Daily Papers

Automatically extracting the geometric content from the hundreds of thousands of diagrams drawn in historical manuscripts would enable historians to study the diffusion of astronomical knowledge on a global scale. However, state-of-the-art vectorization methods, often designed to tackle modern data, are not adapted to the complexity and diversity of historical astronomical diagrams. Our contribution is thus twofold. First, we introduce a unique dataset of 303 astronomical diagrams from diverse traditions, ranging from the XIIth to the XVIIIth century, annotated with more than 3000 line segments, circles and arcs. Second, we develop a model that builds on DINO-DETR to enable the prediction of multiple geometric primitives. We show that it can be trained solely on synthetic data and accurately predict primitives on our challenging dataset. Our approach widely improves over the LETR baseline, which is restricted to lines, by introducing a meaningful parametrization for multiple primitives, jointly training for detection and parameter refinement, using deformable attention and training on rich synthetic data. Our dataset and code are available on our webpage.

Representing complex objects with basic geometric primitives has long been a topic in computer vision. Primitive-based representations have the merits of compactness and computational efficiency in higher-level tasks such as physics simulation, collision checking, and robotic manipulation. Unlike previous works which extract polygonal meshes from a signed distance function (SDF), in this paper, we present a novel method, named Marching-Primitives, to obtain a primitive-based abstraction directly from an SDF. Our method grows geometric primitives (such as superquadrics) iteratively by analyzing the connectivity of voxels while marching at different levels of signed distance. For each valid connected volume of interest, we march on the scope of voxels from which a primitive is able to be extracted in a probabilistic sense and simultaneously solve for the parameters of the primitive to capture the underlying local geometry. We evaluate the performance of our method on both synthetic and real-world datasets. The results show that the proposed method outperforms the state-of-the-art in terms of accuracy, and is directly generalizable among different categories and scales. The code is open-sourced at https://github.com/ChirikjianLab/Marching-Primitives.git.

We present SuperDec, an approach for creating compact 3D scene representations via decomposition into superquadric primitives. While most recent works leverage geometric primitives to obtain photorealistic 3D scene representations, we propose to leverage them to obtain a compact yet expressive representation. We propose to solve the problem locally on individual objects and leverage the capabilities of instance segmentation methods to scale our solution to full 3D scenes. In doing that, we design a new architecture which efficiently decompose point clouds of arbitrary objects in a compact set of superquadrics. We train our architecture on ShapeNet and we prove its generalization capabilities on object instances extracted from the ScanNet++ dataset as well as on full Replica scenes. Finally, we show how a compact representation based on superquadrics can be useful for a diverse range of downstream applications, including robotic tasks and controllable visual content generation and editing.

While haircut indicates distinct personality, existing avatar generation methods fail to model practical hair due to the general or entangled representation. We propose StrandHead, a novel text to 3D head avatar generation method capable of generating disentangled 3D hair with strand representation. Without using 3D data for supervision, we demonstrate that realistic hair strands can be generated from prompts by distilling 2D generative diffusion models. To this end, we propose a series of reliable priors on shape initialization, geometric primitives, and statistical haircut features, leading to a stable optimization and text-aligned performance. Extensive experiments show that StrandHead achieves the state-of-the-art reality and diversity of generated 3D head and hair. The generated 3D hair can also be easily implemented in the Unreal Engine for physical simulation and other applications. The code will be available at https://xiaokunsun.github.io/StrandHead.github.io.

Scale-aware monocular depth estimation poses a significant challenge in computer-aided endoscopic navigation. However, existing depth estimation methods that do not consider the geometric priors struggle to learn the absolute scale from training with monocular endoscopic sequences. Additionally, conventional methods face difficulties in accurately estimating details on tissue and instruments boundaries. In this paper, we tackle these problems by proposing a novel enhanced scale-aware framework that only uses monocular images with geometric modeling for depth estimation. Specifically, we first propose a multi-resolution depth fusion strategy to enhance the quality of monocular depth estimation. To recover the precise scale between relative depth and real-world values, we further calculate the 3D poses of instruments in the endoscopic scenes by algebraic geometry based on the image-only geometric primitives (i.e., boundaries and tip of instruments). Afterwards, the 3D poses of surgical instruments enable the scale recovery of relative depth maps. By coupling scale factors and relative depth estimation, the scale-aware depth of the monocular endoscopic scenes can be estimated. We evaluate the pipeline on in-house endoscopic surgery videos and simulated data. The results demonstrate that our method can learn the absolute scale with geometric modeling and accurately estimate scale-aware depth for monocular scenes.

With the goal of understanding the visual concepts that CLIP associates with text prompts, we show that the latent space of CLIP can be visualized solely in terms of linear transformations on simple geometric primitives like circles and straight lines. Although existing approaches achieve this by sketch-synthesis-through-optimization, they do so on the space of B\'ezier curves, which exhibit a wastefully large set of structures that they can evolve into, as most of them are non-essential for generating meaningful sketches. We present CLIPDrawX, an algorithm that provides significantly better visualizations for CLIP text embeddings, using only simple primitive shapes like straight lines and circles. This constrains the set of possible outputs to linear transformations on these primitives, thereby exhibiting an inherently simpler mathematical form. The synthesis process of CLIPDrawX can be tracked end-to-end, with each visual concept being explained exclusively in terms of primitives. Implementation will be released upon acceptance. Project Page: https://clipdrawx.github.io/{https://clipdrawx.github.io/}.

Current multimodal large language models (MLLMs) often underperform on mathematical problem-solving tasks that require fine-grained visual understanding. The limitation is largely attributable to inadequate perception of geometric primitives during image-level contrastive pre-training (e.g., CLIP). While recent efforts to improve math MLLMs have focused on scaling up mathematical visual instruction datasets and employing stronger LLM backbones, they often overlook persistent errors in visual recognition. In this paper, we systematically evaluate the visual grounding capabilities of state-of-the-art MLLMs and reveal a significant negative correlation between visual grounding accuracy and problem-solving performance, underscoring the critical role of fine-grained visual understanding. Notably, advanced models like GPT-4o exhibit a 70% error rate when identifying geometric entities, highlighting that this remains a key bottleneck in visual mathematical reasoning. To address this, we propose a novel approach, SVE-Math (Selective Vision-Enhanced Mathematical MLLM), featuring a geometric-grounded vision encoder and a feature router that dynamically adjusts the contribution of hierarchical visual feature maps. Our model recognizes accurate visual primitives and generates precise visual prompts tailored to the language model's reasoning needs. In experiments, SVE-Math-Qwen2.5-7B outperforms other 7B models by 15% on MathVerse and is compatible with GPT-4V on MathVista. Despite being trained on smaller datasets, SVE-Math-7B achieves competitive performance on GeoQA, rivaling models trained on significantly larger datasets. Our findings emphasize the importance of incorporating fine-grained visual understanding into MLLMs and provide a promising direction for future research.

Accurate 3D instance segmentation is crucial for high-quality scene understanding in the 3D vision domain. However, 3D instance segmentation based on 2D-to-3D lifting approaches struggle to produce precise instance-level segmentation, due to accumulated errors introduced during the lifting process from ambiguous semantic guidance and insufficient depth constraints. To tackle these challenges, we propose splitting and growing reliable semantic mask for high-fidelity 3D instance segmentation (SGS-3D), a novel "split-then-grow" framework that first purifies and splits ambiguous lifted masks using geometric primitives, and then grows them into complete instances within the scene. Unlike existing approaches that directly rely on raw lifted masks and sacrifice segmentation accuracy, SGS-3D serves as a training-free refinement method that jointly fuses semantic and geometric information, enabling effective cooperation between the two levels of representation. Specifically, for semantic guidance, we introduce a mask filtering strategy that leverages the co-occurrence of 3D geometry primitives to identify and remove ambiguous masks, thereby ensuring more reliable semantic consistency with the 3D object instances. For the geometric refinement, we construct fine-grained object instances by exploiting both spatial continuity and high-level features, particularly in the case of semantic ambiguity between distinct objects. Experimental results on ScanNet200, ScanNet++, and KITTI-360 demonstrate that SGS-3D substantially improves segmentation accuracy and robustness against inaccurate masks from pre-trained models, yielding high-fidelity object instances while maintaining strong generalization across diverse indoor and outdoor environments. Code is available in the supplementary materials.

Despite strong performance on vision-language tasks, Multimodal Large Language Models (MLLMs) struggle with mathematical problem-solving, with both open-source and state-of-the-art models falling short of human performance on visual-math benchmarks. To systematically examine visual-mathematical reasoning in MLLMs, we (1) evaluate their understanding of geometric primitives, (2) test multi-step reasoning, and (3) explore a potential solution to improve visual reasoning capabilities. Our findings reveal fundamental shortcomings in shape recognition, with top models achieving under 50% accuracy in identifying regular polygons. We analyze these failures through the lens of dual-process theory and show that MLLMs rely on System 1 (intuitive, memorized associations) rather than System 2 (deliberate reasoning). Consequently, MLLMs fail to count the sides of both familiar and novel shapes, suggesting they have neither learned the concept of sides nor effectively process visual inputs. Finally, we propose Visually Cued Chain-of-Thought (VC-CoT) prompting, which enhances multi-step mathematical reasoning by explicitly referencing visual annotations in diagrams, boosting GPT-4o's accuracy on an irregular polygon side-counting task from 7% to 93%. Our findings suggest that System 2 reasoning in MLLMs remains an open problem, and visually-guided prompting is essential for successfully engaging visual reasoning. Code available at: https://github.com/rsinghlab/Shape-Blind.

This study introduces a novel framework, G3Reg, for fast and robust global registration of LiDAR point clouds. In contrast to conventional complex keypoints and descriptors, we extract fundamental geometric primitives, including planes, clusters, and lines (PCL) from the raw point cloud to obtain low-level semantic segments. Each segment is represented as a unified Gaussian Ellipsoid Model (GEM), using a probability ellipsoid to ensure the ground truth centers are encompassed with a certain degree of probability. Utilizing these GEMs, we present a distrust-and-verify scheme based on a Pyramid Compatibility Graph for Global Registration (PAGOR). Specifically, we establish an upper bound, which can be traversed based on the confidence level for compatibility testing to construct the pyramid graph. Then, we solve multiple maximum cliques (MAC) for each level of the pyramid graph, thus generating the corresponding transformation candidates. In the verification phase, we adopt a precise and efficient metric for point cloud alignment quality, founded on geometric primitives, to identify the optimal candidate. The algorithm's performance is validated on three publicly available datasets and a self-collected multi-session dataset. Parameter settings remained unchanged during the experiment evaluations. The results exhibit superior robustness and real-time performance of the G3Reg framework compared to state-of-the-art methods. Furthermore, we demonstrate the potential for integrating individual GEM and PAGOR components into other registration frameworks to enhance their efficacy. Code: https://github.com/HKUST-Aerial-Robotics/G3Reg

Human perception of 3D shapes goes beyond reconstructing them as a set of points or a composition of geometric primitives: we also effortlessly understand higher-level shape structure such as the repetition and reflective symmetry of object parts. In contrast, recent advances in 3D shape sensing focus more on low-level geometry but less on these higher-level relationships. In this paper, we propose 3D shape programs, integrating bottom-up recognition systems with top-down, symbolic program structure to capture both low-level geometry and high-level structural priors for 3D shapes. Because there are no annotations of shape programs for real shapes, we develop neural modules that not only learn to infer 3D shape programs from raw, unannotated shapes, but also to execute these programs for shape reconstruction. After initial bootstrapping, our end-to-end differentiable model learns 3D shape programs by reconstructing shapes in a self-supervised manner. Experiments demonstrate that our model accurately infers and executes 3D shape programs for highly complex shapes from various categories. It can also be integrated with an image-to-shape module to infer 3D shape programs directly from an RGB image, leading to 3D shape reconstructions that are both more accurate and more physically plausible.

We introduce CameraBench, a large-scale dataset and benchmark designed to assess and improve camera motion understanding. CameraBench consists of ~3,000 diverse internet videos, annotated by experts through a rigorous multi-stage quality control process. One of our contributions is a taxonomy of camera motion primitives, designed in collaboration with cinematographers. We find, for example, that some motions like "follow" (or tracking) require understanding scene content like moving subjects. We conduct a large-scale human study to quantify human annotation performance, revealing that domain expertise and tutorial-based training can significantly enhance accuracy. For example, a novice may confuse zoom-in (a change of intrinsics) with translating forward (a change of extrinsics), but can be trained to differentiate the two. Using CameraBench, we evaluate Structure-from-Motion (SfM) and Video-Language Models (VLMs), finding that SfM models struggle to capture semantic primitives that depend on scene content, while VLMs struggle to capture geometric primitives that require precise estimation of trajectories. We then fine-tune a generative VLM on CameraBench to achieve the best of both worlds and showcase its applications, including motion-augmented captioning, video question answering, and video-text retrieval. We hope our taxonomy, benchmark, and tutorials will drive future efforts towards the ultimate goal of understanding camera motions in any video.

Vision Language Models (VLMs) achieve strong performance on many vision-language tasks but often struggle with spatial reasoningx2014a prerequisite for many applications. Empirically, we find that a dataset produced by a current training data generation pipeline has a 57.6% human validation rate. These rates stem from current limitations: single-image 3D reconstruction introduces cascading modeling errors and requires wide answer tolerances, while caption-based methods require hyper-detailed annotations and suffer from generative hallucinations. We present GRAID, built on the key insight that qualitative spatial relationships can be reliably determined from 2D geometric primitives alone. By operating exclusively on 2D bounding boxes from standard object detectors, GRAID avoids both 3D reconstruction errors and generative hallucinations, resulting in datasets that are of higher quality than existing tools that produce similar datasets as validated by human evaluations. We apply our framework to the BDD100k, NuImages, and Waymo datasets, generating over 8.5 million high-quality VQA pairs creating questions spanning spatial relations, counting, ranking, and size comparisons. We evaluate one of the datasets and find it achieves 91.16% human-validated accuracyx2014compared to 57.6% on a dataset generated by recent work. Critically, we demonstrate that when trained on GRAID data, models learn spatial reasoning concepts that generalize: models fine-tuned on 6 question types improve on over 10 held-out types, with accuracy gains of 47.5% on BDD and 37.9% on NuImages for Llama 3.2B 11B, and when trained on all questions types, achieve improvements on several existing benchmarks such as BLINK. The GRAID framework, datasets, and additional information can be found this https URL{here}.

Diagrams serve as a fundamental form of visual language, representing complex concepts and their inter-relationships through structured symbols, shapes, and spatial arrangements. Unlike natural images, their inherently symbolic and abstract nature poses significant challenges for Multimodal Large Language Models (MLLMs). However, current benchmarks conflate perceptual and reasoning tasks, making it difficult to assess whether MLLMs genuinely understand mathematical diagrams beyond superficial pattern recognition. To address this gap, we introduce MATHGLANCE, a benchmark specifically designed to isolate and evaluate mathematical perception in MLLMs. MATHGLANCE comprises 1.2K images and 1.6K carefully curated questions spanning four perception tasks: shape classification, object counting, relationship identification, and object grounding, covering diverse domains including plane geometry, solid geometry, and graphical representations. Our evaluation of MLLMs reveals that their ability to understand diagrams is notably limited, particularly in fine-grained grounding tasks. In response, we construct GeoPeP, a perception-oriented dataset of 200K structured geometry image-text pairs explicitly annotated with geometric primitives and precise spatial relationships. Training MLLM on GeoPeP leads to significant gains in perceptual accuracy, which in turn substantially improves mathematical reasoning. Our benchmark and dataset establish critical standards for evaluating and advancing multimodal mathematical understanding, providing valuable resources and insights to foster future MLLM research.

Recent advances in multimodal large language models (MLLMs) have enabled richer perceptual grounding for code policy generation in embodied agents. However, most existing systems lack effective mechanisms to adaptively monitor policy execution and repair codes during task completion. In this work, we introduce HyCodePolicy, a hybrid language-based control framework that systematically integrates code synthesis, geometric grounding, perceptual monitoring, and iterative repair into a closed-loop programming cycle for embodied agents. Technically, given a natural language instruction, our system first decomposes it into subgoals and generates an initial executable program grounded in object-centric geometric primitives. The program is then executed in simulation, while a vision-language model (VLM) observes selected checkpoints to detect and localize execution failures and infer failure reasons. By fusing structured execution traces capturing program-level events with VLM-based perceptual feedback, HyCodePolicy infers failure causes and repairs programs. This hybrid dual feedback mechanism enables self-correcting program synthesis with minimal human supervision. Our results demonstrate that HyCodePolicy significantly improves the robustness and sample efficiency of robot manipulation policies, offering a scalable strategy for integrating multimodal reasoning into autonomous decision-making pipelines.

Articulated objects, as prevalent entities in human life, their 3D representations play crucial roles across various applications. However, achieving both high-fidelity textured surface reconstruction and dynamic generation for articulated objects remains challenging for existing methods. In this paper, we present REArtGS, a novel framework that introduces additional geometric and motion constraints to 3D Gaussian primitives, enabling high-quality textured surface reconstruction and generation for articulated objects. Specifically, given multi-view RGB images of arbitrary two states of articulated objects, we first introduce an unbiased Signed Distance Field (SDF) guidance to regularize Gaussian opacity fields, enhancing geometry constraints and improving surface reconstruction quality. Then we establish deformable fields for 3D Gaussians constrained by the kinematic structures of articulated objects, achieving unsupervised generation of surface meshes in unseen states. Extensive experiments on both synthetic and real datasets demonstrate our approach achieves high-quality textured surface reconstruction for given states, and enables high-fidelity surface generation for unseen states. Codes will be released after acceptance and the project website is at https://sites.google.com/view/reartgs/home.

We propose GGAvatar, a novel 3D avatar representation designed to robustly model dynamic head avatars with complex identities and deformations. GGAvatar employs a coarse-to-fine structure, featuring two core modules: Neutral Gaussian Initialization Module and Geometry Morph Adjuster. Neutral Gaussian Initialization Module pairs Gaussian primitives with deformable triangular meshes, employing an adaptive density control strategy to model the geometric structure of the target subject with neutral expressions. Geometry Morph Adjuster introduces deformation bases for each Gaussian in global space, creating fine-grained low-dimensional representations of deformation behaviors to address the Linear Blend Skinning formula's limitations effectively. Extensive experiments show that GGAvatar can produce high-fidelity renderings, outperforming state-of-the-art methods in visual quality and quantitative metrics.

We introduce GeoDANO, a geometric vision-language model (VLM) with a domain-agnostic vision encoder, for solving plane geometry problems. Although VLMs have been employed for solving geometry problems, their ability to recognize geometric features remains insufficiently analyzed. To address this gap, we propose a benchmark that evaluates the recognition of visual geometric features, including primitives such as dots and lines, and relations such as orthogonality. Our preliminary study shows that vision encoders often used in general-purpose VLMs, e.g., OpenCLIP, fail to detect these features and struggle to generalize across domains. We develop GeoCLIP, a CLIP based model trained on synthetic geometric diagram-caption pairs to overcome the limitation. Benchmark results show that GeoCLIP outperforms existing vision encoders in recognizing geometric features. We then propose our VLM, GeoDANO, which augments GeoCLIP with a domain adaptation strategy for unseen diagram styles. GeoDANO outperforms specialized methods for plane geometry problems and GPT-4o on MathVerse.

Text-to-3D generation has achieved remarkable success via large-scale text-to-image diffusion models. Nevertheless, there is no paradigm for scaling up the methodology to urban scale. Urban scenes, characterized by numerous elements, intricate arrangement relationships, and vast scale, present a formidable barrier to the interpretability of ambiguous textual descriptions for effective model optimization. In this work, we surmount the limitations by introducing a compositional 3D layout representation into text-to-3D paradigm, serving as an additional prior. It comprises a set of semantic primitives with simple geometric structures and explicit arrangement relationships, complementing textual descriptions and enabling steerable generation. Upon this, we propose two modifications -- (1) We introduce Layout-Guided Variational Score Distillation to address model optimization inadequacies. It conditions the score distillation sampling process with geometric and semantic constraints of 3D layouts. (2) To handle the unbounded nature of urban scenes, we represent 3D scene with a Scalable Hash Grid structure, incrementally adapting to the growing scale of urban scenes. Extensive experiments substantiate the capability of our framework to scale text-to-3D generation to large-scale urban scenes that cover over 1000m driving distance for the first time. We also present various scene editing demonstrations, showing the powers of steerable urban scene generation. Website: https://urbanarchitect.github.io.

We propose a Geometry-aware Policy Imitation (GPI) approach that rethinks imitation learning by treating demonstrations as geometric curves rather than collections of state-action samples. From these curves, GPI derives distance fields that give rise to two complementary control primitives: a progression flow that advances along expert trajectories and an attraction flow that corrects deviations. Their combination defines a controllable, non-parametric vector field that directly guides robot behavior. This formulation decouples metric learning from policy synthesis, enabling modular adaptation across low-dimensional robot states and high-dimensional perceptual inputs. GPI naturally supports multimodality by preserving distinct demonstrations as separate models and allows efficient composition of new demonstrations through simple additions to the distance field. We evaluate GPI in simulation and on real robots across diverse tasks. Experiments show that GPI achieves higher success rates than diffusion-based policies while running 20 times faster, requiring less memory, and remaining robust to perturbations. These results establish GPI as an efficient, interpretable, and scalable alternative to generative approaches for robotic imitation learning. Project website: https://yimingli1998.github.io/projects/GPI/

This paper tackles the intricate challenge of object removal to update the radiance field using the 3D Gaussian Splatting. The main challenges of this task lie in the preservation of geometric consistency and the maintenance of texture coherence in the presence of the substantial discrete nature of Gaussian primitives. We introduce a robust framework specifically designed to overcome these obstacles. The key insight of our approach is the enhancement of information exchange among visible and invisible areas, facilitating content restoration in terms of both geometry and texture. Our methodology begins with optimizing the positioning of Gaussian primitives to improve geometric consistency across both removed and visible areas, guided by an online registration process informed by monocular depth estimation. Following this, we employ a novel feature propagation mechanism to bolster texture coherence, leveraging a cross-attention design that bridges sampling Gaussians from both uncertain and certain areas. This innovative approach significantly refines the texture coherence within the final radiance field. Extensive experiments validate that our method not only elevates the quality of novel view synthesis for scenes undergoing object removal but also showcases notable efficiency gains in training and rendering speeds.

Diffusion-based 3D generation has made remarkable progress in recent years. However, existing 3D generative models often produce overly dense and unstructured meshes, which stand in stark contrast to the compact, structured, and sharply-edged Computer-Aided Design (CAD) models crafted by human designers. To address this gap, we introduce CADDreamer, a novel approach for generating boundary representations (B-rep) of CAD objects from a single image. CADDreamer employs a primitive-aware multi-view diffusion model that captures both local geometric details and high-level structural semantics during the generation process. By encoding primitive semantics into the color domain, the method leverages the strong priors of pre-trained diffusion models to align with well-defined primitives. This enables the inference of multi-view normal maps and semantic maps from a single image, facilitating the reconstruction of a mesh with primitive labels. Furthermore, we introduce geometric optimization techniques and topology-preserving extraction methods to mitigate noise and distortion in the generated primitives. These enhancements result in a complete and seamless B-rep of the CAD model. Experimental results demonstrate that our method effectively recovers high-quality CAD objects from single-view images. Compared to existing 3D generation techniques, the B-rep models produced by CADDreamer are compact in representation, clear in structure, sharp in edges, and watertight in topology.

Dynamic view synthesis (DVS) has advanced remarkably in recent years, achieving high-fidelity rendering while reducing computational costs. Despite the progress, optimizing dynamic neural fields from casual videos remains challenging, as these videos do not provide direct 3D information, such as camera trajectories or the underlying scene geometry. In this work, we present RoDyGS, an optimization pipeline for dynamic Gaussian Splatting from casual videos. It effectively learns motion and underlying geometry of scenes by separating dynamic and static primitives, and ensures that the learned motion and geometry are physically plausible by incorporating motion and geometric regularization terms. We also introduce a comprehensive benchmark, Kubric-MRig, that provides extensive camera and object motion along with simultaneous multi-view captures, features that are absent in previous benchmarks. Experimental results demonstrate that the proposed method significantly outperforms previous pose-free dynamic neural fields and achieves competitive rendering quality compared to existing pose-free static neural fields. The code and data are publicly available at https://rodygs.github.io/.

Accurate 3D shape abstraction from a single 2D image is a long-standing problem in computer vision and graphics. By leveraging a set of primitives to represent the target shape, recent methods have achieved promising results. However, these methods either use a relatively large number of primitives or lack geometric flexibility due to the limited expressibility of the primitives. In this paper, we propose a novel bi-channel Transformer architecture, integrated with parameterized deformable models, termed DeFormer, to simultaneously estimate the global and local deformations of primitives. In this way, DeFormer can abstract complex object shapes while using a small number of primitives which offer a broader geometry coverage and finer details. Then, we introduce a force-driven dynamic fitting and a cycle-consistent re-projection loss to optimize the primitive parameters. Extensive experiments on ShapeNet across various settings show that DeFormer achieves better reconstruction accuracy over the state-of-the-art, and visualizes with consistent semantic correspondences for improved interpretability.

Computer-Aided Design (CAD) generative modeling is driving significant innovations across industrial applications. Recent works have shown remarkable progress in creating solid models from various inputs such as point clouds, meshes, and text descriptions. However, these methods fundamentally diverge from traditional industrial workflows that begin with 2D engineering drawings. The automatic generation of parametric CAD models from these 2D vector drawings remains underexplored despite being a critical step in engineering design. To address this gap, our key insight is to reframe CAD generation as a sequence-to-sequence learning problem where vector drawing primitives directly inform the generation of parametric CAD operations, preserving geometric precision and design intent throughout the transformation process. We propose Drawing2CAD, a framework with three key technical components: a network-friendly vector primitive representation that preserves precise geometric information, a dual-decoder transformer architecture that decouples command type and parameter generation while maintaining precise correspondence, and a soft target distribution loss function accommodating inherent flexibility in CAD parameters. To train and evaluate Drawing2CAD, we create CAD-VGDrawing, a dataset of paired engineering drawings and parametric CAD models, and conduct thorough experiments to demonstrate the effectiveness of our method. Code and dataset are available at https://github.com/lllssc/Drawing2CAD.

We introduce SPFSplatV2, an efficient feed-forward framework for 3D Gaussian splatting from sparse multi-view images, requiring no ground-truth poses during training and inference. It employs a shared feature extraction backbone, enabling simultaneous prediction of 3D Gaussian primitives and camera poses in a canonical space from unposed inputs. A masked attention mechanism is introduced to efficiently estimate target poses during training, while a reprojection loss enforces pixel-aligned Gaussian primitives, providing stronger geometric constraints. We further demonstrate the compatibility of our training framework with different reconstruction architectures, resulting in two model variants. Remarkably, despite the absence of pose supervision, our method achieves state-of-the-art performance in both in-domain and out-of-domain novel view synthesis, even under extreme viewpoint changes and limited image overlap, and surpasses recent methods that rely on geometric supervision for relative pose estimation. By eliminating dependence on ground-truth poses, our method offers the scalability to leverage larger and more diverse datasets. Code and pretrained models will be available on our project page: https://ranrhuang.github.io/spfsplatv2/.

We introduce SPFSplat, an efficient framework for 3D Gaussian splatting from sparse multi-view images, requiring no ground-truth poses during training or inference. It employs a shared feature extraction backbone, enabling simultaneous prediction of 3D Gaussian primitives and camera poses in a canonical space from unposed inputs within a single feed-forward step. Alongside the rendering loss based on estimated novel-view poses, a reprojection loss is integrated to enforce the learning of pixel-aligned Gaussian primitives for enhanced geometric constraints. This pose-free training paradigm and efficient one-step feed-forward design make SPFSplat well-suited for practical applications. Remarkably, despite the absence of pose supervision, SPFSplat achieves state-of-the-art performance in novel view synthesis even under significant viewpoint changes and limited image overlap. It also surpasses recent methods trained with geometry priors in relative pose estimation. Code and trained models are available on our project page: https://ranrhuang.github.io/spfsplat/.

Self-supervised learning (SSL) for point cloud pre-training has become a cornerstone for many 3D vision tasks, enabling effective learning from large-scale unannotated data. At the scene level, existing SSL methods often incorporate volume rendering into the pre-training framework, using RGB-D images as reconstruction signals to facilitate cross-modal learning. This strategy promotes alignment between 2D and 3D modalities and enables the model to benefit from rich visual cues in the RGB-D inputs. However, these approaches are limited by their reliance on implicit scene representations and high memory demands. Furthermore, since their reconstruction objectives are applied only in 2D space, they often fail to capture underlying 3D geometric structures. To address these challenges, we propose Gaussian2Scene, a novel scene-level SSL framework that leverages the efficiency and explicit nature of 3D Gaussian Splatting (3DGS) for pre-training. The use of 3DGS not only alleviates the computational burden associated with volume rendering but also supports direct 3D scene reconstruction, thereby enhancing the geometric understanding of the backbone network. Our approach follows a progressive two-stage training strategy. In the first stage, a dual-branch masked autoencoder learns both 2D and 3D scene representations. In the second stage, we initialize training with reconstructed point clouds and further supervise learning using the geometric locations of Gaussian primitives and rendered RGB images. This process reinforces both geometric and cross-modal learning. We demonstrate the effectiveness of Gaussian2Scene across several downstream 3D object detection tasks, showing consistent improvements over existing pre-training methods.

Existing polygonal surface reconstruction methods heavily depend on input completeness and struggle with incomplete point clouds. We argue that while current point cloud completion techniques may recover missing points, they are not optimized for polygonal surface reconstruction, where the parametric representation of underlying surfaces remains overlooked. To address this gap, we introduce parametric completion, a novel paradigm for point cloud completion, which recovers parametric primitives instead of individual points to convey high-level geometric structures. Our presented approach, PaCo, enables high-quality polygonal surface reconstruction by leveraging plane proxies that encapsulate both plane parameters and inlier points, proving particularly effective in challenging scenarios with highly incomplete data. Comprehensive evaluations of our approach on the ABC dataset establish its effectiveness with superior performance and set a new standard for polygonal surface reconstruction from incomplete data. Project page: https://parametric-completion.github.io.

The inference of topological principles is a key problem in structured reconstruction. We observe that wrongly predicted topological relationships are often incurred by the lack of holistic geometry clues in low-level features. Inspired by the fact that massive signals can be compactly described with frequency analysis, we experimentally explore the efficiency and tendency of learning structure geometry in the frequency domain. Accordingly, we propose a frequency-domain feature learning strategy (F-Learn) to fuse scattered geometric fragments holistically for topology-intact structure reasoning. Benefiting from the parsimonious design, the F-Learn strategy can be easily deployed into a deep reconstructor with a lightweight model modification. Experiments demonstrate that the F-Learn strategy can effectively introduce structure awareness into geometric primitive detection and topology inference, bringing significant performance improvement to final structured reconstruction. Code and pre-trained models are available at https://github.com/Geo-Tell/F-Learn.

We introduce TetSphere Splatting, a Lagrangian geometry representation designed for high-quality 3D shape modeling. TetSphere splatting leverages an underused yet powerful geometric primitive -- volumetric tetrahedral meshes. It represents 3D shapes by deforming a collection of tetrahedral spheres, with geometric regularizations and constraints that effectively resolve common mesh issues such as irregular triangles, non-manifoldness, and floating artifacts. Experimental results on multi-view and single-view reconstruction highlight TetSphere splatting's superior mesh quality while maintaining competitive reconstruction accuracy compared to state-of-the-art methods. Additionally, TetSphere splatting demonstrates versatility by seamlessly integrating into generative modeling tasks, such as image-to-3D and text-to-3D generation.

Despite much progress, achieving real-time high-fidelity head avatar animation is still difficult and existing methods have to trade-off between speed and quality. 3DMM based methods often fail to model non-facial structures such as eyeglasses and hairstyles, while neural implicit models suffer from deformation inflexibility and rendering inefficiency. Although 3D Gaussian has been demonstrated to possess promising capability for geometry representation and radiance field reconstruction, applying 3D Gaussian in head avatar creation remains a major challenge since it is difficult for 3D Gaussian to model the head shape variations caused by changing poses and expressions. In this paper, we introduce PSAvatar, a novel framework for animatable head avatar creation that utilizes discrete geometric primitive to create a parametric morphable shape model and employs 3D Gaussian for fine detail representation and high fidelity rendering. The parametric morphable shape model is a Point-based Morphable Shape Model (PMSM) which uses points instead of meshes for 3D representation to achieve enhanced representation flexibility. The PMSM first converts the FLAME mesh to points by sampling on the surfaces as well as off the meshes to enable the reconstruction of not only surface-like structures but also complex geometries such as eyeglasses and hairstyles. By aligning these points with the head shape in an analysis-by-synthesis manner, the PMSM makes it possible to utilize 3D Gaussian for fine detail representation and appearance modeling, thus enabling the creation of high-fidelity avatars. We show that PSAvatar can reconstruct high-fidelity head avatars of a variety of subjects and the avatars can be animated in real-time (ge 25 fps at a resolution of 512 times 512 ).

Shape primitive abstraction, which decomposes complex 3D shapes into simple geometric elements, plays a crucial role in human visual cognition and has broad applications in computer vision and graphics. While recent advances in 3D content generation have shown remarkable progress, existing primitive abstraction methods either rely on geometric optimization with limited semantic understanding or learn from small-scale, category-specific datasets, struggling to generalize across diverse shape categories. We present PrimitiveAnything, a novel framework that reformulates shape primitive abstraction as a primitive assembly generation task. PrimitiveAnything includes a shape-conditioned primitive transformer for auto-regressive generation and an ambiguity-free parameterization scheme to represent multiple types of primitives in a unified manner. The proposed framework directly learns the process of primitive assembly from large-scale human-crafted abstractions, enabling it to capture how humans decompose complex shapes into primitive elements. Through extensive experiments, we demonstrate that PrimitiveAnything can generate high-quality primitive assemblies that better align with human perception while maintaining geometric fidelity across diverse shape categories. It benefits various 3D applications and shows potential for enabling primitive-based user-generated content (UGC) in games. Project page: https://primitiveanything.github.io

This is only a preview version of GauMesh. Recently, primitive-based rendering has been proven to achieve convincing results in solving the problem of modeling and rendering the 3D dynamic scene from 2D images. Despite this, in the context of novel view synthesis, each type of primitive has its inherent defects in terms of representation ability. It is difficult to exploit the mesh to depict the fuzzy geometry. Meanwhile, the point-based splatting (e.g. the 3D Gaussian Splatting) method usually produces artifacts or blurry pixels in the area with smooth geometry and sharp textures. As a result, it is difficult, even not impossible, to represent the complex and dynamic scene with a single type of primitive. To this end, we propose a novel approach, GauMesh, to bridge the 3D Gaussian and Mesh for modeling and rendering the dynamic scenes. Given a sequence of tracked mesh as initialization, our goal is to simultaneously optimize the mesh geometry, color texture, opacity maps, a set of 3D Gaussians, and the deformation field. At a specific time, we perform alpha-blending on the RGB and opacity values based on the merged and re-ordered z-buffers from mesh and 3D Gaussian rasterizations. This produces the final rendering, which is supervised by the ground-truth image. Experiments demonstrate that our approach adapts the appropriate type of primitives to represent the different parts of the dynamic scene and outperforms all the baseline methods in both quantitative and qualitative comparisons without losing render speed.

Given a set of calibrated images of a scene, we present an approach that produces a simple, compact, and actionable 3D world representation by means of 3D primitives. While many approaches focus on recovering high-fidelity 3D scenes, we focus on parsing a scene into mid-level 3D representations made of a small set of textured primitives. Such representations are interpretable, easy to manipulate and suited for physics-based simulations. Moreover, unlike existing primitive decomposition methods that rely on 3D input data, our approach operates directly on images through differentiable rendering. Specifically, we model primitives as textured superquadric meshes and optimize their parameters from scratch with an image rendering loss. We highlight the importance of modeling transparency for each primitive, which is critical for optimization and also enables handling varying numbers of primitives. We show that the resulting textured primitives faithfully reconstruct the input images and accurately model the visible 3D points, while providing amodal shape completions of unseen object regions. We compare our approach to the state of the art on diverse scenes from DTU, and demonstrate its robustness on real-life captures from BlendedMVS and Nerfstudio. We also showcase how our results can be used to effortlessly edit a scene or perform physical simulations. Code and video results are available at https://www.tmonnier.com/DBW .

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges, such as handling thin structures and non-watertight geometries, which limit their flexibility and accuracy. In contrast, we propose a novel geometric data representation that models geometry as distributions-a powerful representation that makes no assumptions about surface genus, connectivity, or boundary conditions. Our approach uses diffusion models with a novel network architecture to learn surface point distributions, capturing fine-grained geometric details. We evaluate our representation qualitatively and quantitatively across various object types, demonstrating its effectiveness in achieving high geometric fidelity. Additionally, we explore applications using our representation, such as textured mesh representation, neural surface compression, dynamic object modeling, and rendering, highlighting its potential to advance 3D geometric learning.

We describe a method to parse a complex, cluttered indoor scene into primitives which offer a parsimonious abstraction of scene structure. Our primitives are simple convexes. Our method uses a learned regression procedure to parse a scene into a fixed number of convexes from RGBD input, and can optionally accept segmentations to improve the decomposition. The result is then polished with a descent method which adjusts the convexes to produce a very good fit, and greedily removes superfluous primitives. Because the entire scene is parsed, we can evaluate using traditional depth, normal, and segmentation error metrics. Our evaluation procedure demonstrates that the error from our primitive representation is comparable to that of predicting depth from a single image.

Problems involving geometric data arise in a variety of fields, including computer vision, robotics, chemistry, and physics. Such data can take numerous forms, such as points, direction vectors, planes, or transformations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric algebra, which offers an efficient 16-dimensional vector space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a transformer, GATr is scalable, expressive, and versatile. In experiments with n-body modeling and robotic planning, GATr shows strong improvements over non-geometric baselines.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

Existing RGB-based imitation learning approaches typically employ traditional vision encoders such as ResNet or ViT, which lack explicit 3D reasoning capabilities. Recent geometry-grounded vision models, such as VGGT~wang2025vggt, provide robust spatial understanding and are promising candidates to address this limitation. This work investigates the integration of geometry-aware visual representations into robotic manipulation. Our results suggest that incorporating the geometry-aware vision encoder into imitation learning frameworks, including ACT and DP, yields up to 6.5% improvement over standard vision encoders in success rate across single- and bi-manual manipulation tasks in both simulation and real-world settings. Despite these benefits, most geometry-grounded models require high computational cost, limiting their deployment in practical robotic systems. To address this challenge, we propose eVGGT, an efficient geometry-aware encoder distilled from VGGT. eVGGT is nearly 9 times faster and 5 times smaller than VGGT, while preserving strong 3D reasoning capabilities. Code and pretrained models will be released to facilitate further research in geometry-aware robotics.

Creating CAD digital twins from the physical world is crucial for manufacturing, design, and simulation. However, current methods typically rely on costly 3D scanning with labor-intensive post-processing. To provide a user-friendly design process, we explore the problem of reverse engineering from unconstrained real-world CAD images that can be easily captured by users of all experiences. However, the scarcity of real-world CAD data poses challenges in directly training such models. To tackle these challenges, we propose CADCrafter, an image-to-parametric CAD model generation framework that trains solely on synthetic textureless CAD data while testing on real-world images. To bridge the significant representation disparity between images and parametric CAD models, we introduce a geometry encoder to accurately capture diverse geometric features. Moreover, the texture-invariant properties of the geometric features can also facilitate the generalization to real-world scenarios. Since compiling CAD parameter sequences into explicit CAD models is a non-differentiable process, the network training inherently lacks explicit geometric supervision. To impose geometric validity constraints, we employ direct preference optimization (DPO) to fine-tune our model with the automatic code checker feedback on CAD sequence quality. Furthermore, we collected a real-world dataset, comprised of multi-view images and corresponding CAD command sequence pairs, to evaluate our method. Experimental results demonstrate that our approach can robustly handle real unconstrained CAD images, and even generalize to unseen general objects.

Real-time rendering and animation of humans is a core function in games, movies, and telepresence applications. Existing methods have a number of drawbacks we aim to address with our work. Triangle meshes have difficulty modeling thin structures like hair, volumetric representations like Neural Volumes are too low-resolution given a reasonable memory budget, and high-resolution implicit representations like Neural Radiance Fields are too slow for use in real-time applications. We present Mixture of Volumetric Primitives (MVP), a representation for rendering dynamic 3D content that combines the completeness of volumetric representations with the efficiency of primitive-based rendering, e.g., point-based or mesh-based methods. Our approach achieves this by leveraging spatially shared computation with a deconvolutional architecture and by minimizing computation in empty regions of space with volumetric primitives that can move to cover only occupied regions. Our parameterization supports the integration of correspondence and tracking constraints, while being robust to areas where classical tracking fails, such as around thin or translucent structures and areas with large topological variability. MVP is a hybrid that generalizes both volumetric and primitive-based representations. Through a series of extensive experiments we demonstrate that it inherits the strengths of each, while avoiding many of their limitations. We also compare our approach to several state-of-the-art methods and demonstrate that MVP produces superior results in terms of quality and runtime performance.

We present a novel generative 3D modeling system, coined CraftsMan, which can generate high-fidelity 3D geometries with highly varied shapes, regular mesh topologies, and detailed surfaces, and, notably, allows for refining the geometry in an interactive manner. Despite the significant advancements in 3D generation, existing methods still struggle with lengthy optimization processes, irregular mesh topologies, noisy surfaces, and difficulties in accommodating user edits, consequently impeding their widespread adoption and implementation in 3D modeling software. Our work is inspired by the craftsman, who usually roughs out the holistic figure of the work first and elaborates the surface details subsequently. Specifically, we employ a 3D native diffusion model, which operates on latent space learned from latent set-based 3D representations, to generate coarse geometries with regular mesh topology in seconds. In particular, this process takes as input a text prompt or a reference image and leverages a powerful multi-view (MV) diffusion model to generate multiple views of the coarse geometry, which are fed into our MV-conditioned 3D diffusion model for generating the 3D geometry, significantly improving robustness and generalizability. Following that, a normal-based geometry refiner is used to significantly enhance the surface details. This refinement can be performed automatically, or interactively with user-supplied edits. Extensive experiments demonstrate that our method achieves high efficacy in producing superior-quality 3D assets compared to existing methods. HomePage: https://craftsman3d.github.io/, Code: https://github.com/wyysf-98/CraftsMan

We present a neural architecture that takes as input a 2D or 3D shape and outputs a program that generates the shape. The instructions in our program are based on constructive solid geometry principles, i.e., a set of boolean operations on shape primitives defined recursively. Bottom-up techniques for this shape parsing task rely on primitive detection and are inherently slow since the search space over possible primitive combinations is large. In contrast, our model uses a recurrent neural network that parses the input shape in a top-down manner, which is significantly faster and yields a compact and easy-to-interpret sequence of modeling instructions. Our model is also more effective as a shape detector compared to existing state-of-the-art detection techniques. We finally demonstrate that our network can be trained on novel datasets without ground-truth program annotations through policy gradient techniques.

Mapping high-fidelity 3D geometry to a representation that allows for intuitive edits remains an elusive goal in computer vision and graphics. The key challenge is the need to model both continuous and discrete shape variations. Current approaches, such as implicit shape representation, lack straightforward interpretable encoding, while others that employ procedural methods output coarse geometry. We present GeoCode, a technique for 3D shape synthesis using an intuitively editable parameter space. We build a novel program that enforces a complex set of rules and enables users to perform intuitive and controlled high-level edits that procedurally propagate at a low level to the entire shape. Our program produces high-quality mesh outputs by construction. We use a neural network to map a given point cloud or sketch to our interpretable parameter space. Once produced by our procedural program, shapes can be easily modified. Empirically, we show that GeoCode can infer and recover 3D shapes more accurately compared to existing techniques and we demonstrate its ability to perform controlled local and global shape manipulations.

Recent years have witnessed significant progress in the field of neural surface reconstruction. While the extensive focus was put on volumetric and implicit approaches, a number of works have shown that explicit graphics primitives such as point clouds can significantly reduce computational complexity, without sacrificing the reconstructed surface quality. However, less emphasis has been put on modeling dynamic surfaces with point primitives. In this work, we present a dynamic point field model that combines the representational benefits of explicit point-based graphics with implicit deformation networks to allow efficient modeling of non-rigid 3D surfaces. Using explicit surface primitives also allows us to easily incorporate well-established constraints such as-isometric-as-possible regularisation. While learning this deformation model is prone to local optima when trained in a fully unsupervised manner, we propose to additionally leverage semantic information such as keypoint dynamics to guide the deformation learning. We demonstrate our model with an example application of creating an expressive animatable human avatar from a collection of 3D scans. Here, previous methods mostly rely on variants of the linear blend skinning paradigm, which fundamentally limits the expressivity of such models when dealing with complex cloth appearances such as long skirts. We show the advantages of our dynamic point field framework in terms of its representational power, learning efficiency, and robustness to out-of-distribution novel poses.

Computer-aided design (CAD) is the digital construction of 2D and 3D objects, and is central to a wide range of engineering and manufacturing applications like automobile and aviation. Despite its importance, CAD modeling remains largely a time-intensive, manual task. Recent works have attempted to automate this process with small transformer-based models and handcrafted CAD sequence representations. However, there has been little effort to leverage the potential of large language models (LLMs) for sequential CAD design. In this work, we introduce a new large-scale dataset of more than 170k CAD models annotated with high-quality, human-like descriptions generated with our pipeline based on GPT-4.1. Using this dataset, we fine-tune powerful code-LLMs to generate CAD sequences represented in a JSON-based format from natural language descriptions, demonstrating the viability and effectiveness of this approach for text-conditioned CAD generation. Because simple metrics often fail to reflect the quality of generated objects, we introduce geometric and topological metrics based on sphericity, mean curvature, and Euler characteristic to provide richer structural insights. Our experiments and ablation studies on both synthetic and human-annotated data demonstrate that CADmium is able to automate CAD design, drastically speeding up the design of new objects. The dataset, code, and fine-tuned models are available online.

Grid-based structures are commonly used to encode explicit features for graphics primitives such as images, signed distance functions (SDF), and neural radiance fields (NeRF) due to their simple implementation. However, in n-dimensional space, calculating the value of a sampled point requires interpolating the values of its 2^n neighboring vertices. The exponential scaling with dimension leads to significant computational overheads. To address this issue, we propose a simplex-based approach for encoding graphics primitives. The number of vertices in a simplex-based structure increases linearly with dimension, making it a more efficient and generalizable alternative to grid-based representations. Using the non-axis-aligned simplicial structure property, we derive and prove a coordinate transformation, simplicial subdivision, and barycentric interpolation scheme for efficient sampling, which resembles transformation procedures in the simplex noise algorithm. Finally, we use hash tables to store multiresolution features of all interest points in the simplicial grid, which are passed into a tiny fully connected neural network to parameterize graphics primitives. We implemented a detailed simplex-based structure encoding algorithm in C++ and CUDA using the methods outlined in our approach. In the 2D image fitting task, the proposed method is capable of fitting a giga-pixel image with 9.4% less time compared to the baseline method proposed by instant-ngp, while maintaining the same quality and compression rate. In the volumetric rendering setup, we observe a maximum 41.2% speedup when the samples are dense enough.

With their meaningful geometry and their omnipresence in the 3D world, edges are extremely useful primitives in computer vision. 3D edges comprise of lines and curves, and methods to reconstruct them use either multi-view images or point clouds as input. State-of-the-art image-based methods first learn a 3D edge point cloud then fit 3D edges to it. The edge point cloud is obtained by learning a 3D neural implicit edge field from which the 3D edge points are sampled on a specific level set (0 or 1). However, such methods present two important drawbacks: i) it is not realistic to sample points on exact level sets due to float imprecision and training inaccuracies. Instead, they are sampled within a range of levels so the points do not lie accurately on the 3D edges and require further processing. ii) Such implicit representations are computationally expensive and require long training times. In this paper, we address these two limitations and propose a 3D edge mapping that is simpler, more efficient, and preserves accuracy. Our method learns explicitly the 3D edge points and their edge direction hence bypassing the need for point sampling. It casts a 3D edge point as the center of a 3D Gaussian and the edge direction as the principal axis of the Gaussian. Such a representation has the advantage of being not only geometrically meaningful but also compatible with the efficient training optimization defined in Gaussian Splatting. Results show that the proposed method produces edges as accurate and complete as the state-of-the-art while being an order of magnitude faster. Code is released at https://github.com/kunalchelani/EdgeGaussians.

Radiance fields have emerged as a predominant representation for modeling 3D scene appearance. Neural formulations such as Neural Radiance Fields provide high expressivity but require costly ray marching for rendering, whereas primitive-based methods such as 3D Gaussian Splatting offer real-time efficiency through splatting, yet at the expense of representational power. Inspired by advances in both these directions, we introduce splattable neural primitives, a new volumetric representation that reconciles the expressivity of neural models with the efficiency of primitive-based splatting. Each primitive encodes a bounded neural density field parameterized by a shallow neural network. Our formulation admits an exact analytical solution for line integrals, enabling efficient computation of perspectively accurate splatting kernels. As a result, our representation supports integration along view rays without the need for costly ray marching. The primitives flexibly adapt to scene geometry and, being larger than prior analytic primitives, reduce the number required per scene. On novel-view synthesis benchmarks, our approach matches the quality and speed of 3D Gaussian Splatting while using 10times fewer primitives and 6times fewer parameters. These advantages arise directly from the representation itself, without reliance on complex control or adaptation frameworks. The project page is https://vcai.mpi-inf.mpg.de/projects/SplatNet/.

This is the first paper in a series of work we have accomplished over the past three years. In this paper, we have constructed a consistent formal plane geometry system. This will serve as a crucial bridge between IMO-level plane geometry challenges and readable AI automated reasoning. Within this formal framework, we have been able to seamlessly integrate modern AI models with our formal system. AI is now capable of providing deductive reasoning solutions to IMO-level plane geometry problems, just like handling other natural languages, and these proofs are readable, traceable, and verifiable. We propose the geometry formalization theory (GFT) to guide the development of the geometry formal system. Based on the GFT, we have established the FormalGeo, which consists of 88 geometric predicates and 196 theorems. It can represent, validate, and solve IMO-level geometry problems. we also have crafted the FGPS (formal geometry problem solver) in Python. It serves as both an interactive assistant for verifying problem-solving processes and an automated problem solver. We've annotated the formalgeo7k and formalgeo-imo datasets. The former contains 6,981 (expand to 133,818 through data augmentation) geometry problems, while the latter includes 18 (expand to 2,627 and continuously increasing) IMO-level challenging geometry problems. All annotated problems include detailed formal language descriptions and solutions. Implementation of the formal system and experiments validate the correctness and utility of the GFT. The backward depth-first search method only yields a 2.42% problem-solving failure rate, and we can incorporate deep learning techniques to achieve lower one. The source code of FGPS and datasets are available at https://github.com/BitSecret/FGPS.

We present DIRECT-3D, a diffusion-based 3D generative model for creating high-quality 3D assets (represented by Neural Radiance Fields) from text prompts. Unlike recent 3D generative models that rely on clean and well-aligned 3D data, limiting them to single or few-class generation, our model is directly trained on extensive noisy and unaligned `in-the-wild' 3D assets, mitigating the key challenge (i.e., data scarcity) in large-scale 3D generation. In particular, DIRECT-3D is a tri-plane diffusion model that integrates two innovations: 1) A novel learning framework where noisy data are filtered and aligned automatically during the training process. Specifically, after an initial warm-up phase using a small set of clean data, an iterative optimization is introduced in the diffusion process to explicitly estimate the 3D pose of objects and select beneficial data based on conditional density. 2) An efficient 3D representation that is achieved by disentangling object geometry and color features with two separate conditional diffusion models that are optimized hierarchically. Given a prompt input, our model generates high-quality, high-resolution, realistic, and complex 3D objects with accurate geometric details in seconds. We achieve state-of-the-art performance in both single-class generation and text-to-3D generation. We also demonstrate that DIRECT-3D can serve as a useful 3D geometric prior of objects, for example to alleviate the well-known Janus problem in 2D-lifting methods such as DreamFusion. The code and models are available for research purposes at: https://github.com/qihao067/direct3d.

In this paper, we present Gaussian Splatting based text-to-3D generation (GSGEN), a novel approach for generating high-quality 3D objects. Previous methods suffer from inaccurate geometry and limited fidelity due to the absence of 3D prior and proper representation. We leverage 3D Gaussian Splatting, a recent state-of-the-art representation, to address existing shortcomings by exploiting the explicit nature that enables the incorporation of 3D prior. Specifically, our method adopts a progressive optimization strategy, which includes a geometry optimization stage and an appearance refinement stage. In geometry optimization, a coarse representation is established under a 3D geometry prior along with the ordinary 2D SDS loss, ensuring a sensible and 3D-consistent rough shape. Subsequently, the obtained Gaussians undergo an iterative refinement to enrich details. In this stage, we increase the number of Gaussians by compactness-based densification to enhance continuity and improve fidelity. With these designs, our approach can generate 3D content with delicate details and more accurate geometry. Extensive evaluations demonstrate the effectiveness of our method, especially for capturing high-frequency components. Video results are provided at https://gsgen3d.github.io. Our code is available at https://github.com/gsgen3d/gsgen

This work studies the problem of panoptic symbol spotting, which is to spot and parse both countable object instances (windows, doors, tables, etc.) and uncountable stuff (wall, railing, etc.) from computer-aided design (CAD) drawings. Existing methods typically involve either rasterizing the vector graphics into images and using image-based methods for symbol spotting, or directly building graphs and using graph neural networks for symbol recognition. In this paper, we take a different approach, which treats graphic primitives as a set of 2D points that are locally connected and use point cloud segmentation methods to tackle it. Specifically, we utilize a point transformer to extract the primitive features and append a mask2former-like spotting head to predict the final output. To better use the local connection information of primitives and enhance their discriminability, we further propose the attention with connection module (ACM) and contrastive connection learning scheme (CCL). Finally, we propose a KNN interpolation mechanism for the mask attention module of the spotting head to better handle primitive mask downsampling, which is primitive-level in contrast to pixel-level for the image. Our approach, named SymPoint, is simple yet effective, outperforming recent state-of-the-art method GAT-CADNet by an absolute increase of 9.6% PQ and 10.4% RQ on the FloorPlanCAD dataset. The source code and models will be available at https://github.com/nicehuster/SymPoint.

Text-to-3D generation by distilling pretrained large-scale text-to-image diffusion models has shown great promise but still suffers from inconsistent 3D geometric structures (Janus problems) and severe artifacts. The aforementioned problems mainly stem from 2D diffusion models lacking 3D awareness during the lifting. In this work, we present GeoDream, a novel method that incorporates explicit generalized 3D priors with 2D diffusion priors to enhance the capability of obtaining unambiguous 3D consistent geometric structures without sacrificing diversity or fidelity. Specifically, we first utilize a multi-view diffusion model to generate posed images and then construct cost volume from the predicted image, which serves as native 3D geometric priors, ensuring spatial consistency in 3D space. Subsequently, we further propose to harness 3D geometric priors to unlock the great potential of 3D awareness in 2D diffusion priors via a disentangled design. Notably, disentangling 2D and 3D priors allows us to refine 3D geometric priors further. We justify that the refined 3D geometric priors aid in the 3D-aware capability of 2D diffusion priors, which in turn provides superior guidance for the refinement of 3D geometric priors. Our numerical and visual comparisons demonstrate that GeoDream generates more 3D consistent textured meshes with high-resolution realistic renderings (i.e., 1024 times 1024) and adheres more closely to semantic coherence.

In this paper, we present a deep learning-based framework for solving geometric construction problems through visual reasoning, which is useful for automated geometry theorem proving. Constructible problems in geometry often ask for the sequence of straightedge-and-compass constructions to construct a given goal given some initial setup. Our EuclidNet framework leverages the neural network architecture Mask R-CNN to extract the visual features from the initial setup and goal configuration with extra points of intersection, and then generate possible construction steps as intermediary data models that are used as feedback in the training process for further refinement of the construction step sequence. This process is repeated recursively until either a solution is found, in which case we backtrack the path for a step-by-step construction guide, or the problem is identified as unsolvable. Our EuclidNet framework is validated on complex Japanese Sangaku geometry problems, demonstrating its capacity to leverage backtracking for deep visual reasoning of challenging problems.

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

Proving geometric theorems constitutes a hallmark of visual reasoning combining both intuitive and logical skills. Therefore, automated theorem proving of Olympiad-level geometry problems is considered a notable milestone in human-level automated reasoning. The introduction of AlphaGeometry, a neuro-symbolic model trained with 100 million synthetic samples, marked a major breakthrough. It solved 25 of 30 International Mathematical Olympiad (IMO) problems whereas the reported baseline based on Wu's method solved only ten. In this note, we revisit the IMO-AG-30 Challenge introduced with AlphaGeometry, and find that Wu's method is surprisingly strong. Wu's method alone can solve 15 problems, and some of them are not solved by any of the other methods. This leads to two key findings: (i) Combining Wu's method with the classic synthetic methods of deductive databases and angle, ratio, and distance chasing solves 21 out of 30 methods by just using a CPU-only laptop with a time limit of 5 minutes per problem. Essentially, this classic method solves just 4 problems less than AlphaGeometry and establishes the first fully symbolic baseline strong enough to rival the performance of an IMO silver medalist. (ii) Wu's method even solves 2 of the 5 problems that AlphaGeometry failed to solve. Thus, by combining AlphaGeometry with Wu's method we set a new state-of-the-art for automated theorem proving on IMO-AG-30, solving 27 out of 30 problems, the first AI method which outperforms an IMO gold medalist.

This paper addresses the challenge of reconstructing dynamic 3D scenes with complex motions. Some recent works define 3D Gaussian primitives in the canonical space and use deformation fields to map canonical primitives to observation spaces, achieving real-time dynamic view synthesis. However, these methods often struggle to handle scenes with complex motions due to the difficulty of optimizing deformation fields. To overcome this problem, we propose FreeTimeGS, a novel 4D representation that allows Gaussian primitives to appear at arbitrary time and locations. In contrast to canonical Gaussian primitives, our representation possesses the strong flexibility, thus improving the ability to model dynamic 3D scenes. In addition, we endow each Gaussian primitive with an motion function, allowing it to move to neighboring regions over time, which reduces the temporal redundancy. Experiments results on several datasets show that the rendering quality of our method outperforms recent methods by a large margin.

3D Gaussian Splatting (3DGS) has made significant strides in novel view synthesis but is limited by the substantial number of Gaussian primitives required, posing challenges for deployment on lightweight devices. Recent methods address this issue by compressing the storage size of densified Gaussians, yet fail to preserve rendering quality and efficiency. To overcome these limitations, we propose ProtoGS to learn Gaussian prototypes to represent Gaussian primitives, significantly reducing the total Gaussian amount without sacrificing visual quality. Our method directly uses Gaussian prototypes to enable efficient rendering and leverage the resulting reconstruction loss to guide prototype learning. To further optimize memory efficiency during training, we incorporate structure-from-motion (SfM) points as anchor points to group Gaussian primitives. Gaussian prototypes are derived within each group by clustering of K-means, and both the anchor points and the prototypes are optimized jointly. Our experiments on real-world and synthetic datasets prove that we outperform existing methods, achieving a substantial reduction in the number of Gaussians, and enabling high rendering speed while maintaining or even enhancing rendering fidelity.

Geometric methods for solving open-world off-road navigation tasks, by learning occupancy and metric maps, provide good generalization but can be brittle in outdoor environments that violate their assumptions (e.g., tall grass). Learning-based methods can directly learn collision-free behavior from raw observations, but are difficult to integrate with standard geometry-based pipelines. This creates an unfortunate conflict -- either use learning and lose out on well-understood geometric navigational components, or do not use it, in favor of extensively hand-tuned geometry-based cost maps. In this work, we reject this dichotomy by designing the learning and non-learning-based components in a way such that they can be effectively combined in a self-supervised manner. Both components contribute to a planning criterion: the learned component contributes predicted traversability as rewards, while the geometric component contributes obstacle cost information. We instantiate and comparatively evaluate our system in both in-distribution and out-of-distribution environments, showing that this approach inherits complementary gains from the learned and geometric components and significantly outperforms either of them. Videos of our results are hosted at https://sites.google.com/view/hybrid-imitative-planning

In this paper we show that visual diffusion models can serve as effective geometric solvers: they can directly reason about geometric problems by working in pixel space. We first demonstrate this on the Inscribed Square Problem, a long-standing problem in geometry that asks whether every Jordan curve contains four points forming a square. We then extend the approach to two other well-known hard geometric problems: the Steiner Tree Problem and the Simple Polygon Problem. Our method treats each problem instance as an image and trains a standard visual diffusion model that transforms Gaussian noise into an image representing a valid approximate solution that closely matches the exact one. The model learns to transform noisy geometric structures into correct configurations, effectively recasting geometric reasoning as image generation. Unlike prior work that necessitates specialized architectures and domain-specific adaptations when applying diffusion to parametric geometric representations, we employ a standard visual diffusion model that operates on the visual representation of the problem. This simplicity highlights a surprising bridge between generative modeling and geometric problem solving. Beyond the specific problems studied here, our results point toward a broader paradigm: operating in image space provides a general and practical framework for approximating notoriously hard problems, and opens the door to tackling a far wider class of challenging geometric tasks.

Existing Large Multimodal Models (LMMs) struggle with mathematical geometric reasoning due to a lack of high-quality image-text paired data. Current geometric data generation approaches, which apply preset templates to generate geometric data or use Large Language Models (LLMs) to rephrase questions and answers (Q&A), unavoidably limit data accuracy and diversity. To synthesize higher-quality data, we propose a two-stage Reverse Chain-of-Thought (R-CoT) geometry problem generation pipeline. First, we introduce GeoChain to produce high-fidelity geometric images and corresponding descriptions highlighting relations among geometric elements. We then design a Reverse A&Q method that reasons step-by-step based on the descriptions and generates questions in reverse from the reasoning results. Experiments demonstrate that the proposed method brings significant and consistent improvements on multiple LMM baselines, achieving new performance records in the 2B, 7B, and 8B settings. Notably, R-CoT-8B significantly outperforms previous state-of-the-art open-source mathematical models by 16.6% on MathVista and 9.2% on GeoQA, while also surpassing the closed-source model GPT-4o by an average of 13% across both datasets. The code is available at https://github.com/dle666/R-CoT.

Recently, methods leveraging diffusion model priors to assist monocular geometric estimation (e.g., depth and normal) have gained significant attention due to their strong generalization ability. However, most existing works focus on estimating geometric properties within the camera coordinate system of individual video frames, neglecting the inherent ability of diffusion models to determine inter-frame correspondence. In this work, we demonstrate that, through appropriate design and fine-tuning, the intrinsic consistency of video generation models can be effectively harnessed for consistent geometric estimation. Specifically, we 1) select geometric attributes in the global coordinate system that share the same correspondence with video frames as the prediction targets, 2) introduce a novel and efficient conditioning method by reusing positional encodings, and 3) enhance performance through joint training on multiple geometric attributes that share the same correspondence. Our results achieve superior performance in predicting global geometric attributes in videos and can be directly applied to reconstruction tasks. Even when trained solely on static video data, our approach exhibits the potential to generalize to dynamic video scenes.

Modern 3D generation methods can rapidly create shapes from sparse or single views, but their outputs often lack geometric detail due to computational constraints. We present DetailGen3D, a generative approach specifically designed to enhance these generated 3D shapes. Our key insight is to model the coarse-to-fine transformation directly through data-dependent flows in latent space, avoiding the computational overhead of large-scale 3D generative models. We introduce a token matching strategy that ensures accurate spatial correspondence during refinement, enabling local detail synthesis while preserving global structure. By carefully designing our training data to match the characteristics of synthesized coarse shapes, our method can effectively enhance shapes produced by various 3D generation and reconstruction approaches, from single-view to sparse multi-view inputs. Extensive experiments demonstrate that DetailGen3D achieves high-fidelity geometric detail synthesis while maintaining efficiency in training.

Deep neural networks are prone to adversarial examples that maliciously alter the network's outcome. Due to the increasing popularity of 3D sensors in safety-critical systems and the vast deployment of deep learning models for 3D point sets, there is a growing interest in adversarial attacks and defenses for such models. So far, the research has focused on the semantic level, namely, deep point cloud classifiers. However, point clouds are also widely used in a geometric-related form that includes encoding and reconstructing the geometry. In this work, we are the first to consider the problem of adversarial examples at a geometric level. In this setting, the question is how to craft a small change to a clean source point cloud that leads, after passing through an autoencoder model, to the reconstruction of a different target shape. Our attack is in sharp contrast to existing semantic attacks on 3D point clouds. While such works aim to modify the predicted label by a classifier, we alter the entire reconstructed geometry. Additionally, we demonstrate the robustness of our attack in the case of defense, where we show that remnant characteristics of the target shape are still present at the output after applying the defense to the adversarial input. Our code is publicly available at https://github.com/itailang/geometric_adv.

Recent advances in radiance field reconstruction, such as 3D Gaussian Splatting (3DGS), have achieved high-quality novel view synthesis and fast rendering by representing scenes with compositions of Gaussian primitives. However, 3D Gaussians present several limitations for scene reconstruction. Accurately capturing hard edges is challenging without significantly increasing the number of Gaussians, creating a large memory footprint. Moreover, they struggle to represent flat surfaces, as they are diffused in space. Without hand-crafted regularizers, they tend to disperse irregularly around the actual surface. To circumvent these issues, we introduce a novel method, named 3D Convex Splatting (3DCS), which leverages 3D smooth convexes as primitives for modeling geometrically-meaningful radiance fields from multi-view images. Smooth convex shapes offer greater flexibility than Gaussians, allowing for a better representation of 3D scenes with hard edges and dense volumes using fewer primitives. Powered by our efficient CUDA-based rasterizer, 3DCS achieves superior performance over 3DGS on benchmarks such as Mip-NeRF360, Tanks and Temples, and Deep Blending. Specifically, our method attains an improvement of up to 0.81 in PSNR and 0.026 in LPIPS compared to 3DGS while maintaining high rendering speeds and reducing the number of required primitives. Our results highlight the potential of 3D Convex Splatting to become the new standard for high-quality scene reconstruction and novel view synthesis. Project page: convexsplatting.github.io.

Geometric deep learning, i.e., designing neural networks to handle the ubiquitous geometric data such as point clouds and graphs, have achieved great successes in the last decade. One critical inductive bias is that the model can maintain invariance towards various transformations such as translation, rotation, and scaling. The existing graph neural network (GNN) approaches can only maintain permutation-invariance, failing to guarantee invariance with respect to other transformations. Besides GNNs, other works design sophisticated transformation-invariant layers, which are computationally expensive and difficult to be extended. To solve this problem, we revisit why the existing neural networks cannot maintain transformation invariance when handling geometric data. Our findings show that transformation-invariant and distance-preserving initial representations are sufficient to achieve transformation invariance rather than needing sophisticated neural layer designs. Motivated by these findings, we propose Transformation Invariant Neural Networks (TinvNN), a straightforward and general framework for geometric data. Specifically, we realize transformation-invariant and distance-preserving initial point representations by modifying multi-dimensional scaling before feeding the representations into neural networks. We prove that TinvNN can strictly guarantee transformation invariance, being general and flexible enough to be combined with the existing neural networks. Extensive experimental results on point cloud analysis and combinatorial optimization demonstrate the effectiveness and general applicability of our proposed method. Based on the experimental results, we advocate that TinvNN should be considered a new starting point and an essential baseline for further studies of transformation-invariant geometric deep learning.

We report on a new, simple, modular, and flexible approach for automated generation of illustrations for (readable) synthetic geometry proofs. The underlying proofs are generated using the Larus automated prover for coherent logic, and corresponding illustrations are generated in the GCLC language. Animated illustrations are also supported.

Novel view synthesis has demonstrated impressive progress recently, with 3D Gaussian splatting (3DGS) offering efficient training time and photorealistic real-time rendering. However, reliance on Cartesian coordinates limits 3DGS's performance on distant objects, which is important for reconstructing unbounded outdoor environments. We found that, despite its ultimate simplicity, using homogeneous coordinates, a concept on the projective geometry, for the 3DGS pipeline remarkably improves the rendering accuracies of distant objects. We therefore propose Homogeneous Gaussian Splatting (HoGS) incorporating homogeneous coordinates into the 3DGS framework, providing a unified representation for enhancing near and distant objects. HoGS effectively manages both expansive spatial positions and scales particularly in outdoor unbounded environments by adopting projective geometry principles. Experiments show that HoGS significantly enhances accuracy in reconstructing distant objects while maintaining high-quality rendering of nearby objects, along with fast training speed and real-time rendering capability. Our implementations are available on our project page https://kh129.github.io/hogs/.

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

We present a computational framework that transforms single images into 3D physical objects. The visual geometry of a physical object in an image is determined by three orthogonal attributes: mechanical properties, external forces, and rest-shape geometry. Existing single-view 3D reconstruction methods often overlook this underlying composition, presuming rigidity or neglecting external forces. Consequently, the reconstructed objects fail to withstand real-world physical forces, resulting in instability or undesirable deformation -- diverging from their intended designs as depicted in the image. Our optimization framework addresses this by embedding physical compatibility into the reconstruction process. We explicitly decompose the three physical attributes and link them through static equilibrium, which serves as a hard constraint, ensuring that the optimized physical shapes exhibit desired physical behaviors. Evaluations on a dataset collected from Objaverse demonstrate that our framework consistently enhances the physical realism of 3D models over existing methods. The utility of our framework extends to practical applications in dynamic simulations and 3D printing, where adherence to physical compatibility is paramount.

Computer-Aided Design (CAD) is a foundational component of industrial prototyping, where models are defined not by raw coordinates but by construction sequences such as sketches and extrusions. This sequential structure enables both efficient prototype initialization and subsequent editing. Text-guided CAD prototyping, which unifies Text-to-CAD generation and CAD editing, has the potential to streamline the entire design pipeline. However, prior work has not explored this setting, largely because standard large language model (LLM) tokenizers decompose CAD sequences into natural-language word pieces, failing to capture primitive-level CAD semantics and hindering attention modules from modeling geometric structure. We conjecture that a multimodal tokenization strategy, aligned with CAD's primitive and structural nature, can provide more effective representations. To this end, we propose CAD-Tokenizer, a framework that represents CAD data with modality-specific tokens using a sequence-based VQ-VAE with primitive-level pooling and constrained decoding. This design produces compact, primitive-aware representations that align with CAD's structural nature. Applied to unified text-guided CAD prototyping, CAD-Tokenizer significantly improves instruction following and generation quality, achieving better quantitative and qualitative performance over both general-purpose LLMs and task-specific baselines.

Geometry problems are a crucial testbed for AI reasoning capabilities. Most existing geometry solving systems cannot express problems within a unified framework, thus are difficult to integrate with other mathematical fields. Besides, since most geometric proofs rely on intuitive diagrams, verifying geometry problems is particularly challenging. To address these gaps, we introduce LeanGeo, a unified formal system for formalizing and solving competition-level geometry problems within the Lean 4 theorem prover. LeanGeo features a comprehensive library of high-level geometric theorems with Lean's foundational logic, enabling rigorous proof verification and seamless integration with Mathlib. We also present LeanGeo-Bench, a formal geometry benchmark in LeanGeo, comprising problems from the International Mathematical Olympiad (IMO) and other advanced sources. Our evaluation demonstrates the capabilities and limitations of state-of-the-art Large Language Models on this benchmark, highlighting the need for further advancements in automated geometric reasoning. We open source the theorem library and the benchmark of LeanGeo at https://github.com/project-numina/LeanGeo/tree/master.

Geometric Deep Learning has recently made striking progress with the advent of continuous deep implicit fields. They allow for detailed modeling of watertight surfaces of arbitrary topology while not relying on a 3D Euclidean grid, resulting in a learnable parameterization that is unlimited in resolution. Unfortunately, these methods are often unsuitable for applications that require an explicit mesh-based surface representation because converting an implicit field to such a representation relies on the Marching Cubes algorithm, which cannot be differentiated with respect to the underlying implicit field. In this work, we remove this limitation and introduce a differentiable way to produce explicit surface mesh representations from Deep Implicit Fields. Our key insight is that by reasoning on how implicit field perturbations impact local surface geometry, one can ultimately differentiate the 3D location of surface samples with respect to the underlying deep implicit field. We exploit this to define DeepMesh - an end-to-end differentiable mesh representation that can vary its topology. We validate our theoretical insight through several applications: Single view 3D Reconstruction via Differentiable Rendering, Physically-Driven Shape Optimization, Full Scene 3D Reconstruction from Scans and End-to-End Training. In all cases our end-to-end differentiable parameterization gives us an edge over state-of-the-art algorithms.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

Recent advances in dense 3D reconstruction have led to significant progress, yet achieving accurate unified geometric prediction remains a major challenge. Most existing methods are limited to predicting a single geometry quantity from input images. However, geometric quantities such as depth, surface normals, and point maps are inherently correlated, and estimating them in isolation often fails to ensure consistency, thereby limiting both accuracy and practical applicability. This motivates us to explore a unified framework that explicitly models the structural coupling among different geometric properties to enable joint regression. In this paper, we present Dens3R, a 3D foundation model designed for joint geometric dense prediction and adaptable to a wide range of downstream tasks. Dens3R adopts a two-stage training framework to progressively build a pointmap representation that is both generalizable and intrinsically invariant. Specifically, we design a lightweight shared encoder-decoder backbone and introduce position-interpolated rotary positional encoding to maintain expressive power while enhancing robustness to high-resolution inputs. By integrating image-pair matching features with intrinsic invariance modeling, Dens3R accurately regresses multiple geometric quantities such as surface normals and depth, achieving consistent geometry perception from single-view to multi-view inputs. Additionally, we propose a post-processing pipeline that supports geometrically consistent multi-view inference. Extensive experiments demonstrate the superior performance of Dens3R across various dense 3D prediction tasks and highlight its potential for broader applications.

In document image rectification, there exist rich geometric constraints between the distorted image and the ground truth one. However, such geometric constraints are largely ignored in existing advanced solutions, which limits the rectification performance. To this end, we present DocGeoNet for document image rectification by introducing explicit geometric representation. Technically, two typical attributes of the document image are involved in the proposed geometric representation learning, i.e., 3D shape and textlines. Our motivation arises from the insight that 3D shape provides global unwarping cues for rectifying a distorted document image while overlooking the local structure. On the other hand, textlines complementarily provide explicit geometric constraints for local patterns. The learned geometric representation effectively bridges the distorted image and the ground truth one. Extensive experiments show the effectiveness of our framework and demonstrate the superiority of our DocGeoNet over state-of-the-art methods on both the DocUNet Benchmark dataset and our proposed DIR300 test set. The code is available at https://github.com/fh2019ustc/DocGeoNet.

A triangular mesh is one of the most popular 3D data representations. As such, the deployment of deep neural networks for mesh processing is widely spread and is increasingly attracting more attention. However, neural networks are prone to adversarial attacks, where carefully crafted inputs impair the model's functionality. The need to explore these vulnerabilities is a fundamental factor in the future development of 3D-based applications. Recently, mesh attacks were studied on the semantic level, where classifiers are misled to produce wrong predictions. Nevertheless, mesh surfaces possess complex geometric attributes beyond their semantic meaning, and their analysis often includes the need to encode and reconstruct the geometry of the shape. We propose a novel framework for a geometric adversarial attack on a 3D mesh autoencoder. In this setting, an adversarial input mesh deceives the autoencoder by forcing it to reconstruct a different geometric shape at its output. The malicious input is produced by perturbing a clean shape in the spectral domain. Our method leverages the spectral decomposition of the mesh along with additional mesh-related properties to obtain visually credible results that consider the delicacy of surface distortions. Our code is publicly available at https://github.com/StolikTomer/SAGA.

Generative models have shown great promise in generating 3D geometric systems, which is a fundamental problem in many natural science domains such as molecule and protein design. However, existing approaches only operate on static structures, neglecting the fact that physical systems are always dynamic in nature. In this work, we propose geometric trajectory diffusion models (GeoTDM), the first diffusion model for modeling the temporal distribution of 3D geometric trajectories. Modeling such distribution is challenging as it requires capturing both the complex spatial interactions with physical symmetries and temporal correspondence encapsulated in the dynamics. We theoretically justify that diffusion models with equivariant temporal kernels can lead to density with desired symmetry, and develop a novel transition kernel leveraging SE(3)-equivariant spatial convolution and temporal attention. Furthermore, to induce an expressive trajectory distribution for conditional generation, we introduce a generalized learnable geometric prior into the forward diffusion process to enhance temporal conditioning. We conduct extensive experiments on both unconditional and conditional generation in various scenarios, including physical simulation, molecular dynamics, and pedestrian motion. Empirical results on a wide suite of metrics demonstrate that GeoTDM can generate realistic geometric trajectories with significantly higher quality.

We introduce PyTorch Geometric, a library for deep learning on irregularly structured input data such as graphs, point clouds and manifolds, built upon PyTorch. In addition to general graph data structures and processing methods, it contains a variety of recently published methods from the domains of relational learning and 3D data processing. PyTorch Geometric achieves high data throughput by leveraging sparse GPU acceleration, by providing dedicated CUDA kernels and by introducing efficient mini-batch handling for input examples of different size. In this work, we present the library in detail and perform a comprehensive comparative study of the implemented methods in homogeneous evaluation scenarios.

Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of geometric deep learning. Often problems in the physical sciences deal with relatively small sets of points in two- or three-dimensional space wherein translation, rotation, and permutation equivariance are important or even vital for models to be useful in practice. In this work, we present rotation- and permutation-equivariant architectures for deep learning on these small point clouds, composed of a set of products of terms from the geometric algebra and reductions over those products using an attention mechanism. The geometric algebra provides valuable mathematical structure by which to combine vector, scalar, and other types of geometric inputs in a systematic way to account for rotation invariance or covariance, while attention yields a powerful way to impose permutation equivariance. We demonstrate the usefulness of these architectures by training models to solve sample problems relevant to physics, chemistry, and biology.

3D Gaussian Splatting (3DGS) integrates the strengths of primitive-based representations and volumetric rendering techniques, enabling real-time, high-quality rendering. However, 3DGS models typically overfit to single-scene training and are highly sensitive to the initialization of Gaussian ellipsoids, heuristically derived from Structure from Motion (SfM) point clouds, which limits both generalization and practicality. To address these limitations, we propose GS-Net, a generalizable, plug-and-play 3DGS module that densifies Gaussian ellipsoids from sparse SfM point clouds, enhancing geometric structure representation. To the best of our knowledge, GS-Net is the first plug-and-play 3DGS module with cross-scene generalization capabilities. Additionally, we introduce the CARLA-NVS dataset, which incorporates additional camera viewpoints to thoroughly evaluate reconstruction and rendering quality. Extensive experiments demonstrate that applying GS-Net to 3DGS yields a PSNR improvement of 2.08 dB for conventional viewpoints and 1.86 dB for novel viewpoints, confirming the method's effectiveness and robustness.

We introduce pi^3, a feed-forward neural network that offers a novel approach to visual geometry reconstruction, breaking the reliance on a conventional fixed reference view. Previous methods often anchor their reconstructions to a designated viewpoint, an inductive bias that can lead to instability and failures if the reference is suboptimal. In contrast, pi^3 employs a fully permutation-equivariant architecture to predict affine-invariant camera poses and scale-invariant local point maps without any reference frames. This design makes our model inherently robust to input ordering and highly scalable. These advantages enable our simple and bias-free approach to achieve state-of-the-art performance on a wide range of tasks, including camera pose estimation, monocular/video depth estimation, and dense point map reconstruction. Code and models are publicly available.

Polygonal meshes are ubiquitous, but have only played a relatively minor role in the deep learning revolution. State-of-the-art neural generative models for 3D shapes learn implicit functions and generate meshes via expensive iso-surfacing. We overcome these challenges by employing a classical spatial data structure from computer graphics, Binary Space Partitioning (BSP), to facilitate 3D learning. The core operation of BSP involves recursive subdivision of 3D space to obtain convex sets. By exploiting this property, we devise BSP-Net, a network that learns to represent a 3D shape via convex decomposition without supervision. The network is trained to reconstruct a shape using a set of convexes obtained from a BSP-tree built over a set of planes, where the planes and convexes are both defined by learned network weights. BSP-Net directly outputs polygonal meshes from the inferred convexes. The generated meshes are watertight, compact (i.e., low-poly), and well suited to represent sharp geometry. We show that the reconstruction quality by BSP-Net is competitive with those from state-of-the-art methods while using much fewer primitives. We also explore variations to BSP-Net including using a more generic decoder for reconstruction, more general primitives than planes, as well as training a generative model with variational auto-encoders. Code is available at https://github.com/czq142857/BSP-NET-original.

The increasing demand for high-quality 3D assets across various industries necessitates efficient and automated 3D content creation. Despite recent advancements in 3D generative models, existing methods still face challenges with optimization speed, geometric fidelity, and the lack of assets for physically based rendering (PBR). In this paper, we introduce 3DTopia-XL, a scalable native 3D generative model designed to overcome these limitations. 3DTopia-XL leverages a novel primitive-based 3D representation, PrimX, which encodes detailed shape, albedo, and material field into a compact tensorial format, facilitating the modeling of high-resolution geometry with PBR assets. On top of the novel representation, we propose a generative framework based on Diffusion Transformer (DiT), which comprises 1) Primitive Patch Compression, 2) and Latent Primitive Diffusion. 3DTopia-XL learns to generate high-quality 3D assets from textual or visual inputs. We conduct extensive qualitative and quantitative experiments to demonstrate that 3DTopia-XL significantly outperforms existing methods in generating high-quality 3D assets with fine-grained textures and materials, efficiently bridging the quality gap between generative models and real-world applications.

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

Neural rendering techniques have made substantial progress in generating photo-realistic 3D scenes. The latest 3D Gaussian Splatting technique has achieved high quality novel view synthesis as well as fast rendering speed. However, 3D Gaussians lack proficiency in defining accurate 3D geometric structures despite their explicit primitive representations. This is due to the fact that Gaussian's attributes are primarily tailored and fine-tuned for rendering diverse 2D images by their anisotropic nature. To pave the way for efficient 3D reconstruction, we present Spherical Gaussians, a simple and effective representation for 3D geometric boundaries, from which we can directly reconstruct 3D feature curves from a set of calibrated multi-view images. Spherical Gaussians is optimized from grid initialization with a view-based rendering loss, where a 2D edge map is rendered at a specific view and then compared to the ground-truth edge map extracted from the corresponding image, without the need for any 3D guidance or supervision. Given Spherical Gaussians serve as intermedia for the robust edge representation, we further introduce a novel optimization-based algorithm called SGCR to directly extract accurate parametric curves from aligned Spherical Gaussians. We demonstrate that SGCR outperforms existing state-of-the-art methods in 3D edge reconstruction while enjoying great efficiency.

Geometric Deep Learning has recently made striking progress with the advent of continuous Deep Implicit Fields. They allow for detailed modeling of watertight surfaces of arbitrary topology while not relying on a 3D Euclidean grid, resulting in a learnable parameterization that is not limited in resolution. Unfortunately, these methods are often not suitable for applications that require an explicit mesh-based surface representation because converting an implicit field to such a representation relies on the Marching Cubes algorithm, which cannot be differentiated with respect to the underlying implicit field. In this work, we remove this limitation and introduce a differentiable way to produce explicit surface mesh representations from Deep Signed Distance Functions. Our key insight is that by reasoning on how implicit field perturbations impact local surface geometry, one can ultimately differentiate the 3D location of surface samples with respect to the underlying deep implicit field. We exploit this to define MeshSDF, an end-to-end differentiable mesh representation which can vary its topology. We use two different applications to validate our theoretical insight: Single-View Reconstruction via Differentiable Rendering and Physically-Driven Shape Optimization. In both cases our differentiable parameterization gives us an edge over state-of-the-art algorithms.

We introduce a new generative approach for synthesizing 3D geometry and images from single-view collections. Most existing approaches predict volumetric density to render multi-view consistent images. By employing volumetric rendering using neural radiance fields, they inherit a key limitation: the generated geometry is noisy and unconstrained, limiting the quality and utility of the output meshes. To address this issue, we propose GeoGen, a new SDF-based 3D generative model trained in an end-to-end manner. Initially, we reinterpret the volumetric density as a Signed Distance Function (SDF). This allows us to introduce useful priors to generate valid meshes. However, those priors prevent the generative model from learning details, limiting the applicability of the method to real-world scenarios. To alleviate that problem, we make the transformation learnable and constrain the rendered depth map to be consistent with the zero-level set of the SDF. Through the lens of adversarial training, we encourage the network to produce higher fidelity details on the output meshes. For evaluation, we introduce a synthetic dataset of human avatars captured from 360-degree camera angles, to overcome the challenges presented by real-world datasets, which often lack 3D consistency and do not cover all camera angles. Our experiments on multiple datasets show that GeoGen produces visually and quantitatively better geometry than the previous generative models based on neural radiance fields.

Recent advances in discriminative and generative pretraining have yielded geometry estimation models with strong generalization capabilities. While discriminative monocular geometry estimation methods rely on large-scale fine-tuning data to achieve zero-shot generalization, several generative-based paradigms show the potential of achieving impressive generalization performance on unseen scenes by leveraging pre-trained diffusion models and fine-tuning on even a small scale of synthetic training data. Frustratingly, these models are trained with different recipes on different datasets, making it hard to find out the critical factors that determine the evaluation performance. Besides, current geometry evaluation benchmarks have two main drawbacks that may prevent the development of the field, i.e., limited scene diversity and unfavorable label quality. To resolve the above issues, (1) we build fair and strong baselines in a unified codebase for evaluating and analyzing the geometry estimation models; (2) we evaluate monocular geometry estimators on more challenging benchmarks for geometry estimation task with diverse scenes and high-quality annotations. Our results reveal that pre-trained using large data, discriminative models such as DINOv2, can outperform generative counterparts with a small amount of high-quality synthetic data under the same training configuration, which suggests that fine-tuning data quality is a more important factor than the data scale and model architecture. Our observation also raises a question: if simply fine-tuning a general vision model such as DINOv2 using a small amount of synthetic depth data produces SOTA results, do we really need complex generative models for depth estimation? We believe this work can propel advancements in geometry estimation tasks as well as a wide range of downstream applications.

Reconstructing 3D scenes and synthesizing novel views has seen rapid progress in recent years. Neural Radiance Fields demonstrated that continuous volumetric radiance fields can achieve high-quality image synthesis, but their long training and rendering times limit practicality. 3D Gaussian Splatting (3DGS) addressed these issues by representing scenes with millions of Gaussians, enabling real-time rendering and fast optimization. However, Gaussian primitives are not natively compatible with the mesh-based pipelines used in VR headsets, and real-time graphics applications. Existing solutions attempt to convert Gaussians into meshes through post-processing or two-stage pipelines, which increases complexity and degrades visual quality. In this work, we introduce Triangle Splatting+, which directly optimizes triangles, the fundamental primitive of computer graphics, within a differentiable splatting framework. We formulate triangle parametrization to enable connectivity through shared vertices, and we design a training strategy that enforces opaque triangles. The final output is immediately usable in standard graphics engines without post-processing. Experiments on the Mip-NeRF360 and Tanks & Temples datasets show that Triangle Splatting+achieves state-of-the-art performance in mesh-based novel view synthesis. Our method surpasses prior splatting approaches in visual fidelity while remaining efficient and fast to training. Moreover, the resulting semi-connected meshes support downstream applications such as physics-based simulation or interactive walkthroughs. The project page is https://trianglesplatting2.github.io/trianglesplatting2/.

The primal sketch is a fundamental representation in Marr's vision theory, which allows for parsimonious image-level processing from 2D to 2.5D perception. This paper takes a further step by computing 3D primal sketch of wireframes from a set of images with known camera poses, in which we take the 2D wireframes in multi-view images as the basis to compute 3D wireframes in a volumetric rendering formulation. In our method, we first propose a NEural Attraction (NEAT) Fields that parameterizes the 3D line segments with coordinate Multi-Layer Perceptrons (MLPs), enabling us to learn the 3D line segments from 2D observation without incurring any explicit feature correspondences across views. We then present a novel Global Junction Perceiving (GJP) module to perceive meaningful 3D junctions from the NEAT Fields of 3D line segments by optimizing a randomly initialized high-dimensional latent array and a lightweight decoding MLP. Benefitting from our explicit modeling of 3D junctions, we finally compute the primal sketch of 3D wireframes by attracting the queried 3D line segments to the 3D junctions, significantly simplifying the computation paradigm of 3D wireframe parsing. In experiments, we evaluate our approach on the DTU and BlendedMVS datasets with promising performance obtained. As far as we know, our method is the first approach to achieve high-fidelity 3D wireframe parsing without requiring explicit matching.

We present Exact Volumetric Ellipsoid Rendering (EVER), a method for real-time differentiable emission-only volume rendering. Unlike recent rasterization based approach by 3D Gaussian Splatting (3DGS), our primitive based representation allows for exact volume rendering, rather than alpha compositing 3D Gaussian billboards. As such, unlike 3DGS our formulation does not suffer from popping artifacts and view dependent density, but still achieves frame rates of sim!30 FPS at 720p on an NVIDIA RTX4090. Since our approach is built upon ray tracing it enables effects such as defocus blur and camera distortion (e.g. such as from fisheye cameras), which are difficult to achieve by rasterization. We show that our method is more accurate with fewer blending issues than 3DGS and follow-up work on view-consistent rendering, especially on the challenging large-scale scenes from the Zip-NeRF dataset where it achieves sharpest results among real-time techniques.

In the realm of digital creativity, our potential to craft intricate 3D worlds from imagination is often hampered by the limitations of existing digital tools, which demand extensive expertise and efforts. To narrow this disparity, we introduce CLAY, a 3D geometry and material generator designed to effortlessly transform human imagination into intricate 3D digital structures. CLAY supports classic text or image inputs as well as 3D-aware controls from diverse primitives (multi-view images, voxels, bounding boxes, point clouds, implicit representations, etc). At its core is a large-scale generative model composed of a multi-resolution Variational Autoencoder (VAE) and a minimalistic latent Diffusion Transformer (DiT), to extract rich 3D priors directly from a diverse range of 3D geometries. Specifically, it adopts neural fields to represent continuous and complete surfaces and uses a geometry generative module with pure transformer blocks in latent space. We present a progressive training scheme to train CLAY on an ultra large 3D model dataset obtained through a carefully designed processing pipeline, resulting in a 3D native geometry generator with 1.5 billion parameters. For appearance generation, CLAY sets out to produce physically-based rendering (PBR) textures by employing a multi-view material diffusion model that can generate 2K resolution textures with diffuse, roughness, and metallic modalities. We demonstrate using CLAY for a range of controllable 3D asset creations, from sketchy conceptual designs to production ready assets with intricate details. Even first time users can easily use CLAY to bring their vivid 3D imaginations to life, unleashing unlimited creativity.

Multimodal large language models (MLLMs) have made rapid progress in recent years, yet continue to struggle with low-level visual perception (LLVP) -- particularly the ability to accurately describe the geometric details of an image. This capability is crucial for applications in areas such as robotics, medical image analysis, and manufacturing. In this paper, we first introduce Geoperception, a benchmark designed to evaluate an MLLM's ability to accurately transcribe 2D geometric information from an image. Using this benchmark, we demonstrate the limitations of leading MLLMs, and then conduct a comprehensive empirical study to explore strategies for improving their performance on geometric tasks. Our findings highlight the benefits of certain model architectures, training techniques, and data strategies, including the use of high-fidelity synthetic data and multi-stage training with a data curriculum. Notably, we find that a data curriculum enables models to learn challenging geometry understanding tasks which they fail to learn from scratch. Leveraging these insights, we develop Euclid, a family of models specifically optimized for strong low-level geometric perception. Although purely trained on synthetic multimodal data, Euclid shows strong generalization ability to novel geometry shapes. For instance, Euclid outperforms the best closed-source model, Gemini-1.5-Pro, by up to 58.56% on certain Geoperception benchmark tasks and 10.65% on average across all tasks.

Geometric diagrams are critical in conveying mathematical and scientific concepts, yet traditional diagram generation methods are often manual and resource-intensive. While text-to-image generation has made strides in photorealistic imagery, creating accurate geometric diagrams remains a challenge due to the need for precise spatial relationships and the scarcity of geometry-specific datasets. This paper presents MagicGeo, a training-free framework for generating geometric diagrams from textual descriptions. MagicGeo formulates the diagram generation process as a coordinate optimization problem, ensuring geometric correctness through a formal language solver, and then employs coordinate-aware generation. The framework leverages the strong language translation capability of large language models, while formal mathematical solving ensures geometric correctness. We further introduce MagicGeoBench, a benchmark dataset of 220 geometric diagram descriptions, and demonstrate that MagicGeo outperforms current methods in both qualitative and quantitative evaluations. This work provides a scalable, accurate solution for automated diagram generation, with significant implications for educational and academic applications.

To address the data scarcity associated with 3D assets, 2D-lifting techniques such as Score Distillation Sampling (SDS) have become a widely adopted practice in text-to-3D generation pipelines. However, the diffusion models used in these techniques are prone to viewpoint bias and thus lead to geometric inconsistencies such as the Janus problem. To counter this, we introduce MT3D, a text-to-3D generative model that leverages a high-fidelity 3D object to overcome viewpoint bias and explicitly infuse geometric understanding into the generation pipeline. Firstly, we employ depth maps derived from a high-quality 3D model as control signals to guarantee that the generated 2D images preserve the fundamental shape and structure, thereby reducing the inherent viewpoint bias. Next, we utilize deep geometric moments to ensure geometric consistency in the 3D representation explicitly. By incorporating geometric details from a 3D asset, MT3D enables the creation of diverse and geometrically consistent objects, thereby improving the quality and usability of our 3D representations.

Mathematics olympiads are prestigious competitions, with problem proposing and solving highly honored. Building artificial intelligence that proposes and solves olympiads presents an unresolved challenge in automated theorem discovery and proving, especially in geometry for its combination of numerical and spatial elements. We introduce TongGeometry, a Euclidean geometry system supporting tree-search-based guided problem proposing and solving. The efficient geometry system establishes the most extensive repository of geometry theorems to date: within the same computational budget as the existing state-of-the-art, TongGeometry discovers 6.7 billion geometry theorems requiring auxiliary constructions, including 4.1 billion exhibiting geometric symmetry. Among them, 10 theorems were proposed to regional mathematical olympiads with 3 of TongGeometry's proposals selected in real competitions, earning spots in a national team qualifying exam or a top civil olympiad in China and the US. Guided by fine-tuned large language models, TongGeometry solved all International Mathematical Olympiad geometry in IMO-AG-30, outperforming gold medalists for the first time. It also surpasses the existing state-of-the-art across a broader spectrum of olympiad-level problems. The full capabilities of the system can be utilized on a consumer-grade machine, making the model more accessible and fostering widespread democratization of its use. By analogy, unlike existing systems that merely solve problems like students, TongGeometry acts like a geometry coach, discovering, presenting, and proving theorems.

Recovering high-quality depth maps from compressed sources has gained significant attention due to the limitations of consumer-grade depth cameras and the bandwidth restrictions during data transmission. However, current methods still suffer from two challenges. First, bit-depth compression produces a uniform depth representation in regions with subtle variations, hindering the recovery of detailed information. Second, densely distributed random noise reduces the accuracy of estimating the global geometric structure of the scene. To address these challenges, we propose a novel framework, termed geometry-decoupled network (GDNet), for compressed depth map super-resolution that decouples the high-quality depth map reconstruction process by handling global and detailed geometric features separately. To be specific, we propose the fine geometry detail encoder (FGDE), which is designed to aggregate fine geometry details in high-resolution low-level image features while simultaneously enriching them with complementary information from low-resolution context-level image features. In addition, we develop the global geometry encoder (GGE) that aims at suppressing noise and extracting global geometric information effectively via constructing compact feature representation in a low-rank space. We conduct experiments on multiple benchmark datasets, demonstrating that our GDNet significantly outperforms current methods in terms of geometric consistency and detail recovery. In the ECCV 2024 AIM Compressed Depth Upsampling Challenge, our solution won the 1st place award. Our codes are available at: https://github.com/Ian0926/GDNet.

The recent developments in neural fields have brought phenomenal capabilities to the field of shape generation, but they lack crucial properties, such as incremental control - a fundamental requirement for artistic work. Triangular meshes, on the other hand, are the representation of choice for most geometry related tasks, offering efficiency and intuitive control, but do not lend themselves to neural optimization. To support downstream tasks, previous art typically proposes a two-step approach, where first a shape is generated using neural fields, and then a mesh is extracted for further processing. Instead, in this paper we introduce a hybrid approach that maintains both a mesh and a Signed Distance Field (SDF) representations consistently. Using this representation, we introduce MagicClay - an artist friendly tool for sculpting regions of a mesh according to textual prompts while keeping other regions untouched. Our framework carefully and efficiently balances consistency between the representations and regularizations in every step of the shape optimization; Relying on the mesh representation, we show how to render the SDF at higher resolutions and faster. In addition, we employ recent work in differentiable mesh reconstruction to adaptively allocate triangles in the mesh where required, as indicated by the SDF. Using an implemented prototype, we demonstrate superior generated geometry compared to the state-of-the-art, and novel consistent control, allowing sequential prompt-based edits to the same mesh for the first time.

Estimating the pose of an object from a monocular image is an inverse problem fundamental in computer vision. The ill-posed nature of this problem requires incorporating deformation priors to solve it. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a powerful and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach to inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and obtains state-of-the-art performance without the need for offline training.

It is inherently ambiguous to lift 2D results from pre-trained diffusion models to a 3D world for text-to-3D generation. 2D diffusion models solely learn view-agnostic priors and thus lack 3D knowledge during the lifting, leading to the multi-view inconsistency problem. We find that this problem primarily stems from geometric inconsistency, and avoiding misplaced geometric structures substantially mitigates the problem in the final outputs. Therefore, we improve the consistency by aligning the 2D geometric priors in diffusion models with well-defined 3D shapes during the lifting, addressing the vast majority of the problem. This is achieved by fine-tuning the 2D diffusion model to be viewpoint-aware and to produce view-specific coordinate maps of canonically oriented 3D objects. In our process, only coarse 3D information is used for aligning. This "coarse" alignment not only resolves the multi-view inconsistency in geometries but also retains the ability in 2D diffusion models to generate detailed and diversified high-quality objects unseen in the 3D datasets. Furthermore, our aligned geometric priors (AGP) are generic and can be seamlessly integrated into various state-of-the-art pipelines, obtaining high generalizability in terms of unseen shapes and visual appearance while greatly alleviating the multi-view inconsistency problem. Our method represents a new state-of-the-art performance with an 85+% consistency rate by human evaluation, while many previous methods are around 30%. Our project page is https://sweetdreamer3d.github.io/

In the realm of vision models, the primary mode of representation is using pixels to rasterize the visual world. Yet this is not always the best or unique way to represent visual content, especially for designers and artists who depict the world using geometry primitives such as polygons. Vector graphics (VG), on the other hand, offer a textual representation of visual content, which can be more concise and powerful for content like cartoons or sketches. Recent studies have shown promising results on processing vector graphics with capable Large Language Models (LLMs). However, such works focus solely on qualitative results, understanding, or a specific type of vector graphics. We propose VGBench, a comprehensive benchmark for LLMs on handling vector graphics through diverse aspects, including (a) both visual understanding and generation, (b) evaluation of various vector graphics formats, (c) diverse question types, (d) wide range of prompting techniques, (e) under multiple LLMs. Evaluating on our collected 4279 understanding and 5845 generation samples, we find that LLMs show strong capability on both aspects while exhibiting less desirable performance on low-level formats (SVG). Both data and evaluation pipeline will be open-sourced at https://vgbench.github.io.

Existing fully-supervised point cloud segmentation methods suffer in the dynamic testing environment with emerging new classes. Few-shot point cloud segmentation algorithms address this problem by learning to adapt to new classes at the sacrifice of segmentation accuracy for the base classes, which severely impedes its practicality. This largely motivates us to present the first attempt at a more practical paradigm of generalized few-shot point cloud segmentation, which requires the model to generalize to new categories with only a few support point clouds and simultaneously retain the capability to segment base classes. We propose the geometric words to represent geometric components shared between the base and novel classes, and incorporate them into a novel geometric-aware semantic representation to facilitate better generalization to the new classes without forgetting the old ones. Moreover, we introduce geometric prototypes to guide the segmentation with geometric prior knowledge. Extensive experiments on S3DIS and ScanNet consistently illustrate the superior performance of our method over baseline methods. Our code is available at: https://github.com/Pixie8888/GFS-3DSeg_GWs.

Design of neural networks that incorporate symmetry is crucial for geometric deep learning. Central to this effort is the development of invariant and equivariant operations. This works presents a systematic method for constructing valid invariant and equivariant operations. It can handle inputs and outputs in the form of Cartesian tensors with different rank, as well as spherical tensors with different types. In addition, our method features a graphical representation utilizing the symmetric tensor network, which simplifies both the proofs and constructions related to invariant and equivariant functions. We also apply this approach to design the equivariant interaction message for the geometry graph neural network, and equivariant machine learning model to learn the constitutive law of materials.

Novel view synthesis has long been a practical but challenging task, although the introduction of numerous methods to solve this problem, even combining advanced representations like 3D Gaussian Splatting, they still struggle to recover high-quality results and often consume too much storage memory and training time. In this paper we propose Swift4D, a divide-and-conquer 3D Gaussian Splatting method that can handle static and dynamic primitives separately, achieving a good trade-off between rendering quality and efficiency, motivated by the fact that most of the scene is the static primitive and does not require additional dynamic properties. Concretely, we focus on modeling dynamic transformations only for the dynamic primitives which benefits both efficiency and quality. We first employ a learnable decomposition strategy to separate the primitives, which relies on an additional parameter to classify primitives as static or dynamic. For the dynamic primitives, we employ a compact multi-resolution 4D Hash mapper to transform these primitives from canonical space into deformation space at each timestamp, and then mix the static and dynamic primitives to produce the final output. This divide-and-conquer method facilitates efficient training and reduces storage redundancy. Our method not only achieves state-of-the-art rendering quality while being 20X faster in training than previous SOTA methods with a minimum storage requirement of only 30MB on real-world datasets. Code is available at https://github.com/WuJH2001/swift4d.

Generating high-quality 3D objects from textual descriptions remains a challenging problem due to computational cost, the scarcity of 3D data, and complex 3D representations. We introduce Geometry Image Diffusion (GIMDiffusion), a novel Text-to-3D model that utilizes geometry images to efficiently represent 3D shapes using 2D images, thereby avoiding the need for complex 3D-aware architectures. By integrating a Collaborative Control mechanism, we exploit the rich 2D priors of existing Text-to-Image models such as Stable Diffusion. This enables strong generalization even with limited 3D training data (allowing us to use only high-quality training data) as well as retaining compatibility with guidance techniques such as IPAdapter. In short, GIMDiffusion enables the generation of 3D assets at speeds comparable to current Text-to-Image models. The generated objects consist of semantically meaningful, separate parts and include internal structures, enhancing both usability and versatility.

Automatic math problem solving has recently attracted increasing attention as a long-standing AI benchmark. In this paper, we focus on solving geometric problems, which requires a comprehensive understanding of textual descriptions, visual diagrams, and theorem knowledge. However, the existing methods were highly dependent on handcraft rules and were merely evaluated on small-scale datasets. Therefore, we propose a Geometric Question Answering dataset GeoQA, containing 4,998 geometric problems with corresponding annotated programs, which illustrate the solving process of the given problems. Compared with another publicly available dataset GeoS, GeoQA is 25 times larger, in which the program annotations can provide a practical testbed for future research on explicit and explainable numerical reasoning. Moreover, we introduce a Neural Geometric Solver (NGS) to address geometric problems by comprehensively parsing multimodal information and generating interpretable programs. We further add multiple self-supervised auxiliary tasks on NGS to enhance cross-modal semantic representation. Extensive experiments on GeoQA validate the effectiveness of our proposed NGS and auxiliary tasks. However, the results are still significantly lower than human performance, which leaves large room for future research. Our benchmark and code are released at https://github.com/chen-judge/GeoQA .

Crystal structures are characterized by atomic bases within a primitive unit cell that repeats along a regular lattice throughout 3D space. The periodic and infinite nature of crystals poses unique challenges for geometric graph representation learning. Specifically, constructing graphs that effectively capture the complete geometric information of crystals and handle chiral crystals remains an unsolved and challenging problem. In this paper, we introduce a novel approach that utilizes the periodic patterns of unit cells to establish the lattice-based representation for each atom, enabling efficient and expressive graph representations of crystals. Furthermore, we propose ComFormer, a SE(3) transformer designed specifically for crystalline materials. ComFormer includes two variants; namely, iComFormer that employs invariant geometric descriptors of Euclidean distances and angles, and eComFormer that utilizes equivariant vector representations. Experimental results demonstrate the state-of-the-art predictive accuracy of ComFormer variants on various tasks across three widely-used crystal benchmarks. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

This paper advocates the use of implicit surface representation in autoencoder-based self-supervised 3D representation learning. The most popular and accessible 3D representation, i.e., point clouds, involves discrete samples of the underlying continuous 3D surface. This discretization process introduces sampling variations on the 3D shape, making it challenging to develop transferable knowledge of the true 3D geometry. In the standard autoencoding paradigm, the encoder is compelled to encode not only the 3D geometry but also information on the specific discrete sampling of the 3D shape into the latent code. This is because the point cloud reconstructed by the decoder is considered unacceptable unless there is a perfect mapping between the original and the reconstructed point clouds. This paper introduces the Implicit AutoEncoder (IAE), a simple yet effective method that addresses the sampling variation issue by replacing the commonly-used point-cloud decoder with an implicit decoder. The implicit decoder reconstructs a continuous representation of the 3D shape, independent of the imperfections in the discrete samples. Extensive experiments demonstrate that the proposed IAE achieves state-of-the-art performance across various self-supervised learning benchmarks.

Semantic distillation in radiance fields has spurred significant advances in open-vocabulary robot policies, e.g., in manipulation and navigation, founded on pretrained semantics from large vision models. While prior work has demonstrated the effectiveness of visual-only semantic features (e.g., DINO and CLIP) in Gaussian Splatting and neural radiance fields, the potential benefit of geometry-grounding in distilled fields remains an open question. In principle, visual-geometry features seem very promising for spatial tasks such as pose estimation, prompting the question: Do geometry-grounded semantic features offer an edge in distilled fields? Specifically, we ask three critical questions: First, does spatial-grounding produce higher-fidelity geometry-aware semantic features? We find that image features from geometry-grounded backbones contain finer structural details compared to their counterparts. Secondly, does geometry-grounding improve semantic object localization? We observe no significant difference in this task. Thirdly, does geometry-grounding enable higher-accuracy radiance field inversion? Given the limitations of prior work and their lack of semantics integration, we propose a novel framework SPINE for inverting radiance fields without an initial guess, consisting of two core components: coarse inversion using distilled semantics, and fine inversion using photometric-based optimization. Surprisingly, we find that the pose estimation accuracy decreases with geometry-grounded features. Our results suggest that visual-only features offer greater versatility for a broader range of downstream tasks, although geometry-grounded features contain more geometric detail. Notably, our findings underscore the necessity of future research on effective strategies for geometry-grounding that augment the versatility and performance of pretrained semantic features.

Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as geometric graphs, i.e., graphs with nodes positioned in the Euclidean space given an arbitrarily chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as Galilean invariance. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.

Remarkable advances have been achieved recently in learning neural representations that characterize object geometry, while generating textured objects suitable for downstream applications and 3D rendering remains at an early stage. In particular, reconstructing textured geometry from images of real objects is a significant challenge -- reconstructed geometry is often inexact, making realistic texturing a significant challenge. We present Mesh2Tex, which learns a realistic object texture manifold from uncorrelated collections of 3D object geometry and photorealistic RGB images, by leveraging a hybrid mesh-neural-field texture representation. Our texture representation enables compact encoding of high-resolution textures as a neural field in the barycentric coordinate system of the mesh faces. The learned texture manifold enables effective navigation to generate an object texture for a given 3D object geometry that matches to an input RGB image, which maintains robustness even under challenging real-world scenarios where the mesh geometry approximates an inexact match to the underlying geometry in the RGB image. Mesh2Tex can effectively generate realistic object textures for an object mesh to match real images observations towards digitization of real environments, significantly improving over previous state of the art.

We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the Pin(p,q,r) group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.

Mathematical geometric reasoning is essential for scientific discovery and educational development, requiring precise logic and rigorous formal verification. While recent advances in Multimodal Large Language Models (MLLMs) have improved reasoning tasks, existing models typically struggle with formal geometric reasoning, particularly when dynamically constructing and verifying auxiliary geometric elements. To address these challenges, we introduce Geoint-R1, a multimodal reasoning framework designed to generate formally verifiable geometric solutions from textual descriptions and visual diagrams. Geoint-R1 uniquely integrates auxiliary elements construction, formal reasoning represented via Lean4, and interactive visualization. To systematically evaluate and advance formal geometric reasoning, we propose the Geoint benchmark, comprising 1,885 rigorously annotated geometry problems across diverse topics such as plane, spatial, and solid geometry. Each problem includes structured textual annotations, precise Lean4 code for auxiliary constructions, and detailed solution steps verified by experts. Extensive experiments demonstrate that Geoint-R1 significantly surpasses existing multimodal and math-specific reasoning models, particularly on challenging problems requiring explicit auxiliary element constructions.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).

Although polygon meshes have been a standard representation in geometry processing, their irregular and combinatorial nature hinders their suitability for learning-based applications. In this work, we introduce a novel learnable mesh representation through a set of local 3D sample Points and their associated Normals and Quadric error metrics (QEM) w.r.t. the underlying shape, which we denote PoNQ. A global mesh is directly derived from PoNQ by efficiently leveraging the knowledge of the local quadric errors. Besides marking the first use of QEM within a neural shape representation, our contribution guarantees both topological and geometrical properties by ensuring that a PoNQ mesh does not self-intersect and is always the boundary of a volume. Notably, our representation does not rely on a regular grid, is supervised directly by the target surface alone, and also handles open surfaces with boundaries and/or sharp features. We demonstrate the efficacy of PoNQ through a learning-based mesh prediction from SDF grids and show that our method surpasses recent state-of-the-art techniques in terms of both surface and edge-based metrics.

This paper introduces a method for simplifying textured surface triangle meshes in the wild while maintaining high visual quality. While previous methods achieve excellent results on manifold meshes by using the quadric error metric, they struggle to produce high-quality outputs for meshes in the wild, which typically contain non-manifold elements and multiple connected components. In this work, we propose a method for simplifying these wild textured triangle meshes. We formulate mesh simplification as a problem of decimating simplicial 2-complexes to handle multiple non-manifold mesh components collectively. Building on the success of quadric error simplification, we iteratively collapse 1-simplices (vertex pairs). Our approach employs a modified quadric error that converges to the original quadric error metric for watertight manifold meshes, while significantly improving the results on wild meshes. For textures, instead of following existing strategies to preserve UVs, we adopt a novel perspective which focuses on computing mesh correspondences throughout the decimation, independent of the UV layout. This combination yields a textured mesh simplification system that is capable of handling arbitrary triangle meshes, achieving to high-quality results on wild inputs without sacrificing the excellent performance on clean inputs. Our method guarantees to avoid common problems in textured mesh simplification, including the prevalent problem of texture bleeding. We extensively evaluate our method on multiple datasets, showing improvements over prior techniques through qualitative, quantitative, and user study evaluations.

Recent advances in 3D X-ray Computed Tomographic (CT) sensors have stimulated research efforts to unveil the extremely complex micro-scale processes that control the activity of soil microorganisms. Voxel-based description (up to hundreds millions voxels) of the pore space can be extracted, from grey level 3D CT scanner images, by means of simple image processing tools. Classical methods for numerical simulation of biological dynamics using mesh of voxels, such as Lattice Boltzmann Model (LBM), are too much time consuming. Thus, the use of more compact and reliable geometrical representations of pore space can drastically decrease the computational cost of the simulations. Several recent works propose basic analytic volume primitives (e.g. spheres, generalized cylinders, ellipsoids) to define a piece-wise approximation of pore space for numerical simulation of draining, diffusion and microbial decomposition. Such approaches work well but the drawback is that it generates approximation errors. In the present work, we study another alternative where pore space is described by means of geometrically relevant connected subsets of voxels (regions) computed from the curvilinear skeleton. Indeed, many works use the curvilinear skeleton (3D medial axis) for analyzing and partitioning 3D shapes within various domains (medicine, material sciences, petroleum engineering, etc.) but only a few ones in soil sciences. Within the context of soil sciences, most studies dealing with 3D medial axis focus on the determination of pore throats. Here, we segment pore space using curvilinear skeleton in order to achieve numerical simulation of microbial decomposition (including diffusion processes). We validate simulation outputs by comparison with other methods using different pore space geometrical representations (balls, voxels).

From a single image, visual cues can help deduce intrinsic and extrinsic camera parameters like the focal length and the gravity direction. This single-image calibration can benefit various downstream applications like image editing and 3D mapping. Current approaches to this problem are based on either classical geometry with lines and vanishing points or on deep neural networks trained end-to-end. The learned approaches are more robust but struggle to generalize to new environments and are less accurate than their classical counterparts. We hypothesize that they lack the constraints that 3D geometry provides. In this work, we introduce GeoCalib, a deep neural network that leverages universal rules of 3D geometry through an optimization process. GeoCalib is trained end-to-end to estimate camera parameters and learns to find useful visual cues from the data. Experiments on various benchmarks show that GeoCalib is more robust and more accurate than existing classical and learned approaches. Its internal optimization estimates uncertainties, which help flag failure cases and benefit downstream applications like visual localization. The code and trained models are publicly available at https://github.com/cvg/GeoCalib.

Despite their proficiency in general tasks, Multi-modal Large Language Models (MLLMs) struggle with automatic Geometry Problem Solving (GPS), which demands understanding diagrams, interpreting symbols, and performing complex reasoning. This limitation arises from their pre-training on natural images and texts, along with the lack of automated verification in the problem-solving process. Besides, current geometric specialists are limited by their task-specific designs, making them less effective for broader geometric problems. To this end, we present GeoX, a multi-modal large model focusing on geometric understanding and reasoning tasks. Given the significant differences between geometric diagram-symbol and natural image-text, we introduce unimodal pre-training to develop a diagram encoder and symbol decoder, enhancing the understanding of geometric images and corpora. Furthermore, we introduce geometry-language alignment, an effective pre-training paradigm that bridges the modality gap between unimodal geometric experts. We propose a Generator-And-Sampler Transformer (GS-Former) to generate discriminative queries and eliminate uninformative representations from unevenly distributed geometric signals. Finally, GeoX benefits from visual instruction tuning, empowering it to take geometric images and questions as input and generate verifiable solutions. Experiments show that GeoX outperforms both generalists and geometric specialists on publicly recognized benchmarks, such as GeoQA, UniGeo, Geometry3K, and PGPS9k.

We present a method for estimating neural scenes representations of objects given only a single image. The core of our method is the estimation of a geometric scaffold for the object and its use as a guide for the reconstruction of the underlying radiance field. Our formulation is based on a generative process that first maps a latent code to a voxelized shape, and then renders it to an image, with the object appearance being controlled by a second latent code. During inference, we optimize both the latent codes and the networks to fit a test image of a new object. The explicit disentanglement of shape and appearance allows our model to be fine-tuned given a single image. We can then render new views in a geometrically consistent manner and they represent faithfully the input object. Additionally, our method is able to generalize to images outside of the training domain (more realistic renderings and even real photographs). Finally, the inferred geometric scaffold is itself an accurate estimate of the object's 3D shape. We demonstrate in several experiments the effectiveness of our approach in both synthetic and real images.

Estimating accurate and temporally consistent 3D human geometry from videos is a challenging problem in computer vision. Existing methods, primarily optimized for single images, often suffer from temporal inconsistencies and fail to capture fine-grained dynamic details. To address these limitations, we present GeoMan, a novel architecture designed to produce accurate and temporally consistent depth and normal estimations from monocular human videos. GeoMan addresses two key challenges: the scarcity of high-quality 4D training data and the need for metric depth estimation to accurately model human size. To overcome the first challenge, GeoMan employs an image-based model to estimate depth and normals for the first frame of a video, which then conditions a video diffusion model, reframing video geometry estimation task as an image-to-video generation problem. This design offloads the heavy lifting of geometric estimation to the image model and simplifies the video model's role to focus on intricate details while using priors learned from large-scale video datasets. Consequently, GeoMan improves temporal consistency and generalizability while requiring minimal 4D training data. To address the challenge of accurate human size estimation, we introduce a root-relative depth representation that retains critical human-scale details and is easier to be estimated from monocular inputs, overcoming the limitations of traditional affine-invariant and metric depth representations. GeoMan achieves state-of-the-art performance in both qualitative and quantitative evaluations, demonstrating its effectiveness in overcoming longstanding challenges in 3D human geometry estimation from videos.

We present MeshLLM, a novel framework that leverages large language models (LLMs) to understand and generate text-serialized 3D meshes. Our approach addresses key limitations in existing methods, including the limited dataset scale when catering to LLMs' token length and the loss of 3D structural information during mesh serialization. We introduce a Primitive-Mesh decomposition strategy, which divides 3D meshes into structurally meaningful subunits. This enables the creation of a large-scale dataset with 1500k+ samples, almost 50 times larger than previous methods, which aligns better with the LLM scaling law principles. Furthermore, we propose inferring face connectivity from vertices and local mesh assembly training strategies, significantly enhancing the LLMs' ability to capture mesh topology and spatial structures. Experiments show that MeshLLM outperforms the state-of-the-art LLaMA-Mesh in both mesh generation quality and shape understanding, highlighting its great potential in processing text-serialized 3D meshes.

High-fidelity 3D asset generation is crucial for various industries. While recent 3D pretrained models show strong capability in producing realistic content, most are built upon diffusion models and follow a two-stage pipeline that first generates geometry and then synthesizes appearance. Such a decoupled design tends to produce geometry-texture misalignment and non-negligible cost. In this paper, we propose UniLat3D, a unified framework that encodes geometry and appearance in a single latent space, enabling direct single-stage generation. Our key contribution is a geometry-appearance Unified VAE, which compresses high-resolution sparse features into a compact latent representation -- UniLat. UniLat integrates structural and visual information into a dense low-resolution latent, which can be efficiently decoded into diverse 3D formats, e.g., 3D Gaussians and meshes. Based on this unified representation, we train a single flow-matching model to map Gaussian noise directly into UniLat, eliminating redundant stages. Trained solely on public datasets, UniLat3D produces high-quality 3D assets in seconds from a single image, achieving superior appearance fidelity and geometric quality. More demos \& code are available at https://unilat3d.github.io/

Gaussian splatting has emerged as a powerful 3D representation that harnesses the advantages of both explicit (mesh) and implicit (NeRF) 3D representations. In this paper, we seek to leverage Gaussian splatting to generate realistic animatable avatars from textual descriptions, addressing the limitations (e.g., flexibility and efficiency) imposed by mesh or NeRF-based representations. However, a naive application of Gaussian splatting cannot generate high-quality animatable avatars and suffers from learning instability; it also cannot capture fine avatar geometries and often leads to degenerate body parts. To tackle these problems, we first propose a primitive-based 3D Gaussian representation where Gaussians are defined inside pose-driven primitives to facilitate animation. Second, to stabilize and amortize the learning of millions of Gaussians, we propose to use neural implicit fields to predict the Gaussian attributes (e.g., colors). Finally, to capture fine avatar geometries and extract detailed meshes, we propose a novel SDF-based implicit mesh learning approach for 3D Gaussians that regularizes the underlying geometries and extracts highly detailed textured meshes. Our proposed method, GAvatar, enables the large-scale generation of diverse animatable avatars using only text prompts. GAvatar significantly surpasses existing methods in terms of both appearance and geometry quality, and achieves extremely fast rendering (100 fps) at 1K resolution.

Empirically validating new 3D-printing related algorithms and implementations requires testing data representative of inputs encountered in the wild. An ideal benchmarking dataset should not only draw from the same distribution of shapes people print in terms of class (e.g., toys, mechanisms, jewelry), representation type (e.g., triangle soup meshes) and complexity (e.g., number of facets), but should also capture problems and artifacts endemic to 3D printing models (e.g., self-intersections, non-manifoldness). We observe that the contextual and geometric characteristics of 3D printing models differ significantly from those used for computer graphics applications, not to mention standard models (e.g., Stanford bunny, Armadillo, Fertility). We present a new dataset of 10,000 models collected from an online 3D printing model-sharing database. Via analysis of both geometric (e.g., triangle aspect ratios, manifoldness) and contextual (e.g., licenses, tags, classes) characteristics, we demonstrate that this dataset represents a more concise summary of real-world models used for 3D printing compared to existing datasets. To facilitate future research endeavors, we also present an online query interface to select subsets of the dataset according to project-specific characteristics. The complete dataset and per-model statistical data are freely available to the public.

In this paper, we propose Img2CAD, the first approach to our knowledge that uses 2D image inputs to generate CAD models with editable parameters. Unlike existing AI methods for 3D model generation using text or image inputs often rely on mesh-based representations, which are incompatible with CAD tools and lack editability and fine control, Img2CAD enables seamless integration between AI-based 3D reconstruction and CAD software. We have identified an innovative intermediate representation called Structured Visual Geometry (SVG), characterized by vectorized wireframes extracted from objects. This representation significantly enhances the performance of generating conditioned CAD models. Additionally, we introduce two new datasets to further support research in this area: ABC-mono, the largest known dataset comprising over 200,000 3D CAD models with rendered images, and KOCAD, the first dataset featuring real-world captured objects alongside their ground truth CAD models, supporting further research in conditioned CAD model generation.

Plane geometry problem solving (PGPS) has recently gained significant attention as a benchmark to assess the multi-modal reasoning capabilities of large vision-language models. Despite the growing interest in PGPS, the research community still lacks a comprehensive overview that systematically synthesizes recent work in PGPS. To fill this gap, we present a survey of existing PGPS studies. We first categorize PGPS methods into an encoder-decoder framework and summarize the corresponding output formats used by their encoders and decoders. Subsequently, we classify and analyze these encoders and decoders according to their architectural designs. Finally, we outline major challenges and promising directions for future research. In particular, we discuss the hallucination issues arising during the encoding phase within encoder-decoder architectures, as well as the problem of data leakage in current PGPS benchmarks.

Object goal navigation aims to navigate an agent to locations of a given object category in unseen environments. Classical methods explicitly build maps of environments and require extensive engineering while lacking semantic information for object-oriented exploration. On the other hand, end-to-end learning methods alleviate manual map design and predict actions using implicit representations. Such methods, however, lack an explicit notion of geometry and may have limited ability to encode navigation history. In this work, we propose an implicit spatial map for object goal navigation. Our implicit map is recursively updated with new observations at each step using a transformer. To encourage spatial reasoning, we introduce auxiliary tasks and train our model to reconstruct explicit maps as well as to predict visual features, semantic labels and actions. Our method significantly outperforms the state of the art on the challenging MP3D dataset and generalizes well to the HM3D dataset. We successfully deploy our model on a real robot and achieve encouraging object goal navigation results in real scenes using only a few real-world demonstrations. Code, trained models and videos are available at https://www.di.ens.fr/willow/research/onav_rim/.

Diffusion models generating images conditionally on text, such as Dall-E 2 and Stable Diffusion, have recently made a splash far beyond the computer vision community. Here, we tackle the related problem of generating point clouds, both unconditionally, and conditionally with images. For the latter, we introduce a novel geometrically-motivated conditioning scheme based on projecting sparse image features into the point cloud and attaching them to each individual point, at every step in the denoising process. This approach improves geometric consistency and yields greater fidelity than current methods relying on unstructured, global latent codes. Additionally, we show how to apply recent continuous-time diffusion schemes. Our method performs on par or above the state of art on conditional and unconditional experiments on synthetic data, while being faster, lighter, and delivering tractable likelihoods. We show it can also scale to diverse indoors scenes.

We propose GeoUni, the first unified geometry expert model capable of generating problem solutions and diagrams within a single framework in a way that enables the creation of unique and individualized geometry problems. Traditionally, solving geometry problems and generating diagrams have been treated as separate tasks in machine learning, with no models successfully integrating both to support problem creation. However, we believe that mastery in geometry requires frictionless integration of all of these skills, from solving problems to visualizing geometric relationships, and finally, crafting tailored problems. Our extensive experiments demonstrate that GeoUni, with only 1.5B parameters, achieves performance comparable to larger models such as DeepSeek-R1 with 671B parameters in geometric reasoning tasks. GeoUni also excels in generating precise geometric diagrams, surpassing both text-to-image models and unified models, including the GPT-4o image generation. Most importantly, GeoUni is the only model capable of successfully generating textual problems with matching diagrams based on specific knowledge points, thus offering a wider range of capabilities that extend beyond current models.

Data heterogeneity in federated learning, characterized by a significant misalignment between local and global distributions, leads to divergent local optimization directions and hinders global model training. Existing studies mainly focus on optimizing local updates or global aggregation, but these indirect approaches demonstrate instability when handling highly heterogeneous data distributions, especially in scenarios where label skew and domain skew coexist. To address this, we propose a geometry-guided data generation method that centers on simulating the global embedding distribution locally. We first introduce the concept of the geometric shape of an embedding distribution and then address the challenge of obtaining global geometric shapes under privacy constraints. Subsequently, we propose GGEUR, which leverages global geometric shapes to guide the generation of new samples, enabling a closer approximation to the ideal global distribution. In single-domain scenarios, we augment samples based on global geometric shapes to enhance model generalization; in multi-domain scenarios, we further employ class prototypes to simulate the global distribution across domains. Extensive experimental results demonstrate that our method significantly enhances the performance of existing approaches in handling highly heterogeneous data, including scenarios with label skew, domain skew, and their coexistence. Code published at: https://github.com/WeiDai-David/2025CVPR_GGEUR

Shape assembly aims to reassemble parts (or fragments) into a complete object, which is a common task in our daily life. Different from the semantic part assembly (e.g., assembling a chair's semantic parts like legs into a whole chair), geometric part assembly (e.g., assembling bowl fragments into a complete bowl) is an emerging task in computer vision and robotics. Instead of semantic information, this task focuses on geometric information of parts. As the both geometric and pose space of fractured parts are exceptionally large, shape pose disentanglement of part representations is beneficial to geometric shape assembly. In our paper, we propose to leverage SE(3) equivariance for such shape pose disentanglement. Moreover, while previous works in vision and robotics only consider SE(3) equivariance for the representations of single objects, we move a step forward and propose leveraging SE(3) equivariance for representations considering multi-part correlations, which further boosts the performance of the multi-part assembly. Experiments demonstrate the significance of SE(3) equivariance and our proposed method for geometric shape assembly. Project page: https://crtie.github.io/SE-3-part-assembly/

Spatial intelligence spans a rich suite of abilities, including visualising and transforming shapes, mentally rotating objects, judging relational positions and containment, and estimating numerosity. However, it still remains a critical unresolved challenge for Multimodal Large Language Models (MLLMs).To fill this gap, we propose to treat Euclidean geometry problem-solving as a surrogate task. Specifically, we meticulously constructed a curated multimodal dataset, called Euclid30K, comprising approximately 30K plane and solid geometry problems. To enable the model to acquire and apply Euclidean principles from these geometry problems, we employed Group Relative Policy Optimization (GRPO) to finetune the Qwen2.5VL family and RoboBrain2.0 family, inspiring the models to identify shapes, count, and relate entities, and perform multi-step deductive reasoning using Euclidean principles. Our experiments demonstrate that the resulting models achieve substantial zero-shot gains across four spatial reasoning benchmarks (Super-CLEVR, Omni3DBench, VSI-Bench, and MindCube) without any task-specific adaptations. Notably, after training on the Euclid30K, the mean VSI-Bench accuracy of all evaluated models rose from 34.5% to 40.5%, improving by 5.5 percentage points. Among them, RoboBrain2.0-Euclid-7B achieves 49.6\% accuracy, surpassing the previous state-of-the-art model, Spatial-MLLM.To our knowledge, this is the first systematic study showing that geometry-centric fine-tuning can confer vision-language models with broadly transferable spatial skills. Code and Euclid30K dataset can be found in https://zgca-ai4edu.github.io/Euclids_Gift.

Automatic 3D content creation has achieved rapid progress recently due to the availability of pre-trained, large language models and image diffusion models, forming the emerging topic of text-to-3D content creation. Existing text-to-3D methods commonly use implicit scene representations, which couple the geometry and appearance via volume rendering and are suboptimal in terms of recovering finer geometries and achieving photorealistic rendering; consequently, they are less effective for generating high-quality 3D assets. In this work, we propose a new method of Fantasia3D for high-quality text-to-3D content creation. Key to Fantasia3D is the disentangled modeling and learning of geometry and appearance. For geometry learning, we rely on a hybrid scene representation, and propose to encode surface normal extracted from the representation as the input of the image diffusion model. For appearance modeling, we introduce the spatially varying bidirectional reflectance distribution function (BRDF) into the text-to-3D task, and learn the surface material for photorealistic rendering of the generated surface. Our disentangled framework is more compatible with popular graphics engines, supporting relighting, editing, and physical simulation of the generated 3D assets. We conduct thorough experiments that show the advantages of our method over existing ones under different text-to-3D task settings. Project page and source codes: https://fantasia3d.github.io/.

We introduce GeoWizard, a new generative foundation model designed for estimating geometric attributes, e.g., depth and normals, from single images. While significant research has already been conducted in this area, the progress has been substantially limited by the low diversity and poor quality of publicly available datasets. As a result, the prior works either are constrained to limited scenarios or suffer from the inability to capture geometric details. In this paper, we demonstrate that generative models, as opposed to traditional discriminative models (e.g., CNNs and Transformers), can effectively address the inherently ill-posed problem. We further show that leveraging diffusion priors can markedly improve generalization, detail preservation, and efficiency in resource usage. Specifically, we extend the original stable diffusion model to jointly predict depth and normal, allowing mutual information exchange and high consistency between the two representations. More importantly, we propose a simple yet effective strategy to segregate the complex data distribution of various scenes into distinct sub-distributions. This strategy enables our model to recognize different scene layouts, capturing 3D geometry with remarkable fidelity. GeoWizard sets new benchmarks for zero-shot depth and normal prediction, significantly enhancing many downstream applications such as 3D reconstruction, 2D content creation, and novel viewpoint synthesis.

Lattice-based planning techniques simplify the motion planning problem for autonomous vehicles by limiting available motions to a pre-computed set of primitives. These primitives are then combined online to generate more complex maneuvers. A set of motion primitives t-span a lattice if, given a real number t at least 1, any configuration in the lattice can be reached via a sequence of motion primitives whose cost is no more than a factor of t from optimal. Computing a minimal t-spanning set balances a trade-off between computed motion quality and motion planning performance. In this work, we formulate this problem for an arbitrary lattice as a mixed integer linear program. We also propose an A*-based algorithm to solve the motion planning problem using these primitives. Finally, we present an algorithm that removes the excessive oscillations from planned motions -- a common problem in lattice-based planning. Our method is validated for autonomous driving in both parking lot and highway scenarios.

As the most common representation for 3D shapes, mesh is often stored discretely with arrays of vertices and faces. However, 3D shapes in the real world are presented continuously. In this paper, we propose to learn a continuous representation for meshes with fixed topology, a common and practical setting in many faces-, hand-, and body-related applications. First, we split the template into multiple closed manifold genus-0 meshes so that each genus-0 mesh can be parameterized onto the unit sphere. Then we learn spherical implicit surface (SIS), which takes a spherical coordinate and a global feature or a set of local features around the coordinate as inputs, predicting the vertex corresponding to the coordinate as an output. Since the spherical coordinates are continuous, SIS can depict a mesh in an arbitrary resolution. SIS representation builds a bridge between discrete and continuous representation in 3D shapes. Specifically, we train SIS networks in a self-supervised manner for two tasks: a reconstruction task and a super-resolution task. Experiments show that our SIS representation is comparable with state-of-the-art methods that are specifically designed for meshes with a fixed resolution and significantly outperforms methods that work in arbitrary resolutions.

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, are inherently unable to scale to high dimensions, or use approximations for limiting quantities that result in biased training objectives. Riemannian Flow Matching bypasses these limitations and offers several advantages over previous approaches: it is simulation-free on simple geometries, does not require divergence computation, and computes its target vector field in closed-form. The key ingredient behind RFM is the construction of a relatively simple premetric for defining target vector fields, which encompasses the existing Euclidean case. To extend to general geometries, we rely on the use of spectral decompositions to efficiently compute premetrics on the fly. Our method achieves state-of-the-art performance on many real-world non-Euclidean datasets, and we demonstrate tractable training on general geometries, including triangular meshes with highly non-trivial curvature and boundaries.

Polygon meshes are an efficient representation of 3D geometry, and are of central importance in computer graphics, robotics and games development. Existing learning-based approaches have avoided the challenges of working with 3D meshes, instead using alternative object representations that are more compatible with neural architectures and training approaches. We present an approach which models the mesh directly, predicting mesh vertices and faces sequentially using a Transformer-based architecture. Our model can condition on a range of inputs, including object classes, voxels, and images, and because the model is probabilistic it can produce samples that capture uncertainty in ambiguous scenarios. We show that the model is capable of producing high-quality, usable meshes, and establish log-likelihood benchmarks for the mesh-modelling task. We also evaluate the conditional models on surface reconstruction metrics against alternative methods, and demonstrate competitive performance despite not training directly on this task.

3D vector graphics play a crucial role in various applications including 3D shape retrieval, conceptual design, and virtual reality interactions due to their ability to capture essential structural information with minimal representation. While recent approaches have shown promise in generating 3D vector graphics, they often suffer from lengthy processing times and struggle to maintain view consistency. To address these limitations, we propose ViewCraft3D (VC3D), an efficient method that leverages 3D priors to generate 3D vector graphics. Specifically, our approach begins with 3D object analysis, employs a geometric extraction algorithm to fit 3D vector graphics to the underlying structure, and applies view-consistent refinement process to enhance visual quality. Our comprehensive experiments demonstrate that VC3D outperforms previous methods in both qualitative and quantitative evaluations, while significantly reducing computational overhead. The resulting 3D sketches maintain view consistency and effectively capture the essential characteristics of the original objects.

To endow machines with the ability to perceive the real-world in a three dimensional representation as we do as humans is a fundamental and long-standing topic in Artificial Intelligence. Given different types of visual inputs such as images or point clouds acquired by 2D/3D sensors, one important goal is to understand the geometric structure and semantics of the 3D environment. Traditional approaches usually leverage hand-crafted features to estimate the shape and semantics of objects or scenes. However, they are difficult to generalize to novel objects and scenarios, and struggle to overcome critical issues caused by visual occlusions. By contrast, we aim to understand scenes and the objects within them by learning general and robust representations using deep neural networks, trained on large-scale real-world 3D data. To achieve these aims, this thesis makes three core contributions from object-level 3D shape estimation from single or multiple views to scene-level semantic understanding.

In recent times, the generation of 3D assets from text prompts has shown impressive results. Both 2D and 3D diffusion models can generate decent 3D objects based on prompts. 3D diffusion models have good 3D consistency, but their quality and generalization are limited as trainable 3D data is expensive and hard to obtain. 2D diffusion models enjoy strong abilities of generalization and fine generation, but the 3D consistency is hard to guarantee. This paper attempts to bridge the power from the two types of diffusion models via the recent explicit and efficient 3D Gaussian splatting representation. A fast 3D generation framework, named as \name, is proposed, where the 3D diffusion model provides point cloud priors for initialization and the 2D diffusion model enriches the geometry and appearance. Operations of noisy point growing and color perturbation are introduced to enhance the initialized Gaussians. Our \name can generate a high-quality 3D instance within 25 minutes on one GPU, much faster than previous methods, while the generated instances can be directly rendered in real time. Demos and code are available at https://taoranyi.com/gaussiandreamer/.

The creation of photorealistic virtual worlds requires the accurate modeling of 3D surface geometry for a wide range of objects. For this, meshes are appealing since they 1) enable fast physics-based rendering with realistic material and lighting, 2) support physical simulation, and 3) are memory-efficient for modern graphics pipelines. Recent work on reconstructing and statistically modeling 3D shape, however, has critiqued meshes as being topologically inflexible. To capture a wide range of object shapes, any 3D representation must be able to model solid, watertight, shapes as well as thin, open, surfaces. Recent work has focused on the former, and methods for reconstructing open surfaces do not support fast reconstruction with material and lighting or unconditional generative modelling. Inspired by the observation that open surfaces can be seen as islands floating on watertight surfaces, we parameterize open surfaces by defining a manifold signed distance field on watertight templates. With this parameterization, we further develop a grid-based and differentiable representation that parameterizes both watertight and non-watertight meshes of arbitrary topology. Our new representation, called Ghost-on-the-Shell (G-Shell), enables two important applications: differentiable rasterization-based reconstruction from multiview images and generative modelling of non-watertight meshes. We empirically demonstrate that G-Shell achieves state-of-the-art performance on non-watertight mesh reconstruction and generation tasks, while also performing effectively for watertight meshes.

Monocular object pose estimation, as a pivotal task in computer vision and robotics, heavily depends on accurate 2D-3D correspondences, which often demand costly CAD models that may not be readily available. Object 3D reconstruction methods offer an alternative, among which recent advancements in 3D Gaussian Splatting (3DGS) afford a compelling potential. Yet its performance still suffers and tends to overfit with fewer input views. Embracing this challenge, we introduce SGPose, a novel framework for sparse view object pose estimation using Gaussian-based methods. Given as few as ten views, SGPose generates a geometric-aware representation by starting with a random cuboid initialization, eschewing reliance on Structure-from-Motion (SfM) pipeline-derived geometry as required by traditional 3DGS methods. SGPose removes the dependence on CAD models by regressing dense 2D-3D correspondences between images and the reconstructed model from sparse input and random initialization, while the geometric-consistent depth supervision and online synthetic view warping are key to the success. Experiments on typical benchmarks, especially on the Occlusion LM-O dataset, demonstrate that SGPose outperforms existing methods even under sparse view constraints, under-scoring its potential in real-world applications.

We present Bidirectional Gaussian Primitives, an image-based novel view synthesis technique designed to represent and render 3D objects with surface and volumetric materials under dynamic illumination. Our approach integrates light intrinsic decomposition into the Gaussian splatting framework, enabling real-time relighting of 3D objects. To unify surface and volumetric material within a cohesive appearance model, we adopt a light- and view-dependent scattering representation via bidirectional spherical harmonics. Our model does not use a specific surface normal-related reflectance function, making it more compatible with volumetric representations like Gaussian splatting, where the normals are undefined. We demonstrate our method by reconstructing and rendering objects with complex materials. Using One-Light-At-a-Time (OLAT) data as input, we can reproduce photorealistic appearances under novel lighting conditions in real time.

We present a learning-based method, namely GeoUDF,to tackle the long-standing and challenging problem of reconstructing a discrete surface from a sparse point cloud.To be specific, we propose a geometry-guided learning method for UDF and its gradient estimation that explicitly formulates the unsigned distance of a query point as the learnable affine averaging of its distances to the tangent planes of neighboring points on the surface. Besides,we model the local geometric structure of the input point clouds by explicitly learning a quadratic polynomial for each point. This not only facilitates upsampling the input sparse point cloud but also naturally induces unoriented normal, which further augments UDF estimation. Finally, to extract triangle meshes from the predicted UDF we propose a customized edge-based marching cube module. We conduct extensive experiments and ablation studies to demonstrate the significant advantages of our method over state-of-the-art methods in terms of reconstruction accuracy, efficiency, and generality. The source code is publicly available at https://github.com/rsy6318/GeoUDF.

General object composition (GOC) aims to seamlessly integrate a target object into a background scene with desired geometric properties, while simultaneously preserving its fine-grained appearance details. Recent approaches derive semantic embeddings and integrate them into advanced diffusion models to enable geometry-editable generation. However, these highly compact embeddings encode only high-level semantic cues and inevitably discard fine-grained appearance details. We introduce a Disentangled Geometry-editable and Appearance-preserving Diffusion (DGAD) model that first leverages semantic embeddings to implicitly capture the desired geometric transformations and then employs a cross-attention retrieval mechanism to align fine-grained appearance features with the geometry-edited representation, facilitating both precise geometry editing and faithful appearance preservation in object composition. Specifically, DGAD builds on CLIP/DINO-derived and reference networks to extract semantic embeddings and appearance-preserving representations, which are then seamlessly integrated into the encoding and decoding pipelines in a disentangled manner. We first integrate the semantic embeddings into pre-trained diffusion models that exhibit strong spatial reasoning capabilities to implicitly capture object geometry, thereby facilitating flexible object manipulation and ensuring effective editability. Then, we design a dense cross-attention mechanism that leverages the implicitly learned object geometry to retrieve and spatially align appearance features with their corresponding regions, ensuring faithful appearance consistency. Extensive experiments on public benchmarks demonstrate the effectiveness of the proposed DGAD framework.

In stark contrast to the case of images, finding a concise, learnable discrete representation of 3D surfaces remains a challenge. In particular, while polygon meshes are arguably the most common surface representation used in geometry processing, their irregular and combinatorial structure often make them unsuitable for learning-based applications. In this work, we present VoroMesh, a novel and differentiable Voronoi-based representation of watertight 3D shape surfaces. From a set of 3D points (called generators) and their associated occupancy, we define our boundary representation through the Voronoi diagram of the generators as the subset of Voronoi faces whose two associated (equidistant) generators are of opposite occupancy: the resulting polygon mesh forms a watertight approximation of the target shape's boundary. To learn the position of the generators, we propose a novel loss function, dubbed VoroLoss, that minimizes the distance from ground truth surface samples to the closest faces of the Voronoi diagram which does not require an explicit construction of the entire Voronoi diagram. A direct optimization of the Voroloss to obtain generators on the Thingi32 dataset demonstrates the geometric efficiency of our representation compared to axiomatic meshing algorithms and recent learning-based mesh representations. We further use VoroMesh in a learning-based mesh prediction task from input SDF grids on the ABC dataset, and show comparable performance to state-of-the-art methods while guaranteeing closed output surfaces free of self-intersections.

Geometry problem solving is a well-recognized testbed for evaluating the high-level multi-modal reasoning capability of deep models. In most existing works, two main geometry problems: calculation and proving, are usually treated as two specific tasks, hindering a deep model to unify its reasoning capability on multiple math tasks. However, in essence, these two tasks have similar problem representations and overlapped math knowledge which can improve the understanding and reasoning ability of a deep model on both two tasks. Therefore, we construct a large-scale Unified Geometry problem benchmark, UniGeo, which contains 4,998 calculation problems and 9,543 proving problems. Each proving problem is annotated with a multi-step proof with reasons and mathematical expressions. The proof can be easily reformulated as a proving sequence that shares the same formats with the annotated program sequence for calculation problems. Naturally, we also present a unified multi-task Geometric Transformer framework, Geoformer, to tackle calculation and proving problems simultaneously in the form of sequence generation, which finally shows the reasoning ability can be improved on both two tasks by unifying formulation. Furthermore, we propose a Mathematical Expression Pretraining (MEP) method that aims to predict the mathematical expressions in the problem solution, thus improving the Geoformer model. Experiments on the UniGeo demonstrate that our proposed Geoformer obtains state-of-the-art performance by outperforming task-specific model NGS with over 5.6% and 3.2% accuracies on calculation and proving problems, respectively.

We present Hunyuan3D 2.0, an advanced large-scale 3D synthesis system for generating high-resolution textured 3D assets. This system includes two foundation components: a large-scale shape generation model -- Hunyuan3D-DiT, and a large-scale texture synthesis model -- Hunyuan3D-Paint. The shape generative model, built on a scalable flow-based diffusion transformer, aims to create geometry that properly aligns with a given condition image, laying a solid foundation for downstream applications. The texture synthesis model, benefiting from strong geometric and diffusion priors, produces high-resolution and vibrant texture maps for either generated or hand-crafted meshes. Furthermore, we build Hunyuan3D-Studio -- a versatile, user-friendly production platform that simplifies the re-creation process of 3D assets. It allows both professional and amateur users to manipulate or even animate their meshes efficiently. We systematically evaluate our models, showing that Hunyuan3D 2.0 outperforms previous state-of-the-art models, including the open-source models and closed-source models in geometry details, condition alignment, texture quality, and etc. Hunyuan3D 2.0 is publicly released in order to fill the gaps in the open-source 3D community for large-scale foundation generative models. The code and pre-trained weights of our models are available at: https://github.com/Tencent/Hunyuan3D-2

We introduce a diffusion-based framework that performs aligned novel view image and geometry generation via a warping-and-inpainting methodology. Unlike prior methods that require dense posed images or pose-embedded generative models limited to in-domain views, our method leverages off-the-shelf geometry predictors to predict partial geometries viewed from reference images, and formulates novel-view synthesis as an inpainting task for both image and geometry. To ensure accurate alignment between generated images and geometry, we propose cross-modal attention distillation, where attention maps from the image diffusion branch are injected into a parallel geometry diffusion branch during both training and inference. This multi-task approach achieves synergistic effects, facilitating geometrically robust image synthesis as well as well-defined geometry prediction. We further introduce proximity-based mesh conditioning to integrate depth and normal cues, interpolating between point cloud and filtering erroneously predicted geometry from influencing the generation process. Empirically, our method achieves high-fidelity extrapolative view synthesis on both image and geometry across a range of unseen scenes, delivers competitive reconstruction quality under interpolation settings, and produces geometrically aligned colored point clouds for comprehensive 3D completion. Project page is available at https://cvlab-kaist.github.io/MoAI.

The topic of stitching images with globally natural structures holds paramount significance. Current methodologies exhibit the ability to preserve local geometric structures, yet fall short in maintaining relationships between these geometric structures. In this paper, we endeavor to safeguard the overall, OBJect-level structures within images based on Global Similarity Prior, while concurrently mitigating distortion and ghosting artifacts with OBJ-GSP. Our approach leverages the Segment Anything Model to extract geometric structures with semantic information, enhancing the algorithm's ability to preserve objects in a manner that aligns more intuitively with human perception. We seek to identify spatial constraints that govern the relationships between various geometric boundaries. Recognizing that multiple geometric boundaries collectively define complete objects, we employ triangular meshes to safeguard not only individual geometric structures but also the overall shapes of objects within the images. Empirical evaluations across multiple image stitching datasets demonstrate that our method establishes a new state-of-the-art benchmark in image stitching. Our implementation and dataset is publicly available at https://github.com/RussRobin/OBJ-GSP .

Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly available tool to manipulate knot diagrams in a real-time, interactive way is yet to be developed. We introduce a method of operating on the geometry of the knot diagram itself without any underlying three-dimensional structure that can underpin such an application. This allows us to directly interact with vector graphics knot diagrams while at the same time computing knot invariants in ways proposed by previous work. An implementation of this method is provided.

We introduce a new approach for generating realistic 3D models with UV maps through a representation termed "Object Images." This approach encapsulates surface geometry, appearance, and patch structures within a 64x64 pixel image, effectively converting complex 3D shapes into a more manageable 2D format. By doing so, we address the challenges of both geometric and semantic irregularity inherent in polygonal meshes. This method allows us to use image generation models, such as Diffusion Transformers, directly for 3D shape generation. Evaluated on the ABO dataset, our generated shapes with patch structures achieve point cloud FID comparable to recent 3D generative models, while naturally supporting PBR material generation.

Nonlinear manifolds are widespread in deep visual features, where Euclidean distances often fail to capture true similarity. This limitation becomes particularly severe in prototype-based interpretable fine-grained recognition, where subtle semantic distinctions are essential. To address this challenge, we propose a novel paradigm for prototype-based recognition that anchors similarity within the intrinsic geometry of deep features. Specifically, we distill the latent manifold structure of each class into a diffusion space and introduce a differentiable Nystr\"om interpolation, making the geometry accessible to both unseen samples and learnable prototypes. To ensure efficiency, we employ compact per-class landmark sets with periodic updates. This design keeps the embedding aligned with the evolving backbone, enabling fast and scalable inference. Extensive experiments on the CUB-200-2011 and Stanford Cars datasets show that our GeoProto framework produces prototypes focusing on semantically aligned parts, significantly outperforming Euclidean prototype networks.

Recent advancements in implicit 3D representations and generative models have markedly propelled the field of 3D object generation forward. However, it remains a significant challenge to accurately model geometries with defined sharp features under parametric controls, which is crucial in fields like industrial design and manufacturing. To bridge this gap, we introduce a framework that employs Large Language Models (LLMs) to generate text-driven 3D shapes, manipulating 3D software via program synthesis. We present 3D-PreMise, a dataset specifically tailored for 3D parametric modeling of industrial shapes, designed to explore state-of-the-art LLMs within our proposed pipeline. Our work reveals effective generation strategies and delves into the self-correction capabilities of LLMs using a visual interface. Our work highlights both the potential and limitations of LLMs in 3D parametric modeling for industrial applications.

The integration of geometric reconstruction and generative modeling remains a critical challenge in developing AI systems capable of human-like spatial reasoning. This paper proposes Aether, a unified framework that enables geometry-aware reasoning in world models by jointly optimizing three core capabilities: (1) 4D dynamic reconstruction, (2) action-conditioned video prediction, and (3) goal-conditioned visual planning. Through task-interleaved feature learning, Aether achieves synergistic knowledge sharing across reconstruction, prediction, and planning objectives. Building upon video generation models, our framework demonstrates unprecedented synthetic-to-real generalization despite never observing real-world data during training. Furthermore, our approach achieves zero-shot generalization in both action following and reconstruction tasks, thanks to its intrinsic geometric modeling. Remarkably, even without real-world data, its reconstruction performance far exceeds that of domain-specific models. Additionally, Aether leverages a geometry-informed action space to seamlessly translate predictions into actions, enabling effective autonomous trajectory planning. We hope our work inspires the community to explore new frontiers in physically-reasonable world modeling and its applications.

Several first order stochastic optimization methods commonly used in the Euclidean domain such as stochastic gradient descent (SGD), accelerated gradient descent or variance reduced methods have already been adapted to certain Riemannian settings. However, some of the most popular of these optimization tools - namely Adam , Adagrad and the more recent Amsgrad - remain to be generalized to Riemannian manifolds. We discuss the difficulty of generalizing such adaptive schemes to the most agnostic Riemannian setting, and then provide algorithms and convergence proofs for geodesically convex objectives in the particular case of a product of Riemannian manifolds, in which adaptivity is implemented across manifolds in the cartesian product. Our generalization is tight in the sense that choosing the Euclidean space as Riemannian manifold yields the same algorithms and regret bounds as those that were already known for the standard algorithms. Experimentally, we show faster convergence and to a lower train loss value for Riemannian adaptive methods over their corresponding baselines on the realistic task of embedding the WordNet taxonomy in the Poincare ball.

We describe Generative Blocks World to interact with the scene of a generated image by manipulating simple geometric abstractions. Our method represents scenes as assemblies of convex 3D primitives, and the same scene can be represented by different numbers of primitives, allowing an editor to move either whole structures or small details. Once the scene geometry has been edited, the image is generated by a flow-based method which is conditioned on depth and a texture hint. Our texture hint takes into account the modified 3D primitives, exceeding texture-consistency provided by existing key-value caching techniques. These texture hints (a) allow accurate object and camera moves and (b) largely preserve the identity of objects depicted. Quantitative and qualitative experiments demonstrate that our approach outperforms prior works in visual fidelity, editability, and compositional generalization.

We present En3D, an enhanced generative scheme for sculpting high-quality 3D human avatars. Unlike previous works that rely on scarce 3D datasets or limited 2D collections with imbalanced viewing angles and imprecise pose priors, our approach aims to develop a zero-shot 3D generative scheme capable of producing visually realistic, geometrically accurate and content-wise diverse 3D humans without relying on pre-existing 3D or 2D assets. To address this challenge, we introduce a meticulously crafted workflow that implements accurate physical modeling to learn the enhanced 3D generative model from synthetic 2D data. During inference, we integrate optimization modules to bridge the gap between realistic appearances and coarse 3D shapes. Specifically, En3D comprises three modules: a 3D generator that accurately models generalizable 3D humans with realistic appearance from synthesized balanced, diverse, and structured human images; a geometry sculptor that enhances shape quality using multi-view normal constraints for intricate human anatomy; and a texturing module that disentangles explicit texture maps with fidelity and editability, leveraging semantical UV partitioning and a differentiable rasterizer. Experimental results show that our approach significantly outperforms prior works in terms of image quality, geometry accuracy and content diversity. We also showcase the applicability of our generated avatars for animation and editing, as well as the scalability of our approach for content-style free adaptation.

The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are invariant or equivariant to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at https://github.com/chaitjo/geometric-gnn-dojo

3D object generation has undergone significant advancements, yielding high-quality results. However, fall short of achieving precise user control, often yielding results that do not align with user expectations, thus limiting their applicability. User-envisioning 3D object generation faces significant challenges in realizing its concepts using current generative models due to limited interaction capabilities. Existing methods mainly offer two approaches: (i) interpreting textual instructions with constrained controllability, or (ii) reconstructing 3D objects from 2D images. Both of them limit customization to the confines of the 2D reference and potentially introduce undesirable artifacts during the 3D lifting process, restricting the scope for direct and versatile 3D modifications. In this work, we introduce Interactive3D, an innovative framework for interactive 3D generation that grants users precise control over the generative process through extensive 3D interaction capabilities. Interactive3D is constructed in two cascading stages, utilizing distinct 3D representations. The first stage employs Gaussian Splatting for direct user interaction, allowing modifications and guidance of the generative direction at any intermediate step through (i) Adding and Removing components, (ii) Deformable and Rigid Dragging, (iii) Geometric Transformations, and (iv) Semantic Editing. Subsequently, the Gaussian splats are transformed into InstantNGP. We introduce a novel (v) Interactive Hash Refinement module to further add details and extract the geometry in the second stage. Our experiments demonstrate that Interactive3D markedly improves the controllability and quality of 3D generation. Our project webpage is available at https://interactive-3d.github.io/.

We present DreamPolisher, a novel Gaussian Splatting based method with geometric guidance, tailored to learn cross-view consistency and intricate detail from textual descriptions. While recent progress on text-to-3D generation methods have been promising, prevailing methods often fail to ensure view-consistency and textural richness. This problem becomes particularly noticeable for methods that work with text input alone. To address this, we propose a two-stage Gaussian Splatting based approach that enforces geometric consistency among views. Initially, a coarse 3D generation undergoes refinement via geometric optimization. Subsequently, we use a ControlNet driven refiner coupled with the geometric consistency term to improve both texture fidelity and overall consistency of the generated 3D asset. Empirical evaluations across diverse textual prompts spanning various object categories demonstrate the efficacy of DreamPolisher in generating consistent and realistic 3D objects, aligning closely with the semantics of the textual instructions.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Due to high utility in many applications, from social networks to blockchain to power grids, deep learning on non-Euclidean objects such as graphs and manifolds, coined Geometric Deep Learning (GDL), continues to gain an ever increasing interest. We propose a new L\'evy Flights Graph Convolutional Networks (LFGCN) method for semi-supervised learning, which casts the L\'evy Flights into random walks on graphs and, as a result, allows both to accurately account for the intrinsic graph topology and to substantially improve classification performance, especially for heterogeneous graphs. Furthermore, we propose a new preferential P-DropEdge method based on the Girvan-Newman argument. That is, in contrast to uniform removing of edges as in DropEdge, following the Girvan-Newman algorithm, we detect network periphery structures using information on edge betweenness and then remove edges according to their betweenness centrality. Our experimental results on semi-supervised node classification tasks demonstrate that the LFGCN coupled with P-DropEdge accelerates the training task, increases stability and further improves predictive accuracy of learned graph topology structure. Finally, in our case studies we bring the machinery of LFGCN and other deep networks tools to analysis of power grid networks - the area where the utility of GDL remains untapped.

Self-supervised learning has emerged as a promising approach for acquiring transferable 3D representations from unlabeled 3D point clouds. Unlike 2D images, which are widely accessible, acquiring 3D assets requires specialized expertise or professional 3D scanning equipment, making it difficult to scale and raising copyright concerns. To address these challenges, we propose learning 3D representations from procedural 3D programs that automatically generate 3D shapes using simple primitives and augmentations. Remarkably, despite lacking semantic content, the 3D representations learned from this synthesized dataset perform on par with state-of-the-art representations learned from semantically recognizable 3D models (e.g., airplanes) across various downstream 3D tasks, including shape classification, part segmentation, and masked point cloud completion. Our analysis further suggests that current self-supervised learning methods primarily capture geometric structures rather than high-level semantics.

Reconstructing dynamic 3D scenes from 2D images and generating diverse views over time is challenging due to scene complexity and temporal dynamics. Despite advancements in neural implicit models, limitations persist: (i) Inadequate Scene Structure: Existing methods struggle to reveal the spatial and temporal structure of dynamic scenes from directly learning the complex 6D plenoptic function. (ii) Scaling Deformation Modeling: Explicitly modeling scene element deformation becomes impractical for complex dynamics. To address these issues, we consider the spacetime as an entirety and propose to approximate the underlying spatio-temporal 4D volume of a dynamic scene by optimizing a collection of 4D primitives, with explicit geometry and appearance modeling. Learning to optimize the 4D primitives enables us to synthesize novel views at any desired time with our tailored rendering routine. Our model is conceptually simple, consisting of a 4D Gaussian parameterized by anisotropic ellipses that can rotate arbitrarily in space and time, as well as view-dependent and time-evolved appearance represented by the coefficient of 4D spherindrical harmonics. This approach offers simplicity, flexibility for variable-length video and end-to-end training, and efficient real-time rendering, making it suitable for capturing complex dynamic scene motions. Experiments across various benchmarks, including monocular and multi-view scenarios, demonstrate our 4DGS model's superior visual quality and efficiency.

Faithful visualizations of data residing on manifolds must take the underlying geometry into account when producing a flat planar view of the data. In this paper, we extend the classic stochastic neighbor embedding (SNE) algorithm to data on general Riemannian manifolds. We replace standard Gaussian assumptions with Riemannian diffusion counterparts and propose an efficient approximation that only requires access to calculations of Riemannian distances and volumes. We demonstrate that the approach also allows for mapping data from one manifold to another, e.g. from a high-dimensional sphere to a low-dimensional one.

The rapid development of Large Multimodal Models (LMMs) has led to remarkable progress in 2D visual understanding; however, extending these capabilities to 3D scene understanding remains a significant challenge. Existing approaches predominantly rely on text-only supervision, which fails to provide the geometric constraints required for learning robust 3D spatial representations. In this paper, we introduce Reg3D, a novel Reconstructive Geometry Instruction Tuning framework that addresses this limitation by incorporating geometry-aware supervision directly into the training process. Our key insight is that effective 3D understanding necessitates reconstructing underlying geometric structures rather than merely describing them. Unlike existing methods that inject 3D information solely at the input level, Reg3D adopts a dual-supervision paradigm that leverages 3D geometric information both as input and as explicit learning targets. Specifically, we design complementary object-level and frame-level reconstruction tasks within a dual-encoder architecture, enforcing geometric consistency to encourage the development of spatial reasoning capabilities. Extensive experiments on ScanQA, Scan2Cap, ScanRefer, and SQA3D demonstrate that Reg3D delivers substantial performance improvements, establishing a new training paradigm for spatially aware multimodal models.

Geometry is a fundamental branch of mathematics and plays a crucial role in evaluating the reasoning capabilities of multimodal large language models (MLLMs). However, existing multimodal mathematics benchmarks mainly focus on plane geometry and largely ignore solid geometry, which requires spatial reasoning and is more challenging than plane geometry. To address this critical gap, we introduce SolidGeo, the first large-scale benchmark specifically designed to evaluate the performance of MLLMs on mathematical reasoning tasks in solid geometry. SolidGeo consists of 3,113 real-world K-12 and competition-level problems, each paired with visual context and annotated with difficulty levels and fine-grained solid geometry categories. Our benchmark covers a wide range of 3D reasoning subjects such as projection, unfolding, spatial measurement, and spatial vector, offering a rigorous testbed for assessing solid geometry. Through extensive experiments, we observe that MLLMs encounter substantial challenges in solid geometry math tasks, with a considerable performance gap relative to human capabilities on SolidGeo. Moreover, we analyze the performance, inference efficiency and error patterns of various models, offering insights into the solid geometric mathematical reasoning capabilities of MLLMs. We hope SolidGeo serves as a catalyst for advancing MLLMs toward deeper geometric reasoning and spatial intelligence.

This paper presents the computational challenge on differential geometry and topology that happened within the ICLR 2021 workshop "Geometric and Topological Representation Learning". The competition asked participants to provide creative contributions to the fields of computational geometry and topology through the open-source repositories Geomstats and Giotto-TDA. The challenge attracted 16 teams in its two month duration. This paper describes the design of the challenge and summarizes its main findings.

Generative models for 3D object synthesis have seen significant advancements with the incorporation of prior knowledge distilled from 2D diffusion models. Nevertheless, challenges persist in the form of multi-view geometric inconsistencies and slow generation speeds within the existing 3D synthesis frameworks. This can be attributed to two factors: firstly, the deficiency of abundant geometric a priori knowledge in optimization, and secondly, the entanglement issue between geometry and texture in conventional 3D generation methods.In response, we introduce MetaDreammer, a two-stage optimization approach that leverages rich 2D and 3D prior knowledge. In the first stage, our emphasis is on optimizing the geometric representation to ensure multi-view consistency and accuracy of 3D objects. In the second stage, we concentrate on fine-tuning the geometry and optimizing the texture, thereby achieving a more refined 3D object. Through leveraging 2D and 3D prior knowledge in two stages, respectively, we effectively mitigate the interdependence between geometry and texture. MetaDreamer establishes clear optimization objectives for each stage, resulting in significant time savings in the 3D generation process. Ultimately, MetaDreamer can generate high-quality 3D objects based on textual prompts within 20 minutes, and to the best of our knowledge, it is the most efficient text-to-3D generation method. Furthermore, we introduce image control into the process, enhancing the controllability of 3D generation. Extensive empirical evidence confirms that our method is not only highly efficient but also achieves a quality level that is at the forefront of current state-of-the-art 3D generation techniques.

We target a 3D generative model for general natural scenes that are typically unique and intricate. Lacking the necessary volumes of training data, along with the difficulties of having ad hoc designs in presence of varying scene characteristics, renders existing setups intractable. Inspired by classical patch-based image models, we advocate for synthesizing 3D scenes at the patch level, given a single example. At the core of this work lies important algorithmic designs w.r.t the scene representation and generative patch nearest-neighbor module, that address unique challenges arising from lifting classical 2D patch-based framework to 3D generation. These design choices, on a collective level, contribute to a robust, effective, and efficient model that can generate high-quality general natural scenes with both realistic geometric structure and visual appearance, in large quantities and varieties, as demonstrated upon a variety of exemplar scenes.

Vision-Language Models (VLMs) have shown remarkable performance on diverse visual and linguistic tasks, yet they remain fundamentally limited in their understanding of 3D spatial structures. We propose Geometric Distillation, a lightweight, annotation-free fine-tuning framework that injects human-inspired geometric cues into pretrained VLMs without modifying their architecture. By distilling (1) sparse correspondences, (2) relative depth relations, and (3) dense cost volumes from off-the-shelf 3D foundation models (e.g., MASt3R, VGGT), our method shapes representations to be geometry-aware while remaining compatible with natural image-text inputs. Through extensive evaluations on 3D vision-language reasoning and 3D perception benchmarks, our method consistently outperforms prior approaches, achieving improved 3D spatial reasoning with significantly lower computational cost. Our work demonstrates a scalable and efficient path to bridge 2D-trained VLMs with 3D understanding, opening up wider use in spatially grounded multimodal tasks.

For robots to operate in general environments like households, they must be able to perform non-prehensile manipulation actions such as toppling and rolling to manipulate ungraspable objects. However, prior works on non-prehensile manipulation cannot yet generalize across environments with diverse geometries. The main challenge lies in adapting to varying environmental constraints: within a cabinet, the robot must avoid walls and ceilings; to lift objects to the top of a step, the robot must account for the step's pose and extent. While deep reinforcement learning (RL) has demonstrated impressive success in non-prehensile manipulation, accounting for such variability presents a challenge for the generalist policy, as it must learn diverse strategies for each new combination of constraints. To address this, we propose a modular and reconfigurable architecture that adaptively reconfigures network modules based on task requirements. To capture the geometric variability in environments, we extend the contact-based object representation (CORN) to environment geometries, and propose a procedural algorithm for generating diverse environments to train our agent. Taken together, the resulting policy can zero-shot transfer to novel real-world environments and objects despite training entirely within a simulator. We additionally release a simulation-based benchmark featuring nine digital twins of real-world scenes with 353 objects to facilitate non-prehensile manipulation research in realistic domains.

Recent progress in Sign Language Translation (SLT) has focussed primarily on improving the representational capacity of large language models to incorporate Sign Language features. This work explores an alternative direction: enhancing the geometric properties of skeletal representations themselves. We propose Geo-Sign, a method that leverages the properties of hyperbolic geometry to model the hierarchical structure inherent in sign language kinematics. By projecting skeletal features derived from Spatio-Temporal Graph Convolutional Networks (ST-GCNs) into the Poincar\'e ball model, we aim to create more discriminative embeddings, particularly for fine-grained motions like finger articulations. We introduce a hyperbolic projection layer, a weighted Fr\'echet mean aggregation scheme, and a geometric contrastive loss operating directly in hyperbolic space. These components are integrated into an end-to-end translation framework as a regularisation function, to enhance the representations within the language model. This work demonstrates the potential of hyperbolic geometry to improve skeletal representations for Sign Language Translation, improving on SOTA RGB methods while preserving privacy and improving computational efficiency. Code available here: https://github.com/ed-fish/geo-sign.

G-code (Geometric code) or RS-274 is the most widely used computer numerical control (CNC) and 3D printing programming language. G-code provides machine instructions for the movement of the 3D printer, especially for the nozzle, stage, and extrusion of material for extrusion-based additive manufacturing. Currently there does not exist a large repository of curated CAD models along with their corresponding G-code files for additive manufacturing. To address this issue, we present SLICE-100K, a first-of-its-kind dataset of over 100,000 G-code files, along with their tessellated CAD model, LVIS (Large Vocabulary Instance Segmentation) categories, geometric properties, and renderings. We build our dataset from triangulated meshes derived from Objaverse-XL and Thingi10K datasets. We demonstrate the utility of this dataset by finetuning GPT-2 on a subset of the dataset for G-code translation from a legacy G-code format (Sailfish) to a more modern, widely used format (Marlin). SLICE-100K will be the first step in developing a multimodal foundation model for digital manufacturing.

Symmetry is one of the most fundamental geometric cues in computer vision, and detecting it has been an ongoing challenge. With the recent advances in vision-language models,~i.e., CLIP, we investigate whether a pre-trained CLIP model can aid symmetry detection by leveraging the additional symmetry cues found in the natural image descriptions. We propose CLIPSym, which leverages CLIP's image and language encoders and a rotation-equivariant decoder based on a hybrid of Transformer and G-Convolution to detect rotation and reflection symmetries. To fully utilize CLIP's language encoder, we have developed a novel prompting technique called Semantic-Aware Prompt Grouping (SAPG), which aggregates a diverse set of frequent object-based prompts to better integrate the semantic cues for symmetry detection. Empirically, we show that CLIPSym outperforms the current state-of-the-art on three standard symmetry detection datasets (DENDI, SDRW, and LDRS). Finally, we conduct detailed ablations verifying the benefits of CLIP's pre-training, the proposed equivariant decoder, and the SAPG technique. The code is available at https://github.com/timyoung2333/CLIPSym.

Text-to-3D generation from a single-view image is a popular but challenging task in 3D vision. Although numerous methods have been proposed, existing works still suffer from the inconsistency issues, including 1) semantic inconsistency, 2) geometric inconsistency, and 3) saturation inconsistency, resulting in distorted, overfitted, and over-saturated generations. In light of the above issues, we present Consist3D, a three-stage framework Chasing for semantic-, geometric-, and saturation-Consistent Text-to-3D generation from a single image, in which the first two stages aim to learn parameterized consistency tokens, and the last stage is for optimization. Specifically, the semantic encoding stage learns a token independent of views and estimations, promoting semantic consistency and robustness. Meanwhile, the geometric encoding stage learns another token with comprehensive geometry and reconstruction constraints under novel-view estimations, reducing overfitting and encouraging geometric consistency. Finally, the optimization stage benefits from the semantic and geometric tokens, allowing a low classifier-free guidance scale and therefore preventing oversaturation. Experimental results demonstrate that Consist3D produces more consistent, faithful, and photo-realistic 3D assets compared to previous state-of-the-art methods. Furthermore, Consist3D also allows background and object editing through text prompts.