Many 3D generative models rely on variational autoencoders (VAEs) to learn compact shape representations. However, existing methods encode all shapes into a fixed-size token, disregarding the inherent variations in scale and complexity across 3D data. This leads to inefficient latent representations that can compromise downstream generation. We address this challenge by introducing Octree-based Adaptive Tokenization, a novel framework that adjusts the dimension of latent representations according to shape complexity. Our approach constructs an adaptive octree structure guided by a quadric-error-based subdivision criterion and allocates a shape latent vector to each octree cell using a query-based transformer. Building upon this tokenization, we develop an octree-based autoregressive generative model that effectively leverages these variable-sized representations in shape generation. Extensive experiments demonstrate that our approach reduces token counts by 50% compared to fixed-size methods while maintaining comparable visual quality. When using a similar token length, our method produces significantly higher-quality shapes. When incorporated with our downstream generative model, our method creates more detailed and diverse 3D content than existing approaches.

We present GEOPARD, a transformer-based architecture for predicting articulation from a single static snapshot of a 3D shape. The key idea of our method is a pretraining strategy that allows our transformer to learn plausible candidate articulations for 3D shapes based on a geometric-driven search without manual articulation annotation. The search automatically discovers physically valid part motions that do not cause detachments or collisions with other shape parts. Our experiments indicate that this geometric pretraining strategy, along with carefully designed choices in our transformer architecture, yields state-of-the-art results in articulation inference in the PartNet-Mobility dataset.

Artist-created meshes in-the-wild often do not have a well defined interior. We observe that they typically consist of a mix of solid elements, faces that bound a volume, and shell elements that represent the medial surface of a thin shell. The lack of a well-defined interior prevents downstream applications, such as solid-modeling, simulation, and manufacturing. We present a method that takes as input a surface mesh and assigns to each face a label determining whether it belongs to a solid or shell. These labels reduce ambiguity by defining the interior for solid faces through thresholding the generalized winding number field, and for shell faces as the volume within an offset. We cast the labeling problem as an optimization that outputs a solid/shell label for each face, guided by a sparse set of user inputs. Once labeling is complete, we show how the shape can be volume meshed by passing the shell faces through an offset mesher and the solid faces to an off-the-shelf tetrahedral mesher, producing a final volumetric mesh by taking their union. Experiments on diverse meshes with defects and multiple solid and shell components demonstrate that our approach delivers the desired labels, enabling modeling and simulation on wild meshes in a way that respects the user intent.

Abstract This work presents a method to control the output of mesh simplification algorithms based on iterative edge collapses. Traditional mesh simplification focuses on preserving the visual appearance. Despite still being an important criterion, other geometric properties also play critical roles in different applications, such as triangle quality for computations. This motivates our work to stay under the umbrella of the popular quadric error mesh simplification, while proposing different ways to control the simplified mesh to possess other geometric properties. The key ingredient of our work is another quadric error, called line quadrics, which can be seamlessly added to the vanilla quadric error metric. We show that, theoretically and empirically, adding our line quadrics can improve the numerics and encourage the simplified mesh to have uniformly distributed vertices. If we spread the line quadric adaptively to different regions, it can easily lead to soft preservation of feature vertices and edges. Our method is simple to implement, requiring only a few lines of code change on top of the original quadric error simplification, and can lead to a variety of user controls.

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.

This paper introduces a method to synthesize a 3D tensor field within a constrained geometric domain represented as a tetrahedral mesh. Whereas previous techniques optimize for isotropic fields, we focus on anisotropic tensor fields that are smooth and aligned with the domain boundary or user guidance. The key ingredient of our method is a novel computational design framework, built on top of the symmetric orthogonally decomposable (odeco) tensor representation, to optimize the stretching ratios and orientations for each tensor in the domain. In contrast to past techniques designed only for isotropic tensors, we demonstrate the efficacy of our approach in generating smooth volumetric tensor fields with high anisotropy and shape conformity, especially for the domain with complex shapes. We apply these anisotropic tensor fields to various applications, such as anisotropic meshing, structural mechanics, and fabrication.

Vector fields are widely used to represent and model flows for many science and engineering applications. This paper introduces a novel neural network architecture for learning tangent vector fields that are intrinsically defined on manifold surfaces embedded in 3D. Previous approaches to learning vector fields on surfaces treat vectors as multi-dimensional scalar fields, using traditional scalar-valued architectures to process channels individually, thus fail to preserve fundamental intrinsic properties of the vector field. The core idea of this work is to introduce a trainable vector heat diffusion module to spatially propagate vector-valued feature data across the surface, which we incorporate into our proposed architecture that consists of vector-valued neurons. Our architecture is invariant to rigid motion of the input, isometric deformation, and choice of local tangent bases, and is robust to discretizations of the surface. We evaluate our Vector Heat Network on triangle meshes, and empirically validate its invariant properties. We also demonstrate the effectiveness of our method on the useful industrial application of quadrilateral mesh generation.

Signed Distance Fields (SDFs) parameterized by neural networks have recently gained popularity as a fundamental geometric representation. However, editing the shape encoded by a neural SDF remains an open challenge. A tempting approach is to leverage common geometric operators (e.g., boolean operations), but such edits often lead to incorrect non-SDF outputs (which we call Pseudo-SDFs), preventing them from being used for downstream tasks. In this paper, we characterize the space of Pseudo-SDFs, which are eikonal yet not true distance functions, and derive the closest point loss, a novel regularizer that encourages the output to be an exact SDF. We demonstrate the applicability of our regularization to many operations in which traditional methods cause a Pseudo-SDF to arise, such as CSG and swept volumes, and produce a true (neural) SDF for the result of these operations.

This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a coarse intrinsic triangulation of the input domain. In the spirit of the quadric error metric (QEM) , we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature "drifts" during simplification. This process also yields a bijective map between the fine and coarse mesh, and prolongation operators for both scalar- and vector-valued data. Moreover, we obtain hard guarantees on element quality via intrinsic retriangulation---a feature unique to the intrinsic setting. The overall payoff is a "black box" approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations. We show how our method benefits several fundamental tasks, including geometric multigrid, all-pairs geodesic distance, mean curvature flow, geodesic Voronoi diagrams, and the discrete exponential map.

Data is the driving force of machine learning, with the amount and quality of training data often being more important for the performance of a system than architecture and training details. But collecting, processing and annotating real data at scale is difficult, expensive, and frequently raises additional privacy, fairness and legal concerns. Synthetic data is a powerful tool with the potential to address these shortcomings: 1) it is cheap 2) supports rich ground-truth annotations 3) offers full control over data and 4) can circumvent or mitigate problems regarding bias, privacy and licensing. Unfortunately, software tools for effective data generation are less mature than those for architecture design and training, which leads to fragmented generation efforts. To address these problems we introduce Kubric, an open-source Python framework that interfaces with PyBullet and Blender to generate photo-realistic scenes, with rich annotations, and seamlessly scales to large jobs distributed over thousands of machines, and generating TBs of data. We demonstrate the effectiveness of Kubric by presenting a series of 13 different generated datasets for tasks ranging from studying 3D NeRF models to optical flow estimation. We release Kubric, the used assets, all of the generation code, as well as the rendered datasets for reuse and modification.

Abstract This paper introduces a new method to stylize 3D geometry. The key observation is that the surface normal is an effective instrument to capture different geometric styles. Centered around this observation, we cast stylization as a shape analogy problem, where the analogy relationship is defined on the surface normal. This formulation can deform a 3D shape into different styles within a single framework. One can plug‐and‐play different target styles by providing an exemplar shape or an energy‐based style description (e.g., developable surfaces). Our surface stylization methodology enables Normal Captures as a geometric counterpart to material captures (MatCaps) used in rendering, and the prototypical concept of Spherical Shape Analogies as a geometric counterpart to image analogies in image processing.

This paper introduces a novel geometric multigrid solver for unstructured curved surfaces. Multigrid methods are highly efficient iterative methods for solving systems of linear equations. Despite the success in solving problems defined on structured domains, generalizing multigrid to unstructured curved domains remains a challenging problem. The critical missing ingredient is a prolongation operator to transfer functions across different multigrid levels. We propose a novel method for computing the prolongation for triangulated surfaces based on intrinsic geometry, enabling an efficient geometric multigrid solver for curved surfaces. Our surface multigrid solver achieves better convergence than existing multigrid methods. Compared to direct solvers, our solver is orders of magnitude faster. We evaluate our method on many geometry processing applications and a wide variety of complex shapes with and without boundaries. By simply replacing the direct solver, we upgrade existing algorithms to interactive frame rates, and shift the computational bottleneck away from solving linear systems.

Abstract The spectrum of the Laplace‐Beltrami operator is instrumental for a number of geometric modeling applications, from processing to analysis. Recently, multiple methods were developed to retrieve an approximation of a shape that preserves its eigenvectors as much as possible, but these techniques output a subset of input points with no connectivity, which limits their potential applications. Furthermore, the obtained Laplacian results from an optimization procedure, implying its storage alongside the selected points. Focusing on keeping a mesh instead of an operator would allow to retrieve the latter using the standard cotangent formulation, enabling easier processing afterwards. Instead, we propose to simplify the input mesh using a spectrum‐preserving mesh decimation scheme, so that the Laplacian computed on the simplified mesh is spectrally close to the one of the input mesh. We illustrate the benefit of our approach for quickly approximating spectral distances and functional maps on low resolution proxies of potentially high resolution input meshes.

We introduce a novel solver to significantly reduce the size of a geometric operator while preserving its spectral properties at the lowest frequencies. We use chordal decomposition to formulate a convex optimization problem which allows the user to control the operator sparsity pattern. This allows for a trade-off between the spectral accuracy of the operator and the cost of its application. We efficiently minimize the energy with a change of variables and achieve state-of-the-art results on spectral coarsening. Our solver further enables novel applications including volume-to-surface approximation and detaching the operator from the mesh, i.e., one can produce a mesh tailor-made for visualization and optimize an operator separately for computation.

This paper introduces Neural Subdivision , a novel framework for data-driven coarse-to-fine geometry modeling. During inference, our method takes a coarse triangle mesh as input and recursively subdivides it to a finer geometry by applying the fixed topological updates of Loop Subdivision, but predicting vertex positions using a neural network conditioned on the local geometry of a patch. This approach enables us to learn complex non-linear subdivision schemes, beyond simple linear averaging used in classical techniques. One of our key contributions is a novel self-supervised training setup that only requires a set of high-resolution meshes for learning network weights. For any training shape, we stochastically generate diverse low-resolution discretizations of coarse counterparts, while maintaining a bijective mapping that prescribes the exact target position of every new vertex during the subdivision process. This leads to a very efficient and accurate loss function for conditional mesh generation, and enables us to train a method that generalizes across discretizations and favors preserving the manifold structure of the output. During training we optimize for the same set of network weights across all local mesh patches, thus providing an architecture that is not constrained to a specific input mesh, fixed genus, or category. Our network encodes patch geometry in a local frame in a rotation- and translation-invariant manner. Jointly, these design choices enable our method to generalize well, and we demonstrate that even when trained on a single high-resolution mesh our method generates reasonable subdivisions for novel shapes.

We present a 3D stylization algorithm that can turn an input shape into the style of a cube while maintaining the content of the original shape. The key insight is that cubic style sculptures can be captured by the as-rigid-as-possible energy with an ℓ 1 -regularization on rotated surface normals. Minimizing this energy naturally leads to a detail-preserving, cubic geometry. Our optimization can be solved efficiently without any mesh surgery. Our method serves as a non-realistic modeling tool where one can incorporate many artistic controls to create stylized geometries.

We introduce a novel approach to measure the behavior of a geometric operator before and after coarsening. By comparing eigenvectors of the input operator and its coarsened counterpart, we can quantitatively and visually analyze how well the spectral properties of the operator are maintained. Using this measure, we show that standard mesh simplification and algebraic coarsening techniques fail to maintain spectral properties. In response, we introduce a novel approach forspectral coarsening.We show that it is possible to significantly reduce the sampling density of an operator derived from a 3D shape without affecting the low-frequency eigenvectors. By marrying techniques developed within the algebraic multigrid and the functional maps literatures, we successfully coarsen a variety of isotropic and anisotropic operators while maintaining sparsity and positive semi-definiteness. We demonstrate the utility of this approach for applications including operatorsensitive sampling, shape matching, and graph pooling for convolutional neural networks.

The image processing pipeline boasts a wide variety of complex filters and effects. Translating an individual effect to operate on 3D surface geometry inevitably results in a bespoke algorithm. Instead, we propose a general-purpose back-end optimization that allows users to edit an input 3D surface by simply selecting an off-the-shelf image processing filter. We achieve this by constructing a differentiable triangle mesh renderer, with which we can back propagate changes in the image domain to the 3D mesh vertex positions. The given image processing technique is applied to the entire shape via stochastic snapshots of the shape: hence, we call our method Paparazzi. We provide simple yet important design considerations to construct the Paparazzi renderer and optimization algorithms. The power of this rendering-based surface editing is demonstrated via the variety of image processing filters we apply. Each application uses an off-the-shelf implementation of an image processing method without requiring modification to the core Paparazzi algorithm.

Abstract The eigenfunctions and eigenvalues of the Laplace‐Beltrami operator have proven to be a powerful tool for digital geometry processing, providing a description of geometry that is essentially independent of coordinates or the choice of discretization. However, since Laplace‐Beltrami is purely intrinsic it struggles to capture important phenomena such as extrinsic bending, sharp edges, and fine surface texture. We introduce a new extrinsic differential operator called the relative Dirac operator, leading to a family of operators with a continuous trade‐off between intrinsic and extrinsic features. Previous operators are either fully or partially intrinsic. In contrast, the proposed family spans the entire spectrum: from completely intrinsic (depending only on the metric) to completely extrinsic (depending only on the Gauss map). By adding an infinite potential well to this (or any) operator we can also robustly handle surface patches with irregular boundary. We explore use of these operators for a variety of shape analysis tasks, and study their performance relative to operators previously found in the geometry processing literature.