We introduce Geometric Multigrid Neural Networks (GMNN), a novel network structure for geometric deep learning on point clouds and surfaces. Convolutional neural networks face a common challenge: how can relevant features be communicated over longer distances? Our architecture facilitates long-distance communication with Geometric Multigrid Convolution (GMC) blocks, which apply convolutions in parallel to features defined on each level of a multigrid representation of the surface, and enable communication all the way up and down the hierarchy. We observe two major structural advantages of such a network: First, because each GMC operates on all levels of the multigrid hierarchy, even early stages can make use of coarse-scale information and receptive field grows rapidly with depth. Second, networks built with this backbone have the freedom to route information between different scales, including in ways not possible for other architectures. Because of these advantages, we find that a GMNN can combine the fast convergence of a shallow network with the greater expressiveness of a deeper, larger network. We build a GMNN from the components of a state-of-the-art U-Net, and find that on real tasks it can match or exceed the accuracy of the base network while using fewer epochs and roughly half the parameter count.

Geometric multigrid (GMG) methods are a fundamental tool for efficiently solving large sparse linear systems. A requirement for GMG is a hierarchy of grids; however, many practical volumetric domains are available only as single, irregular tetrahedral meshes, making the construction of a multigrid hierarchy necessary. Existing approaches often trade off speed against hierarchy quality: remeshing- or coarsening-based methods can be expensive to construct, whereas graph-based techniques are fast but often yield weaker multigrid performance. We introduce GravoTet, which bridges this gap by combining geometric structure with graph-based efficiency to construct fast and effective multigrid hierarchies. GravoTet builds a vertex hierarchy and then generates graph-Voronoi diagrams whose dual cells define coarse tetrahedra, enabling rapid construction of multigrid levels. Boundary elements are explicitly prioritized during both sampling and tet generation to preserve boundary. In our evaluation, we solve Poisson and biharmonic problems on irregular tetrahedral meshes and compare GravoTet against state-of-the-art geometric multigrid, algebraic multigrid and direct solvers, demonstrating superior performance, particularly on large meshes.

We introduce TetWeave, a novel isosurface representation for gradient-based mesh optimization that jointly optimizes the placement of a tetrahedral grid used for Marching Tetrahedra and a novel directional signed distance at each point. TetWeave constructs tetrahedral grids on-the-fly via Delaunay triangulation, enabling increased flexibility compared to predefined grids. The extracted meshes are guaranteed to be watertight, two-manifold and intersection-free. The flexibility of TetWeave enables a resampling strategy that places new points where reconstruction error is high and allows to encourage mesh fairness without compromising on reconstruction error. This leads to high-quality, adaptive meshes that require minimal memory usage and few parameters to optimize. Consequently, TetWeave exhibits near-linear memory scaling relative to the vertex count of the output mesh — a substantial improvement over predefined grids. We demonstrate the applicability of TetWeave to a broad range of challenging tasks in computer graphics and vision, such as multi-view 3D reconstruction, mesh compression and geometric texture generation. Our code is available at https://github.com/AlexandreBinninger/TetWeave.

Deep learning has improved vanishing point detection in images. Yet, deep networks require expensive annotated datasets trained on costly hardware and do not generalize to even slightly different domains, and minor problem variants. Here, we address these issues by injecting deep vanishing point detection networks with prior knowledge. This prior knowledge no longer needs to be learned from data, saving valuable annotation efforts and compute, unlocking realistic few-sample scenarios, and reducing the impact of domain changes. Moreover, the interpretability of the priors allows to adapt deep networks to minor problem variations such as switching between Manhattan and non-Manhattan worlds. We seamlessly incorporate two geometric priors: (i) Hough Transform – mapping image pixels to straight lines, and (ii) Gaussian sphere – mapping lines to great circles whose intersections denote vanishing points. Experimentally, we ablate our choices and show comparable accuracy to existing models in the large-data setting. We validate our model's improved data efficiency, robustness to domain changes, adaptability to non-Manhattan settings.

Learning from 3D point-cloud data has rapidly gained momentum, motivated by the success of deep learning on images and the increased availability of 3D data. In this paper, we aim to construct anisotropic convolution layers that work directly on the surface derived from a point cloud. This is challenging because of the lack of a global coordinate system for tangential directions on surfaces. We introduce DeltaConv, a convolution layer that combines geometric operators from vector calculus to enable the construction of anisotropic filters on point clouds. Because these operators are defined on scalar- and vector-fields, we separate the network into a scalar- and a vector-stream, which are connected by the operators. The vector stream enables the network to explicitly represent, evaluate, and process directional information. Our convolutions are robust and simple to implement and match or improve on state-of-the-art approaches on several benchmarks, while also speeding up training and inference.

Many universities digitize exams or the process of grading the exams. This potentially allows for faster grading, is less labor intensive and less error-prone. But are the grades produced by online grading consistent with how we grade on paper? In this paper we present preliminary results of the comparison between scores given by grading online and grading on paper.

This paper is concerned with a fundamental problem in geometric deep learning that arises in the construction of convolutional neural networks on surfaces. Due to curvature, the transport of filter kernels on surfaces results in a rotational ambiguity, which prevents a uniform alignment of these kernels on the surface. We propose a network architecture for surfaces that consists of vector-valued, rotation-equivariant features. The equivariance property makes it possible to locally align features, which were computed in arbitrary coordinate systems, when aggregating features in a convolution layer. The resulting network is agnostic to the choices of coordinate systems for the tangent spaces on the surface. We implement our approach for triangle meshes. Based on circular harmonic functions, we introduce convolution filters for meshes that are rotation-equivariant at the discrete level. We evaluate the resulting networks on shape correspondence and shape classifications tasks and compare their performance to other approaches.