for sharing implementations and results; Zih-Yin Chen for early discussions.

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.

Fig. 1. Cubic stylization deforms a given 3D shape into the style of a cube while maintaining textures and geometric features. This can be used as a non-realistic modeling tool for creating stylized 3D virtual world. We obtain 3D assets from sketchfab.com by smeerws and Jesús Orgaz licensed under CC BY 4.0.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-rigidas-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.

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.

Figure 1: We propose to simplify a mesh using edge collapses while aiming to preserve the input eigenvectors and eigenvalues as much as possible. While different strategies exist to reduce a mesh (here, from 25,727 vertices to 771 vertices, or 3% of its initial size), such as enforcing uniform edge lengths or using the Quadric Error Metric [GH97], they do not focus on keeping the spectral properties of the mesh. Reducing a mesh can be spectrally described using functional maps [OBCS * 12], shown here with the output meshes, and which should ideally be diagonal. We also evaluate functional maps using two norms, the Laplacian commutativity • L and the orthogonality • D .

Fig. 1. We introduce a novel geometric multigrid solver for curved surfaces. Our key ingredient is an intrinsic prolongation operator computed via parameterizing the high resolution shape via its coarsened counterpart, visualized using colored triangles. By recursively applying this self-parameterization, we obtain a hierarchy (from left to right) for our multigrid method (e.g., to solve heat geodesics [Crane et al. 2017], far left). ©model by Benoît Rogez under CC BY-NC.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.

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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.