one-step student using teacher-guided endpoint supervision and geometric matching losses, while avoiding explicit path straightening. We evaluate TED on the SkullFix and SkullBreak benchmarks. TED achieves the best overall performance on the SkullBreak dataset, remains competitive on SkullFix, and provides the strongest Chamfer distance performance among the compared one-step methods. In addition, TED generates implants in approximately 0.04s per sample. These results show that one-step distillation can substantially accelerate conditional point cloud implant generation without sacrificing reconstruction quality.

Scene-level point cloud understanding remains challenging due to diverse geometries and spatial layouts. While pre-trained 3D point cloud foundation models (PFMs) offer strong transferability, full fine-tuning (FFT) incurs substantial computational and storage costs. Parameter-efficient fine-tuning (PEFT) provides a promising alternative, but existing PEFT methods largely focus on object-level point clouds and overlook serialization-induced partition variations in large-scale scenes. To address this issue, we propose PointPiT, a partition-invariant tuning framework for scene-level point clouds. Specifically, a Scene-aware Structural Adapter (SSA) integrates local geometric patterns with global scene context to mitigate partition-induced representation shifts. Moreover, Gradient Subspace Optimization (GSO) selects informative and partition-stable update directions, suppressing partition-dependent variations during optimization. Extensive experiments across multiple scene-level benchmarks demonstrate that PointPiT achieves competitive or even superior performance to full fine-tuning with less than 1% of backbone's parameters, while achieving consistent state-of-the-art performance among representative PEFT methods.

Bridging the simulation-to-reality gap in roadside LiDAR requires addressing several coupled discrepancies, including scene geometry, sampling density, return patterns, and pedestrian scale. This report presents a multi-source collaborative training and class-aware fusion framework for Sim2Real 3D detection. The method organizes digital-twin scans, diffusion-redrawn scans, density-stabilized scans, and pedestrian morphology-aligned samples into a unified training pool with complementary roles. Within a common DSVT detection formulation, source-specialized expert branches preserve those roles while optimizing for the same detection objective. At inference, a predefined class-aware fusion pathway integrates geometry-stable and calibration-aware branches for vehicles, sampling-complementary branches for trucks, and morphology-consistent evidence for pedestrians. A label-free point-cloud center blend then refines geometric localization. On the UrbanTwin V2X-Real hidden test set, the unified system achieves a combined score of 0.7421, with 3D mAP@0.5 of 0.4518 and a realism score of 0.8871. The results indicate that a stable, interpretable collaboration among data sources is more valuable than unconstrained aggregation of model outputs.

Advances in neural rendering have enabled high-fidelity multi-view reconstruction of 3D scenes. However, free-form non-rigid shape editing remains a significant challenge. Point-based neural representations are highly desirable for multi-view reconstruction because they lack fixed connectivity, which does not constrain the learned surface topology to that of the initialization. Yet this same property causes point-based representations to struggle with holes and surface discontinuities under large deformations. To address this, we propose a novel self-supervised method to enable point-based representations to adapt to large deformations without requiring ground truth multi-view images of deformed geometry. The key idea is to generate random deformations and to ensure consistency in the predicted surface before and after deformation. In particular, the surface prediction from the deformed point cloud should be the same as the deformation applied to the surface prediction from the original point cloud. We incorporate our approach into attention-based point representations, which differ from splatting-based point representations in their use of a learned interpolation kernel between points as opposed to a Gaussian kernel around each point. This learned interpolation kernel can learn to adapt to large deformations, without requiring addition or removal of points. We show that our framework significantly enhances its robustness to large deformations. Experiments on synthetic geometry editing benchmarks (Neural Editor, Objaverse) demonstrate that our approach outperforms existing point-based methods in zero-shot editing and significantly reduces artifacts. Furthermore, qualitative results on the DTU and Mip-NeRF 360 datasets demonstrate our method's effectiveness on real-world scenes.

Certifiable methods have arisen as a means to guarantee global optimality of solutions to non-convex problems using convex semidefinite programming (SDP) relaxations. The most performant of these methods use a local solver to obtain the candidate solution, and then certify its optimality using efficient linear algebra techniques. However, for many problems of interest in robotics, this local-solve-then-certify approach is impeded by a form of degeneracy in the relaxation, leaving a costly optimization of the relaxation as the only recourse. In this paper, we introduce our Central-Path Certifier (CP-Cert), a certifiable method explicitly tailored to certify candidate optima to problems that exhibit this form of degeneracy. Using a candidate as a starting point, our approach seeks a nearby region of the feasible space -- known as the central path -- where a valid certificate can be readily obtained. The approach is kept efficient by exploiting indirect linear algebra techniques, problem sparsity, and parallelism. We apply CP-Cert to both matrix-weighted pose registration and pointcloud data association, whose novel SDP relaxation is of independent interest. On simulated examples, we explore the properties of this novel relaxation and show that CP-Cert is fast and scalable, achieving runtimes that are up to three orders of magnitude faster than state-of-the-art direct solvers. Finally, we combine these contributions into a certifiable, outlier-robust pose-estimation pipeline, which we apply to real-world data.

Plenoptic point clouds (PPC) are novel data structures that represent the light from different viewing directions in order to provide a higher degree of realism to regular point clouds. This is achieved by associating each point to multiple colors instead of a single one. Here, we present a method to efficiently compress the attributes of a PPC, consisting of a Karhunen-Loève transform over the color attributes followed by multiple attribute coders with intra prediction capability. This compression scheme can be incorporated within the MPEG's geometry-based PCC (G-PCC) standard, using any of G-PCC's existing solutions for attribute coding. Compression performance assessment using PPCs of different spatial resolutions reveals competitive results in comparison to existing methods, such as RAHT-based or video-based PCC solutions. We believe our coder to be the new state of the art.

Transformers have had a profound impact on the world of language processing and computer vision. As efforts to answer the million-dollar question of ``How does a Transformer learn?" have been increasing, existing interpretability studies primarily analyze representations at isolated layers or the network as a whole, while the developmental evolution of individual representations and its manifolds across transformer layers remains underexplored. With this work, we aim at providing a comprehensive analysis of the evolution of representations as the representation point cloud transforms across the layers; thereby attempting to isolate layers or establish a trend which comes closer to justifying how and when raw input representations evolve into task-relevant feature representations. Thus, Transformer Geometry Observatory-TGO-IV introduces a topological framework for analysing the evolution of Transformer representations through the lens of Persistent Homology. Rather than studying local geometric properties alone, TGO-IV constructs Vietoris--Rips simplicial complexes from token-level representation point clouds and investigates the evolution of their persistent topological signatures across Transformer layers. The proposed framework comprises complementary topological observatories including Persistence Diagrams, Barcode Diagrams, Betti Curves, Persistence Landscapes, Bottleneck Distance, and Wasserstein Distance, enabling a comprehensive analysis of how the global topology of representation point clouds develops throughout the forward pass.

Dense colored LiDAR maps provide accurate city-scale geometry, but lifting them into 3D Gaussian Splatting (3DGS) retains millions of primitives, making the resulting models costly to store, transmit, render, and adapt. Aggressive primitive reduction alleviates this burden, but can remove the local surface support needed for stable novel-view synthesis and downstream geometric use. We introduce G$^2$ARD-GS, a geometry-guided distillation method that converts a dense Gaussian prior instantiated either as a training-free point-cloud lift or a trained GS model into a compact, reusable representation. G$^2$ARD-GS progressively consolidates the prior into surface-aware representatives, then recovers appearance on the resulting fixed topology under construction-time anchor constraints, with no primitives added or removed during recovery. Under limited supervision, geometry-aware view selection allocates the available view budget. On MatrixCity, G$^2$ARD-GS achieves the best PSNR, SSIM, and LPIPS across matched $5\times$--$30\times$ compression budgets, outperforming PUP by $3.2$--$6.8$,dB in PSNR. When reused as frozen geometry, the compact model improves off-trajectory appearance adaptation by $3.7$--$4.9$,dB over PUP 3D-GS and preserves image-to-model registration accuracy on Cambridge KingsCollege at $30\times$ compression. Project page: https://patrick1159.github.io/gardGS-page/.

Dataset distillation compresses large-scale datasets into compact synthetic sets while preserving their training utility, enabling efficient 3D point cloud training. Current point cloud dataset distillation methods only tackle geometric and representation challenges while ignoring the distributional imbalance prevalent in point cloud datasets where both training and test splits follow long-tailed class distributions. To our knowledge, we present the first study on long-tailed point cloud dataset distillation. Rather than focusing primarily on geometric and representation properties or simply constructing a class-balanced synthetic set, our framework explicitly accounts for long-tailed class distributions via two core modules. First, we design Adaptive Synthetic Budgeting to allocate class-wise synthetic budgets according to class quantity and the expected benefit of additional synthetic samples. Given the allocated budgets, we further design 3D Long-Tailed Distribution Matching to optimize synthetic point clouds through Global-Local Feature Alignment and Prior-Aware Supervision. The former preserves both global class distributions and diverse intra-class structures, while the latter provides class-dependent expert supervision to keep tail-class samples recognizable while maintaining diverse head-class patterns. Extensive experiments demonstrate the effectiveness of our method, lifting classification accuracy by 7.0 points on ShapeNet55 against state-of-the-art methods.

This work proposes an adaptation of the attention mechanism for triangle meshes. The core observation is that endowing the attention mechanism with critical properties for learning over meshes -- intrinsicality and triangulation-agnosticism -- enables it to attain state-of-the-art results over several learning-based tasks in geometry-processing. The above is achieved by modifying the attention mechanism from the bottom up based on simple principles from geometry-processing. Namely, the quantities used within attention -- queries, keys and values -- are created by an intrinsic, triangulation-agnostic network, and treated as discretizations of continuous functions. From that, we devise an appropriate attention mechanism that operates over triangle meshes through standard FEM discretization of the resulting integrals of the above functions. Surprisingly, as far as we know, this straightforward approach has not been utilized for learning over meshes. Experiments show our method exceeds current state of the art, including both mesh-based architectures as well as point cloud transformers. Namely, we show significant improvements on several common benchmarks and tasks -- predicting canonical high-frequency signals; predicting deformations; computing dense correspondences, both between full shapes and partial ones; and predicting feature descriptors.

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 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.

Learning high-fidelity point-cloud features in the 3D space poses significant challenges, including permutation invariance, lack of local context, difficulty in fine-grained surface reconstruction, and high computational cost. In this article, we propose Trans-Unet, a novel framework that addresses these issues by first tansforming 3D point-cloud data into a 2D grid domain and then employing a U-shaped hybrid model that integrates Convolutional Neural Networks, and self-attention mechanisms. The proposed Trans-Unet effectively learns and reconstructs precise features from high-resolution 3D point-cloud data (with 40,401 points in surface and 2,382 points in fiber) derived from a predefined finite element brain patch growth model, enabling accurate prediction of brain folding patterns. By combining multiple techniques, Trans-Unet leverages the complementary strengths: the 3D-to-2D transformation preserves fine-grained structural information while significantly reducing computational cost and the curse of dimensionality; convolutional blocks capture hierarchical, low-level local representations; and the self-attention mechanism models global, high-level semantics and long-range dependencies. The dataset consists of 3D point-clouds containing both brain surface patches and fiber information generated by a large-scale finite element model. Trans-Unet is applied to predict brain surface folding from the initial state (state 0 or states 0-2) to the final state (state 3). Experimental results demonstrate that Trans-Unet achieves high-resolution predictions of brain patch growth, surpassing existing methods in both fidelity and accuracy.

Multi-view 3D surface reconstruction is a longstanding challenge in computer vision. Although recent large-scale reconstruction methods based on 3D Gaussian Splatting (3DGS) achieve impressive novel-view synthesis, producing high-quality surfaces over large scenes remains difficult, due to complex geometry, long optimization, and limited memory. In this paper, we propose a novel yet simple partitioning method to efficiently and faithfully reconstruct large-scale scene surfaces. Our key insight lies in a scene partitioning method based on viewpoint orientation. This partitioning approach ensures that views with similar orientations are jointly involved for more accurate depth estimations, leading to precise surface reconstructions and balanced computation on multiple GPUs in parallel. In addition, we propose a strategy to detect and repair missing regions in the initial point cloud caused by sparse viewpoints or insufficient textures, thereby further improving the geometric quality. Extensive experiments on the GauU-Scene, MatrixCity, and UrbanScene3D datasets demonstrate that our method outperforms the state-of-the-art approaches in surface reconstruction for large-scale scenes. Project page: https://hanl2010.github.io/VOP-GS.

Point cloud compression relies on techniques to compress both geometry and attributes. Motion-based approaches for dynamic solid point cloud geometry compression within the geometry-based point cloud compression (G-PCC) framework have achieved significant reductions in geometry rate. However, motion-based techniques for attribute compression remain underexplored, making it challenging to achieve significant reductions in the temporal redundancy of attributes. Firstly, this paper proposes a geometry-based inter-coding scheme to compress the attributes of dynamic solid point clouds. Secondly, a graph-based motion-estimation scheme for point-cloud attribute compression is proposed. Thirdly, an interpolation-free fractional-voxel motion estimation method is proposed to refine motion accuracy to fractional-voxel precision. Our experimental results on the MPEG point cloud dataset show that the proposed scheme outperforms G-PCC, GeS-TM, and V-PCC in lossless and lossy geometry conditions. We achieve average bitrate savings of $55.3\%$, $42.3\%$, and $16.5\%$ over G-PCC, GeS-TM, and V-PCC, respectively, under lossy-geometry conditions.

This paper introduces a fast method for high-quality 3D Gaussian Splatting (3DGS) reconstruction without traditional Structure-from-Motion (SfM). The proposed approach leverages 3D Foundation Models (3DFMs) for camera pose and point-cloud initialization, then jointly optimizes both camera poses and Gaussian primitives using a depth-guided loss function. This enables fast convergence even from rough initialization with as few as 50-60 input views. To further improve reconstruction quality in sparse-view scenarios, an MLP-based pose refinement module is introduced alongside depth-guided supervision from the foundation model. Extensive experiments on Mip-NeRF 360, Tanks and Temples, and RobustNeRF demonstrate that the proposed method achieves competitive reconstruction quality (23.61 dB PSNR, 0.19 LPIPS) while reducing training time to approximately three minutes per scene. The proposed method produces ready-to-use 3DGS models at a fraction of the time required by existing pipelines, making it suitable for near real-time applications in robotics, VR, and autonomous navigation.

Deploying 3D point cloud analysis in privacy-sensitive, resource-constrained settings faces two barriers: data cannot be centralized, and models must run on limited edge hardware. We present a multi-seed benchmark jointly evaluating federated learning (FL) and knowledge distillation (KD) for 3D point cloud classification. It spans 13 FL algorithms and 10 KD objectives (a 130-pair cross-product) across 504 training runs, evaluated on ModelNet40 and a clinical craniosynostosis dataset. We report three findings. First, under extreme non-IID label skew, standalone FL degrades sharply: on ModelNet40, the strongest method reaches 76.32% against a 92.26% centralized reference; on clinical data, the best reaches 75.83% against 100%. Second, distillation successfully compresses the teacher into a student 74.51% smaller and roughly twice as fast at inference, often matching or surpassing the teacher. Third, the combined pipeline exposes an evaluation pitfall: when distillation keeps a hard-label cross-entropy term on a labeled proxy split, a collapsed federated teacher (8.50%) paired with Logit-MSE still yields a 92.94% student. This 84.4-point gap reflects the proxy labels rather than the federated model, reusing the very labels whose privacy motivated federation. Objectives without hard labels instead track teacher quality ($r \approx 0.99$) and collapse when the teacher does. We therefore recommend evaluating FL-KD pipelines with label-free distillation so reported accuracy reflects the federated teacher, not the proxy.

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.

3D point cloud models suffer significant performance degradation under distribution shifts caused by sensor noise, occlusions, and environmental changes. Test-time adaptation (TTA) has emerged as a practical paradigm for mitigating this issue during inference. Recently, leveraging multi-view augmentation has shown promise in improving 3D TTA performance. However, existing multi-view approaches are often constrained by sequential optimization that treats each view independently. This sequential optimization leads to substantial inference latency due to repetitive optimization steps, making real-time adaptation impractical. To address this, we propose Masked Multi-View Test-Time Adaptation (MAMVI), which replaces sequential optimization with a unified single-step adaptation. Specifically, MAMVI utilizes a hybrid masking strategy that combines fixed ratios for stability with Beta-distributed sampling for diversity. By aggregating losses across multiple views, MAMVI performs adaptation through a single backward pass based on multi-view consensus. Additionally, a confidence-based adaptive learning rate is used to dynamically adjust the adaptation intensity for each sample. Extensive experiments on ModelNet-40C, ShapeNet-C, and ScanObjectNN-C demonstrate that MAMVI achieves state-of-the-art accuracy on ShapeNet-C and ScanObjectNN-C. Moreover, it remains competitive on ModelNet-40C while delivering 4.9-8.9 times faster inference, making it highly suitable for real-time applications. Our code is available at https://github.com/Inseok-kong/MAMVI

In 3D environments, Embodied Agents answer spatially relevant questions through reasoning from a mixture of modalities including natural language, RGB images, point clouds, depth maps and camera poses. Existing Vision-Language models (VLMs) are fine-tuned over a single modality. This completely ignores the question semantics which may favor a different modality than the finetuned modality. To address this, we propose MASER (Modality-Adaptive SpEcialist Routing), a lightweight framework that trains five different modality adapters of a shared VLM backbone and learns a neural routing policy that selects the best adapter based on the question during inference. We encode each question with a frozen sentence transformer and pass the embedding through a small Multi-layer Perceptron (MLP) trained on oracle adapter-accuracy labels. We evaluate our methodology over the Open3D-VQA benchmark and our evaluations show that no single modality is universally optimal -- point-cloud answers are best in 51.5% of cases. MASER routes with 51.3% oracle agreement, outperforming a Random-Forest ablation (43.5%), with only a single adapter call per question.

3D Gaussian Splatting (3DGS) is a technique for creating and rendering 3D scenes, however its performance depends heavily on the quality of initial seed points. To improve 3DGS initialization, this study presents and evaluates several point cloud upsampling approaches: linear interpolation, triangular interpolation, spline-based surface reconstruction, moving least squares surface fitting, and Voronoi-based point generation. Additionally, this research introduces a depth-guided point lifting method that leverages depth maps to maintain geometric consistency with Structure-from-Motion (SfM) reconstructions. Through extensive experiments on the Mip-NeRF360 and Replica datasets, the proposed methods demonstrate improvements in reconstruction quality across diverse scene types. Results indicate that different upsampling strategies excel in different scenarios: surface reconstruction methods perform better with organic, detailed scenes, while simpler interpolation approaches are more suited for scenes dominated by piecewise-smooth geometries. In comparison, the depth-guided approach shows promise for adding geometry-aware points across the entire scene, importantly in texture-less regions. These findings, which provide preliminary practical guidelines for selecting appropriate upsampling methods based on scene characteristics and computational constraints, advances the understanding of how point cloud initialization affects 3DGS quality.

The eigendecomposition of the Laplace--Beltrami Operator (LBO) is fundamental to geometric analysis, yet computing its low-frequency eigenmodes remains a significant bottleneck due to the high cost of iterative solvers on large-scale data. To amortize this cost, we introduce the Neural Eigenspace Operator (NEO), a feed-forward framework designed to predict the spectrum directly from point clouds. Crucially, NEO circumvents the ill-posed nature of standard eigenvector regression, which suffers from intrinsic sign flips and rotation ambiguities, by learning the stable, invariant low-frequency subspace instead. Specifically, the network predicts a redundant set of basis functions whose span robustly covers the target eigenspace, allowing for the recovery of accurate eigenpairs via a lightweight Rayleigh--Ritz refinement. To handle irregular sampling, we propose a mass-aware neural operator that incorporates per-point area weights into attention-based aggregation, improving robustness to non-uniform densities and enabling zero-shot generalization across resolutions. Our approach achieves near-linear runtime scaling and substantial wall-clock speedups over iterative solvers at comparable accuracy, and exhibits strong zero-shot transfer to high-resolution point clouds. The resulting eigenpairs support standard spectral geometry tasks, while the raw basis functions provide effective point-wise features for downstream learning. Code: https://github.com/Adversarr/NEO.

Understanding 3D point clouds through language remains a fundamental challenge in computer graphics and visual computing, due to the irregular structure of point cloud data and the lack of explicit reasoning in existing 3D multimodal models. While Chain-of-Thought (CoT) reasoning has shown strong effectiveness in LLMs and image-based MLLMs, its extension to 3D understanding remains largely underexplored. In this paper, we propose a data-centric framework for constructing large-scale CoT supervision tailored to 3D point cloud understanding. Our framework consists of a two-stage pipeline that first refines point-text instruction data via vision-language-model-based quality evaluation and reference-guided refinement, and then synthesizes high-quality reasoning paths through Human-in-the-Loop Prompt Optimization (HiLPO). Using this approach, we build PoCoTI, a CoT-enhanced point-text instruction-following dataset containing 55K samples with explicit reasoning paths. Fine-tuning PointLLM on PoCoTI yields PointLLM-R, a reasoning-capable 3D multimodal language model. Extensive experiments on generative 3D classification and captioning demonstrate that PointLLM-R achieves state-of-the-art performance and generalizes robustly to real-world scanned point clouds and multi-turn dialogue scenarios.

Point cloud stands as the most widely adopted format for representing 3D shapes and scenes due to its simplicity and geometric fidelity. However, its inherent unordered and irregular nature, exacerbated by sensor noise and occlusions, introduces unique challenges for machine learning based methodologies. To combat these issues, diverse strategies have been developed, including converting to a format that has orderliness, extracting local geometry, and permutation-invariant or self-attention-based processing. In this paper, our focus is directed towards deep learning models for three fundamental tasks in 3D vision: point cloud classification, part segmentation, and semantic segmentation. We begin by formally defining point cloud data, followed by an in-depth discussion on its structural characteristics. Then, we categorize notable works based on their backbone structure and evaluate their performance on popular benchmarks. Beyond empirical comparison, we offer insights into architectural innovations and limitations. We also outline open challenges and promising future directions for 3D point cloud understanding.

This work presents a novel method for fitting superquadrics to point clouds under the contamination of noise and outliers, which has many applications for shape modeling across diverse fields. Unlike prior approaches that either exclusively focus on fitting rigid or deformable superquadrics, or suffer from robustness and numerical instability issues, our method redefines the problem from a new unsupervised clustering perspective, enabling the holistic fitting of both rigid and deformable superquadrics within a unified framework. Central to our approach is a stable optimization function inspired by unsupervised clustering analysis, where we formulate the point cloud data and samples from the potential parametric surface as clustering members and centroids, respectively. Then, the clustering process with dynamic updates to centroid locations serves as a direct proxy for optimizing superquadric parameters, establishing a principled link between geometric fitting and clustering dynamics. We further derive the relationship between pairwise computations of clustering centroids and clustering members to orthogonal distances, effectively eliminating the need for the time-consuming surface sampling process. Moreover, our formulation provides closed-form analytical solutions for both the fuzzy membership degree vector and the covariance matrix, ensuring efficient iteration optimization and enabling more effective handling of geometric deformations. In addition, we provide a theoretical certificate of convergence analysis and demonstrate that the clustering-inspired fitting method can escape local minima by inherently increasing the convexity of the objective function. The implementation is publicly available at https://github.com/zikai1/SuperquadricFitting.

Point cloud learning often rests on the premise that observed samples are noisy traces of an underlying geometric object, such as a manifold embedded in a high-dimensional feature space. Yet much of this geometry is not captured directly by coordinates, pairwise distances, or learned graph neighborhoods alone. In the smooth setting, differential forms are devices to encode higher order tangency information. In this work, we introduce a new family of principled learnable geometric features for point clouds called neural point-forms (NPFs). In the absence of a natural tangency structure, we instead use Laplacian-based techniques from Diffusion Geometry to build a discrete model for comparing differential forms on point clouds via inner products. In the continuum, submanifolds of a shared ambient feature space are represented as comparison matrices, whose entries describe how pairs of feature forms interact with extrinsic tangency information. We make this intuition precise by proving the long-run consistency of comparison matrices under standard sampling, bandwidth, density, and manifold-hypothesis assumptions. This yields a compact, efficient and permutation-invariant neural layer whose output is a learned form-comparison matrix. Across synthetic and biologically relevant experiments, we show that NPFs provide a competitive, and interpretable representation, with the strongest benefits appearing when labels depend on sampling density, manifold-like structure, or response-relevant population geometry.

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.

While invariant architectures are standard for processing symmetric data, there is growing interest in achieving invariance by applying group averaging or canonization to non-invariant backbones. However, the theoretical generalization properties of these alternative strategies remain poorly understood. We introduce a theoretical framework to analyze the generalization error of these methods by bounding their covering numbers. We establish a rigorous generalization hierarchy: the error bounds of canonized models are at best equal to the error bounds of structurally invariant and group-averaged models, and at worst equal to the bounds of non-invariant baselines. Furthermore, we show that there exist optimal canonizations which attain the optimal error bounds, and poor canonizations which attain the non-invariant error bounds, and that this depends on the regularity of the canonization. Finally, applying this framework to permutation groups in point cloud processing, we rigorously prove that the covering number of lexicographical sorting grows exponentially with point cloud dimension, whereas Hilbert curve canonization guarantees polynomial growth. This provides the first formal theoretical justification for the empirical success of Hilbert curve serialization in state-of-the-art point cloud architectures. We conclude with experiments that support our theoretical claims. Code is available at https://github.com/yonatansverdlov/Canonization

We introduce PALACE (Persistence Adaptive-Landmark Analytic Classification Engine), the data-adaptive companion to PLACE, paying a small cross-validation tier on three knobs (budget, radii, bandwidth; $\leq 5$ choices each). A cover-theoretic core (Lebesgue-number criterion on the landmark cover) yields four closed-form guarantees. (i) A structural lower distortion bound $λ(τ;ν)$ on $\mathcal{D}_n$ under cross-diagram non-interference, with a $(D/L)^2$ budget reduction over the uniform grid when diagrams concentrate. (ii) Equal weights $w_k = K^{-1/2}$ maximizing $λ$, and farthest-point-sampling positions $2$-approximating the optimal $k$-center covering radius; both derived from training labels alone, no gradient training. (iii) A kernel-RKHS classification rate $O((k-1)\sqrt{K}/(γ\sqrt{m_{\min}}))$ with binary necessity threshold $m = Ω(\sqrt K/γ)$ from a matching Le Cam lower bound, and a closed-form filtration-selection rule. The kernel-Mahalanobis margin $\hatρ_{\mathrm{Mah}}$ is the strongest closed-form ranker across the chemical-graph pool (mean Spearman $ρ\approx +0.60$); the isotropic surrogate $\hatγ/\sqrt{K}$ admits a selection-consistency rate, and $\widehatλ$ from (i) provides an independent data-level signal (positive on COX2 and PTC). (iv) A per-prediction certificate, in non-asymptotic Pinelis and asymptotic Gaussian forms, with no calibration split. Empirically, PALACE is the strongest closed-form diagram-based method on Orbit5k ($91.3 \pm 1.0\%$, matching Persformer), leads every diagram-based competitor on COX2 and MUTAG, and is competitive on DHFR (within 1 pp of ECP). At $8\times$ domain inflation, adaptive placement maintains $94\%$ while the uniform grid collapses to chance ($25\%$ on 4-class data).

Pre-trained 3D point cloud foundation models (PFMs) have demonstrated strong transferability across diverse downstream tasks. However, full fine-tuning these models is computationally expensive and storage-intensive. Parameter-efficient fine-tuning (PEFT) offers a promising alternative, but existing PEFT approaches are primarily designed for Transformer-based backbones and rely on token-level prompting or feature transformation. Mamba-based backbones introduce a granularity mismatch between token-level adaptation and state-level sequence dynamics. Consequently, straightforward transfer of existing PEFT approaches to frozen Mamba backbones leads to substantial accuracy degradation and unstable optimization. To address this issue, we propose Mantis, the first Mamba-native PEFT framework for 3D PFMs. Specifically, a State-Aware Adapter (SAA) is introduced to inject lightweight task-conditioned control signals into selective state-space updates, enabling state-level adaptation while keeping the pre-trained backbone frozen. Moreover, different valid point cloud serializations are regularized by Dual-Serialization Consistency Distillation (DSCD), thereby reducing serialization-induced instability. Extensive experiments across multiple benchmarks demonstrate that our Mantis achieves competitive performance with only about 5% trainable parameters. Our code is available at https://github.com/gzhhhhhhh/Mantis.

We introduce PLACE (Persistence-Landmark Analytic Classification Engine), a closed-form pipeline for classifying point clouds and graphs through their persistent-homology signatures. Three quantitative guarantees -- a margin-based excess-risk rate, a closed-form descriptor-selection rule, and a per-prediction certificate -- are derived from training labels alone, with no learned weights or held-out calibration. The embedding sums Mitra-Virk single-point coordinate functions over a sparse landmark grid; the closed-form weight rule $w_k^2 \propto (d_{k+1}^2 - d_k^2)/R_k^2$ maximizes the distortion slope in Mitra-Virk's affine certificate under $ν$-coherence. (i) An $O(kR/(Δ\sqrt{m_{\min}}))$ margin bound, driven by class-mean separation $Δ$ and embedding radius $R$, matched in the sample-starved regime $m \lesssim R/Δ$ by a Le Cam minimax lower bound. (ii) The Mahalanobis margin under Ledoit-Wolf-shrunk covariance is the strongest closed-form ranker on a 64-descriptor chemical-graph pool (mean Spearman $ρ= +0.56$ across 11 benchmarks, positive on 10 of 11); the isotropic surrogate $Δ/\sqrt{\ell}$ admits a closed-form selection-consistency rate on the homogeneous protein/social pools. (iii) A training-time-decided certificate, with no per-prediction overhead, in three concrete radii (Pinelis, Gaussian plug-in, and variance-aware Pinelis-Bernstein). Empirically, PLACE is the strongest diagram-based method on Orbit5k and matches the strongest topology-based baseline within statistical noise on MUTAG and COX2; remaining gaps fall into two diagnosable regimes (descriptor blindness on NCI1/NCI109; pool-coverage limits elsewhere). The Pinelis-Bernstein radius fires on 8 of the 12 benchmarks; on MUTAG the empirical and population nearest-centroid rules agree on every one of 940 held-out test predictions, validating the certificate's mechanism.

Clinical fusion of Single Photon Emission Computed Tomography Myocardial Perfusion Imaging (SPECT MPI) and Computed Tomography Angiography (CTA) remains limited by cross-modality misregistration and reliance on manual landmarks, which can hinder accurate ischemia localization and lesion-level functional assessment. To address this issue, we propose a registration and fusion framework for SPECT MPI and CTA that integrates functional and structural information for comprehensive cardiac evaluation. The proposed pipeline performs U-Net-based segmentation on both modalities. On SPECT MPI, only the left ventricle (LV) is extracted, and anatomical landmarks are automatically derived from characteristic LV structures. On CTA, both ventricles are segmented, and their spatial relationship is used to automatically define landmarks at the interventricular septal junction. Scale-space consistency preprocessing and landmark-driven coarse registration are applied to mitigate initial misalignment. Based on this initialization, multiple fine registration methods are evaluated on LV epicardial surface point clouds, including ICP, SICP, CPD, CluReg, FFD, and BCPD-plus-plus. The resulting transformations are then propagated to voxel-level resampling for high-precision SPECT-CTA fusion. In a retrospective cohort of 60 patients, the proposed framework preserved sub-millimeter coronary detail from CTA while accurately overlaying quantitative SPECT perfusion. Among the evaluated methods, BCPD-plus-plus achieved the highest accuracy with a mean point cloud distance of 1.7 mm. By combining robust initialization, comparative fine registration, and voxel-level fusion, the proposed approach provides a practical solution for myocardial ischemia localization and functional evaluation of coronary lesions, while remaining independent of any specific fine registration algorithm.

Each scanner possesses its unique characteristics and exhibits its distinct sampling error distribution. Training a network on a dataset that includes data collected from different scanners is less effective than training it on data specific to a single scanner. Therefore, we present a novel one-shot learning method allowing for edge extraction on point clouds, by learning the specific data distribution of the target point cloud, and thus achieve superior results compared to networks that were trained on general data distributions. More specifically, we present how to train a lightweight network named OSFENet (One-Shot edge Feature Extraction Network), by designing a filtered-KNN-based surface patch representation that supports a one-shot learning framework. Additionally, we introduce an RBF_DoS module, which integrates Radial Basis Function-based Descriptor of the Surface patch, highly beneficial for the edge extraction on point clouds. The advantage of the proposed OSFENet is demonstrated through comparative analyses against 7 baselines on the ABC dataset, and its practical utility is validated by results across diverse real-scanned datasets, including indoor scenes like S3DIS dataset, and outdoor scenes such as the Semantic3D dataset and UrbanBIS dataset.

Recent advances in pretraining 3D point cloud encoders (e.g., Point-BERT, Point-MAE) have produced powerful models, whose abilities are typically evaluated on geometric or semantic tasks. At the same time, topological descriptors have been shown to provide informative summaries of a shape's multiscale structure. In this paper we pose the question whether topological information can be derived from features produced by 3D encoders. To address this question, we first introduce DONUT, a synthetic benchmark with controlled topological complexity, and propose FILTR (Filtration Transformer), a learnable framework to predict persistence diagrams directly from frozen encoders. FILTR adapts a transformer decoder to treat diagram generation as a set prediction task. Our analysis on DONUT reveals that existing encoders retain only limited global topological signals, yet FILTR successfully leverages information produced by these encoders to approximate persistence diagrams. Our approach enables, for the first time, data-driven extraction of persistence diagrams from raw point clouds through an efficient learnable feed-forward mechanism.

Alzheimer's disease (AD) is a progressive neurodegenerative disorder and a major cause of dementia. Structural MRI is widely used to analyze AD-related brain atrophy; however, most deep learning methods rely on computationally expensive 3D convolutional neural networks (CNNs), limiting deployment in resource-constrained settings. This work introduces two main contributions. First, we propose a pipeline that converts T1-weighted MRI into anatomically informed 2D point clouds using Anatomical Priority Sampling (APS), producing ADNI-2DPC, the first neuroanatomically labeled MRI-derived point cloud dataset. Second, we present NeuroAPS-Net, a lightweight geometric deep learning model that incorporates anatomical priors via region-aware feature encoding and ROI token aggregation. Experiments on ADNI-2DPC demonstrate that NeuroAPS-Net achieves competitive classification accuracy while significantly reducing inference latency and GPU memory compared to state-of-the-art point cloud methods. These results highlight the potential of anatomically guided point cloud learning as an efficient and interpretable alternative to voxel-based CNNs for AD classification.

With the rise of pre-trained models in the 3D point cloud domain for a wide range of real-world applications, adapting them to downstream tasks has become increasingly important. However, conventional full fine-tuning methods are computationally expensive and storage-intensive. Although prompt tuning has emerged as an efficient alternative, it often suffers from overfitting, thereby compromising generalization capability. To address this issue, we propose Prototypical Point-level Prompt Tuning (P$^3$T), a parameter-efficient prompt tuning method designed for pre-trained 3D vision-language models (VLMs). P$^3$T consists of two components: 1) \textit{Point Prompter}, which generates instance-aware point-level prompts for the input point cloud, and 2) \textit{Text Prompter}, which employs learnable prompts into the input text instead of hand-crafted ones. Since both prompters operate directly on input data, P$^3$T enables task-specific adaptation of 3D VLMs without sacrificing generalizability. Furthermore, to enhance embedding space alignment, which is key to fine-tuning 3D VLMs, we introduce a prototypical loss that reduces intra-category variance. Extensive experiments demonstrate that our method matches or outperforms full fine-tuning in classification and few-shot learning, and further exhibits robust generalization under data shift in the cross-dataset setting. The code is available at \textcolor{violet}{https://github.com/gyjung975/P3T}.

In this paper, we present a network structure for classifying metadata of cuneiform tablets. The problem is of practical importance, as the size of the existing corpus far exceeds the number of experts available to analyze it. But the task is made difficult by the combination of limited annotated datasets and the high-resolution point-cloud representation of each tablet. To address this, we develop a convolution-inspired architecture that gradually down-scales the point cloud while integrating local neighbor information. The final down-scaled point cloud is then processed by computing neighbors in the feature space to include global information. Our method is compared with the state-of-the-art transformer-based network Point-BERT, and consistently obtains the best performance. Source code and data available at github.com/fhagelskjaer/cuneiform3d

Few-shot 3D point cloud semantic segmentation aims to segment novel categories using a minimal number of annotated support samples. However, prototypes derived from the limited non-structural point cloud support set are often misaligned and have a small capacity, hindering effective gen eralization to novel categories. This stems from two core issues: i) the prototype possess limited representational capacity fails to cover the full intra-class diversity of a novel category, and ii) the prototypes suffer from misalignment with the query space due to the inter-set inconsistency between support and query sets. To address these issues, our work focuses on leveraging the few support samples to construct a well-aligned big-capacity prototype. Motivated by the powerful generative capabilities of diffusion models, we re-purpose its pre-trained conditional encoder to provide rich feature components for prototype ex pansion. Subsequently, a push-pull force aligns this expanded prototype towards the query feature space. Under this setup, we introduce the Prototype Expansion Network (PENet), a framework that constructs aligned big-capacity prototypes from two complementary feature sources. Specifically, PENet employs a dual-stream learner architecture: it retains a conventional fully supervised Intrinsic Learner (IL) to distill representative features, while introducing a novel Diffusion Learner (DL) to provide rich generalizable features. The resulting dual prototypes are then processed by a Prototype Assimilation Module (PAM), which adopts a push-pull attention block to align the prototypes with the query space. Furthermore, a Prototype Calibration Mechanism (PCM) regularizes the final big-capacity prototype to prevent semantic drift. Extensive experiments on the S3DIS and ScanNet datasets demonstrate that PENet outperforms state-of-the-art methods across various few-shot settings.

Given encoded 3D point cloud geometry available at the decoder, we study the problem of lossy attribute compression in a multi-resolution B-spline projection framework. A target continuous 3D attribute function is first projected onto a sequence of nested subspaces $\mathcal{F}^{(p)}_{l_0} \subseteq \cdots \subseteq \mathcal{F}^{(p)}_{L}$, where $\mathcal{F}^{(p)}_{l}$ is a family of functions spanned by a B-spline basis function of order $p$ at a chosen scale and its integer shifts. The projected low-pass coefficients $F_l^*$ are computed by variable-complexity unrolling of a rate-distortion (RD) optimization algorithm into a feed-forward network, where the rate term is the sparsity-promoting $\ell_1$-norm. Thus, the projection operation is end-to-end differentiable. For a chosen coarse-to-fine predictor, the coefficients are then adjusted to account for the prediction from a lower-resolution to a higher-resolution, which is also optimized in a data-driven manner.