Denis Zorin

Leyi Zhu, Michael Tao, Yixin Hu et al.

We introduce BijectiveRemesh, a robust algorithm for maintaining a continuous, bijective mapping across complex remeshing sequences on both 2D triangle surfaces and 3D tetrahedral meshes. Unlike traditional data transfer methods that rely on interpolation or projection, our approach constructs a mathematically rigorous composite map from the input mesh to the output mesh by chaining local bijective atlases defined for each primitive remeshing operation. Our framework represents the overall mapping as a composition of local bijective atlases, one per remeshing operation. Building upon successive self-parameterization, we introduce a Shared Scaffold structure for 2D triangle meshes that enforces global bijectivity through local orientation preservation. We extend this approach to handle edge splits, edge swaps, and vertex smoothing beyond the original edge collapses. For 3D tetrahedral meshes, we generalize the local atlas construction using Steinitz's Theorem and Maxwell-Cremona lifting to ensure valid embeddings. This enables exact tracking of geometric entities, including points, curves, and surfaces, across remeshing, with applications from texture transfer to volumetric simulations.

Ehssan Nazockdast, Abtin Rahimian, Denis Zorin et al.

We present a novel platform for the large-scale simulation of fibrous structures immersed in a Stokesian fluid and evolving under confinement or in free-space. One of the main motivations for this work is to study the dynamics of fiber assemblies within biological cells. For this, we also incorporate the key biophysical elements that determine the dynamics of these assemblies, which include the polymerization and depolymerization kinetics of fibers, their interactions with molecular motors and other objects, their flexibility, and hydrodynamic coupling. This work, to our knowledge, is the first technique to include many-body hydrodynamic interactions (HIs), and the resulting fluid flows, in cellular fiber assemblies. We use the non-local slender body theory to compute the fluid-structure interactions of the fibers and a second-kind boundary integral formulation for other rigid bodies and the confining boundary. A kernel-independent implementation of the fast multiple method is utilized for efficient evaluation of HIs. The deformation of the fibers is described by the nonlinear Euler--Bernoulli beam theory and their polymerization is modeled by the reparametrization of the dynamic equations in the appropriate non-Lagrangian frame. We use a pseudo-spectral representation of fiber positions and implicit HIs in the time-stepping to resolve large fiber deformations, and to allow time-steps not constrained by temporal stiffness or fiber-fiber interactions. The entire computational scheme is parallelized, which enables simulating assemblies of thousands of fibers. We use our method to investigate two important questions in the mechanics of cell division: (i) the effect of confinement on the hydrodynamic mobility of microtubule asters; and (ii) the dynamics of the positioning of mitotic spindle in complex cell geometries.

Teseo Schneider, Yixin Hu, Xifeng Gao et al.

The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. The popularity of FEM led to the development of a large family of variants, most of which require a tetrahedral or hexahedral mesh to construct the basis. While the theoretical properties of FEM basis (such as convergence rate, stability, etc.) are well understood under specific assumptions on the mesh quality, their practical performance, influenced both by the choice of the basis construction and quality of mesh generation, have not been systematically documented for large collections of automatically meshed 3D geometries. We introduce a set of benchmark problems involving most commonly solved elliptic PDEs, starting from simple cases with an analytical solution, moving to commonly used test problem setups, and using manufactured solutions for thousands of real-world, automatically meshed geometries. For all these cases, we use state-of-the-art meshing tools to create both tetrahedral and hexahedral meshes, and compare the performance of different element types for common elliptic PDEs. The goal of his benchmark is to enable comparison of complete FEM pipelines, from mesh generation to algebraic solver, and exploration of relative impact of different factors on the overall system performance.

Teseo Schneider, Jeremie Dumas, Xifeng Gao et al.

We introduce an integrated meshing and finite element method pipeline enabling black-box solution of partial differential equations in the volume enclosed by a boundary representation. We construct a hybrid hexahedral-dominant mesh, which contains a small number of star-shaped polyhedra, and build a set of high-order basis on its elements, combining triquadratic B-splines, triquadratic hexahedra (27 degrees of freedom), and harmonic elements. We demonstrate that our approach converges cubically under refinement, while requiring around 50% of the degrees of freedom than a similarly dense hexahedral mesh composed of triquadratic hexahedra. We validate our approach solving Poisson's equation on a large collection of models, which are automatically processed by our algorithm, only requiring the user to provide boundary conditions on their surface.

Abtin Rahimian, Alex Barnett, Denis Zorin

We introduce a quadrature scheme--QBKIX--for the high-order accurate evaluation of layer potentials associated with general elliptic PDEs near to and on the domain boundary. Relying solely on point evaluations of the underlying kernel, our scheme is essentially PDE-independent; in particular, no analytic expansion nor addition theorem is required. Moreover, it applies to boundary integrals with singular, weakly singular, and hypersingular kernels. Our work builds upon Quadrature by Expansion (QBX), which approximates the potential by an analytic expansion in the neighborhood of each expansion center. In contrast, we use a sum of fundamental solutions lying on a ring enclosing the neighborhood, and solve a small dense linear system for their coefficients to match the potential on a smaller concentric ring. We test the new method with Laplace, Helmholtz, Yukawa, Stokes, and Navier (elastostatic) kernels in two dimensions (2D) using adaptive, panel-based boundary quadratures on smooth and corner domains. Advantages of the algorithm include its relative simplicity of implementation, immediate extension to new kernels, dimension-independence (allowing simple generalization to 3D), and compatibility with fast algorithms such as the kernel-independent FMM.

Oleg Voynov, Gleb Bobrovskikh, Pavel Karpyshev et al.

We present a new multi-sensor dataset for multi-view 3D surface reconstruction. It includes registered RGB and depth data from sensors of different resolutions and modalities: smartphones, Intel RealSense, Microsoft Kinect, industrial cameras, and structured-light scanner. The scenes are selected to emphasize a diverse set of material properties challenging for existing algorithms. We provide around 1.4 million images of 107 different scenes acquired from 100 viewing directions under 14 lighting conditions. We expect our dataset will be useful for evaluation and training of 3D reconstruction algorithms and for related tasks. The dataset is available at skoltech3d.appliedai.tech.

Libin Lu, Abtin Rahimian, Denis Zorin

We present an efficient, accurate, and robust method for simulation of dense suspensions of deformable and rigid particles immersed in Stokesian fluid in two dimensions. We use a well-established boundary integral formulation for the problem as the foundation of our approach. This type of formulations, with a high-order spatial discretization and an implicit and adaptive time discretization, have been shown to be able to handle complex interactions between particles with high accuracy. Yet, for dense suspensions, very small time-steps or expensive implicit solves as well as a large number of discretization points are required to avoid non-physical contact and intersections between particles, leading to infinite forces and numerical instability. Our method maintains the accuracy of previous methods at a significantly lower cost for dense suspensions. The key idea is to ensure interference-free configuration by introducing explicit contact constraints into the system. While such constraints are unnecessary in the formulation, in the discrete form of the problem, they make it possible to eliminate catastrophic loss of accuracy by preventing contact explicitly. Introducing contact constraints results in a significant increase in stable time-step size for explicit time-stepping, and a reduction in the number of points adequate for stability.

Libin Lu, Abtin Rahimian, Denis Zorin

We present a parallel-scalable method for simulating non-dilute suspensions of deformable particles immersed in Stokesian fluid in three dimensions. A critical component in these simulations is robust and accurate collision handling. This work complements our previous work [L. Lu, A. Rahimian, and D. Zorin. Contact-aware simulations of particulate Stokesian suspensions. Journal of Computational Physics 347C: 160-182] by extending it to 3D and by introducing new parallel algorithms for collision detection and handling. We use a well-established boundary integral formulation with spectral Galerkin method to solve the fluid flow. The key idea is to ensure an interference-free particle configuration by introducing explicit contact constraints into the system. While such constraints are typically unnecessary in the formulation they make it possible to eliminate catastrophic loss of accuracy in the discretized problem by preventing contact explicitly. The incorporation of contact constraints results in a significant increase in stable time-step size for locally-implicit time-stepping and a reduction in the necessary number of discretization points for stability. Our method maintains the accuracy of previous methods at a significantly lower cost for dense suspensions and the time step size is independent from the volume fraction. Our method permits simulations with high volume fractions; we report results with up to 60% volume fraction. We demonstrated the parallel scaling of the algorithms on up to 16K CPU cores.

Igor Ostanin, Denis Zorin, Ivan Oseledets

Wide variety of engineering design tasks can be formulated as constrained optimization problems where the shape and topology of the domain are optimized to reduce costs while satisfying certain constraints. Several mathematical approaches were developed to address the problem of finding optimal design of an engineered structure. Recent works have demonstrated the feasibility of boundary element method as a tool for topological-shape optimization. However, it was noted that the approach has certain drawbacks, and in particular high computational cost of the iterative optimization process. In this short note we suggest ways to address critical limitations of boundary element method as a tool for topological-shape optimization. We validate our approaches by supplementing the existing complex variables boundary element code for elastostatic problems with robust tools for fast topological-shape optimization. The efficiency of the approach is illustrated with a numerical example.

Igor Ostanin, Ivan Tsybulin, Mikhail Litsarev et al.

The paper presents a new method for shape and topology optimization based on an efficient and scalable boundary integral formulation for elasticity. To optimize topology, our approach uses iterative extraction of isosurfaces of a topological derivative. The numerical solution of the elasticity boundary value problem at every iteration is performed with the boundary element formulation and the kernel-independent fast multipole method. Providing excellent single node performance, scalable parallelization and the best available asymptotic complexity, our method is among the fastest optimization tools available today. The performance of our approach is studied on few illustrative examples, including the optimization of engineered constructions for the minimum compliance and the optimization of the microstructure of a metamaterial for the desired macroscopic tensor of elasticity.

Natalia Soboleva, Olga Gorbunova, Maria Ivanova et al.

Surface reconstruction with preservation of geometric features is a challenging computer vision task. Despite significant progress in implicit shape reconstruction, state-of-the-art mesh extraction methods often produce aliased, perceptually distorted surfaces and lack scalability to high-resolution 3D shapes. We present a data-driven approach for automatic feature detection and remeshing that requires only a coarse, aliased mesh as input and scales to arbitrary resolution reconstructions. We define and learn a collection of surface-based fields to (1) capture sharp geometric features in the shape with an implicit vertexwise model and (2) approximate improvements in normals alignment obtained by applying edge-flips with an edgewise model. To support scaling to arbitrary complexity shapes, we learn our fields using local triangulated patches, fusing estimates on complete surface meshes. Our feature remeshing algorithm integrates the learned fields as sharp feature priors and optimizes vertex placement and mesh connectivity for maximum expected surface improvement. On a challenging collection of high-resolution shape reconstructions in the ABC dataset, our algorithm improves over state-of-the-art by 26% normals F-score and 42% perceptual $\text{RMSE}_{\text{v}}$.

Venkatesh Pattabiraman, Zizhou Huang, Daniele Panozzo et al.

If human experience is any guide, operating effectively in unstructured environments -- like homes and offices -- requires robots to sense the forces during physical interaction. Yet, the lack of a versatile, accessible, and easily customizable tactile sensor has led to fragmented, sensor-specific solutions in robotic manipulation -- and in many cases, to force-unaware, sensorless approaches. With eFlesh, we bridge this gap by introducing a magnetic tactile sensor that is low-cost, easy to fabricate, and highly customizable. Building an eFlesh sensor requires only four components: a hobbyist 3D printer, off-the-shelf magnets (<$5), a CAD model of the desired shape, and a magnetometer circuit board. The sensor is constructed from tiled, parameterized microstructures, which allow for tuning the sensor's geometry and its mechanical response. We provide an open-source design tool that converts convex OBJ/STL files into 3D-printable STLs for fabrication. This modular design framework enables users to create application-specific sensors, and to adjust sensitivity depending on the task. Our sensor characterization experiments demonstrate the capabilities of eFlesh: contact localization RMSE of 0.5 mm, and force prediction RMSE of 0.27 N for normal force and 0.12 N for shear force. We also present a learned slip detection model that generalizes to unseen objects with 95% accuracy, and visuotactile control policies that improve manipulation performance by 40% over vision-only baselines -- achieving 91% average success rate for four precise tasks that require sub-mm accuracy for successful completion. All design files, code and the CAD-to-eFlesh STL conversion tool are open-sourced and available on https://e-flesh.com.

Eduardo Corona, Per-Gunnar Martinsson, Denis Zorin

An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal blocks of certain dense matrices have numerically low rank. Technically, the solver is inspired by previously developed direct solvers for integral equations based on "recursive skeletonization" and "Hierarchically Semi-Separable" (HSS) matrices, but it improves on the asymptotic complexity of existing solvers by incorporating an additional level of compression. The resulting solver has optimal O(N) complexity for all stages of the computation, as demonstrated by both theoretical analysis and numerical examples. The computational examples further display good practical performance in terms of both speed and memory usage. In particular, it is demonstrated that even problems involving 10^{7} unknowns can be solved to precision 10^{-10} using a simple Matlab implementation of the algorithm executed on a single core.

Francis Williams, Zan Gojcic, Sameh Khamis et al.

We present Neural Kernel Fields: a novel method for reconstructing implicit 3D shapes based on a learned kernel ridge regression. Our technique achieves state-of-the-art results when reconstructing 3D objects and large scenes from sparse oriented points, and can reconstruct shape categories outside the training set with almost no drop in accuracy. The core insight of our approach is that kernel methods are extremely effective for reconstructing shapes when the chosen kernel has an appropriate inductive bias. We thus factor the problem of shape reconstruction into two parts: (1) a backbone neural network which learns kernel parameters from data, and (2) a kernel ridge regression that fits the input points on-the-fly by solving a simple positive definite linear system using the learned kernel. As a result of this factorization, our reconstruction gains the benefits of data-driven methods under sparse point density while maintaining interpolatory behavior, which converges to the ground truth shape as input sampling density increases. Our experiments demonstrate a strong generalization capability to objects outside the train-set category and scanned scenes. Source code and pretrained models are available at https://nv-tlabs.github.io/nkf.

Karl Otness, Arvi Gjoka, Joan Bruna et al.

Simulating physical systems is a core component of scientific computing, encompassing a wide range of physical domains and applications. Recently, there has been a surge in data-driven methods to complement traditional numerical simulations methods, motivated by the opportunity to reduce computational costs and/or learn new physical models leveraging access to large collections of data. However, the diversity of problem settings and applications has led to a plethora of approaches, each one evaluated on a different setup and with different evaluation metrics. We introduce a set of benchmark problems to take a step towards unified benchmarks and evaluation protocols. We propose four representative physical systems, as well as a collection of both widely used classical time integrators and representative data-driven methods (kernel-based, MLP, CNN, nearest neighbors). Our framework allows evaluating objectively and systematically the stability, accuracy, and computational efficiency of data-driven methods. Additionally, it is configurable to permit adjustments for accommodating other learning tasks and for establishing a foundation for future developments in machine learning for scientific computing.

Albert Matveev, Alexey Artemov, Denis Zorin et al.

We present a pipeline for parametric wireframe extraction from densely sampled point clouds. Our approach processes a scalar distance field that represents proximity to the nearest sharp feature curve. In intermediate stages, it detects corners, constructs curve segmentation, and builds a topological graph fitted to the wireframe. As an output, we produce parametric spline curves that can be edited and sampled arbitrarily. We evaluate our method on 50 complex 3D shapes and compare it to the novel deep learning-based technique, demonstrating superior quality.

Aleksandr Safin, Maxim Kan, Nikita Drobyshev et al.

Depth maps captured with commodity sensors are often of low quality and resolution; these maps need to be enhanced to be used in many applications. State-of-the-art data-driven methods of depth map super-resolution rely on registered pairs of low- and high-resolution depth maps of the same scenes. Acquisition of real-world paired data requires specialized setups. Another alternative, generating low-resolution maps from high-resolution maps by subsampling, adding noise and other artificial degradation methods, does not fully capture the characteristics of real-world low-resolution images. As a consequence, supervised learning methods trained on such artificial paired data may not perform well on real-world low-resolution inputs. We consider an approach to depth super-resolution based on learning from unpaired data. While many techniques for unpaired image-to-image translation have been proposed, most fail to deliver effective hole-filling or reconstruct accurate surfaces using depth maps. We propose an unpaired learning method for depth super-resolution, which is based on a learnable degradation model, enhancement component and surface normal estimates as features to produce more accurate depth maps. We propose a benchmark for unpaired depth SR and demonstrate that our method outperforms existing unpaired methods and performs on par with paired.

Gal Metzer, Rana Hanocka, Denis Zorin et al.

Establishing a consistent normal orientation for point clouds is a notoriously difficult problem in geometry processing, requiring attention to both local and global shape characteristics. The normal direction of a point is a function of the local surface neighborhood; yet, point clouds do not disclose the full underlying surface structure. Even assuming known geodesic proximity, calculating a consistent normal orientation requires the global context. In this work, we introduce a novel approach for establishing a globally consistent normal orientation for point clouds. Our solution separates the local and global components into two different sub-problems. In the local phase, we train a neural network to learn a coherent normal direction per patch (i.e., consistently oriented normals within a single patch). In the global phase, we propagate the orientation across all coherent patches using a dipole propagation. Our dipole propagation decides to orient each patch using the electric field defined by all previously orientated patches. This gives rise to a global propagation that is stable, as well as being robust to nearby surfaces, holes, sharp features and noise.

Alexey Bokhovkin, Vladislav Ishimtsev, Emil Bogomolov et al.

Recent advances in 3D semantic scene understanding have shown impressive progress in 3D instance segmentation, enabling object-level reasoning about 3D scenes; however, a finer-grained understanding is required to enable interactions with objects and their functional understanding. Thus, we propose the task of part-based scene understanding of real-world 3D environments: from an RGB-D scan of a scene, we detect objects, and for each object predict its decomposition into geometric part masks, which composed together form the complete geometry of the observed object. We leverage an intermediary part graph representation to enable robust completion as well as building of part priors, which we use to construct the final part mask predictions. Our experiments demonstrate that guiding part understanding through part graph to part prior-based predictions significantly outperforms alternative approaches to the task of semantic part completion.

Albert Matveev, Ruslan Rakhimov, Alexey Artemov et al.

We propose Deep Estimators of Features (DEFs), a learning-based framework for predicting sharp geometric features in sampled 3D shapes. Differently from existing data-driven methods, which reduce this problem to feature classification, we propose to regress a scalar field representing the distance from point samples to the closest feature line on local patches. Our approach is the first that scales to massive point clouds by fusing distance-to-feature estimates obtained on individual patches. We extensively evaluate our approach against related state-of-the-art methods on newly proposed synthetic and real-world 3D CAD model benchmarks. Our approach not only outperforms these (with improvements in Recall and False Positives Rates), but generalizes to real-world scans after training our model on synthetic data and fine-tuning it on a small dataset of scanned data. We demonstrate a downstream application, where we reconstruct an explicit representation of straight and curved sharp feature lines from range scan data.

Vladislav Ishimtsev, Alexey Bokhovkin, Alexey Artemov et al.

Shape retrieval and alignment are a promising avenue towards turning 3D scans into lightweight CAD representations that can be used for content creation such as mobile or AR/VR gaming scenarios. Unfortunately, CAD model retrieval is limited by the availability of models in standard 3D shape collections (e.g., ShapeNet). In this work, we address this shortcoming by introducing CAD-Deform, a method which obtains more accurate CAD-to-scan fits by non-rigidly deforming retrieved CAD models. Our key contribution is a new non-rigid deformation model incorporating smooth transformations and preservation of sharp features, that simultaneously achieves very tight fits from CAD models to the 3D scan and maintains the clean, high-quality surface properties of hand-modeled CAD objects. A series of thorough experiments demonstrate that our method achieves significantly tighter scan-to-CAD fits, allowing a more accurate digital replica of the scanned real-world environment while preserving important geometric features present in synthetic CAD environments.

Albert Matveev, Alexey Artemov, Denis Zorin et al.

Estimation of differential geometric quantities in discrete 3D data representations is one of the crucial steps in the geometry processing pipeline. Specifically, estimating normals and sharp feature lines from raw point cloud helps improve meshing quality and allows us to use more precise surface reconstruction techniques. When designing a learnable approach to such problems, the main difficulty is selecting neighborhoods in a point cloud and incorporating geometric relations between the points. In this study, we present a geometric attention mechanism that can provide such properties in a learnable fashion. We establish the usefulness of the proposed technique with several experiments on the prediction of normal vectors and the extraction of feature lines.

Ruslan Rakhimov, Emil Bogomolov, Alexandr Notchenko et al.

DensePose estimation task is a significant step forward for enhancing user experience computer vision applications ranging from augmented reality to cloth fitting. Existing neural network models capable of solving this task are heavily parameterized and a long way from being transferred to an embedded or mobile device. To enable Dense Pose inference on the end device with current models, one needs to support an expensive server-side infrastructure and have a stable internet connection. To make things worse, mobile and embedded devices do not always have a powerful GPU inside. In this work, we target the problem of redesigning the DensePose R-CNN model's architecture so that the final network retains most of its accuracy but becomes more light-weight and fast. To achieve that, we tested and incorporated many deep learning innovations from recent years, specifically performing an ablation study on 23 efficient backbone architectures, multiple two-stage detection pipeline modifications, and custom model quantization methods. As a result, we achieved $17\times$ model size reduction and $2\times$ latency improvement compared to the baseline model.

Francis Williams, Matthew Trager, Joan Bruna et al.

We present Neural Splines, a technique for 3D surface reconstruction that is based on random feature kernels arising from infinitely-wide shallow ReLU networks. Our method achieves state-of-the-art results, outperforming recent neural network-based techniques and widely used Poisson Surface Reconstruction (which, as we demonstrate, can also be viewed as a type of kernel method). Because our approach is based on a simple kernel formulation, it is easy to analyze and can be accelerated by general techniques designed for kernel-based learning. We provide explicit analytical expressions for our kernel and argue that our formulation can be seen as a generalization of cubic spline interpolation to higher dimensions. In particular, the RKHS norm associated with Neural Splines biases toward smooth interpolants.

Ruslan Rakhimov, Denis Volkhonskiy, Alexey Artemov et al.

The video generation task can be formulated as a prediction of future video frames given some past frames. Recent generative models for videos face the problem of high computational requirements. Some models require up to 512 Tensor Processing Units for parallel training. In this work, we address this problem via modeling the dynamics in a latent space. After the transformation of frames into the latent space, our model predicts latent representation for the next frames in an autoregressive manner. We demonstrate the performance of our approach on BAIR Robot Pushing and Kinetics-600 datasets. The approach tends to reduce requirements to 8 Graphical Processing Units for training the models while maintaining comparable generation quality.

Vage Egiazarian, Oleg Voynov, Alexey Artemov et al.

We present a new method for vectorization of technical line drawings, such as floor plans, architectural drawings, and 2D CAD images. Our method includes (1) a deep learning-based cleaning stage to eliminate the background and imperfections in the image and fill in missing parts, (2) a transformer-based network to estimate vector primitives, and (3) optimization procedure to obtain the final primitive configurations. We train the networks on synthetic data, renderings of vector line drawings, and manually vectorized scans of line drawings. Our method quantitatively and qualitatively outperforms a number of existing techniques on a collection of representative technical drawings.

Francis Williams, Matthew Trager, Claudio Silva et al.

We present a theoretical and empirical study of the gradient dynamics of overparameterized shallow ReLU networks with one-dimensional input, solving least-squares interpolation. We show that the gradient dynamics of such networks are determined by the gradient flow in a non-redundant parameterization of the network function. We examine the principal qualitative features of this gradient flow. In particular, we determine conditions for two learning regimes:kernel and adaptive, which depend both on the relative magnitude of initialization of weights in different layers and the asymptotic behavior of initialization coefficients in the limit of large network widths. We show that learning in the kernel regime yields smooth interpolants, minimizing curvature, and reduces to cubic splines for uniform initializations. Learning in the adaptive regime favors instead linear splines, where knots cluster adaptively at the sample points.

Oleg Voynov, Alexey Artemov, Vage Egiazarian et al.

RGBD images, combining high-resolution color and lower-resolution depth from various types of depth sensors, are increasingly common. One can significantly improve the resolution of depth maps by taking advantage of color information; deep learning methods make combining color and depth information particularly easy. However, fusing these two sources of data may lead to a variety of artifacts. If depth maps are used to reconstruct 3D shapes, e.g., for virtual reality applications, the visual quality of upsampled images is particularly important. The main idea of our approach is to measure the quality of depth map upsampling using renderings of resulting 3D surfaces. We demonstrate that a simple visual appearance-based loss, when used with either a trained CNN or simply a deep prior, yields significantly improved 3D shapes, as measured by a number of existing perceptual metrics. We compare this approach with a number of existing optimization and learning-based techniques.

Sebastian Koch, Albert Matveev, Zhongshi Jiang et al.

We introduce ABC-Dataset, a collection of one million Computer-Aided Design (CAD) models for research of geometric deep learning methods and applications. Each model is a collection of explicitly parametrized curves and surfaces, providing ground truth for differential quantities, patch segmentation, geometric feature detection, and shape reconstruction. Sampling the parametric descriptions of surfaces and curves allows generating data in different formats and resolutions, enabling fair comparisons for a wide range of geometric learning algorithms. As a use case for our dataset, we perform a large-scale benchmark for estimation of surface normals, comparing existing data driven methods and evaluating their performance against both the ground truth and traditional normal estimation methods.

Francis Williams, Teseo Schneider, Claudio Silva et al.

The reconstruction of a discrete surface from a point cloud is a fundamental geometry processing problem that has been studied for decades, with many methods developed. We propose the use of a deep neural network as a geometric prior for surface reconstruction. Specifically, we overfit a neural network representing a local chart parameterization to part of an input point cloud using the Wasserstein distance as a measure of approximation. By jointly fitting many such networks to overlapping parts of the point cloud, while enforcing a consistency condition, we compute a manifold atlas. By sampling this atlas, we can produce a dense reconstruction of the surface approximating the input cloud. The entire procedure does not require any training data or explicit regularization, yet, we show that it is able to perform remarkably well: not introducing typical overfitting artifacts, and approximating sharp features closely at the same time. We experimentally show that this geometric prior produces good results for both man-made objects containing sharp features and smoother organic objects, as well as noisy inputs. We compare our method with a number of well-known reconstruction methods on a standard surface reconstruction benchmark.

Ilya Kostrikov, Zhongshi Jiang, Daniele Panozzo et al.

We study data-driven representations for three-dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curvature directions --- this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models \emph{Surface Networks (SN)}. We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs.