Search

We introduce a novel adaptive eigenvalue filtering strategy to stabilize and accelerate the optimization of Neo-Hookean energy and its variants under the Projected Newton framework. For the first time, we show that Newton's method, Projected Newton with eigenvalue clamping and Projected Newton with absolute eigenvalue filtering can be unified using ideas from the generalized trust region method. Based on the trust-region fit, our model adaptively chooses the correct eigenvalue filtering strategy to apply during the optimization. Our method is simple but effective, requiring only two lines of code change in the existing Projected Newton framework. We validate our model outperforms stand-alone variants across a number of experiments on quasistatic simulation of deformable solids over a large dataset., Comment: SIGGRAPH Asia 2024 (Conference track). Project page: https://www.cs.columbia.edu/cg/trust-region/

This paper introduces a fast and robust method for simplifying 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 suffer from producing high-quality outputs for user-created meshes, which often contain non-manifold elements and multiple connected components. In this work, we begin by outlining the pitfalls of existing mesh simplification techniques and highlighting the discrepancy in their formulations with existing mesh data. We then propose a method for simplifying these (non-manifold) triangle meshes, while maintaining quality comparable to the existing methods for manifold inputs. Our key idea is to reformulate mesh simplification as a problem of decimating simplicial 2-complexes. This involves a novel construction to turn a triangle soup into a simplicial 2-complex, followed by iteratively collapsing 1-simplices (vertex pairs) with our modified quadric error metric tailored for topology changes. Besides, we also tackle textured mesh simplification. Instead of following existing strategies to preserve mesh UVs, we propose a novel perspective that only focuses on preserving texture colors defined on the surface, regardless of the layout in the texture UV space. This leads to a more robust method for textured mesh simplification that is free from the texture bleeding artifact. Our mesh simplification enables level-of-detail algorithms to operate on arbitrary triangle meshes in the wild. We demonstrate improvements over prior techniques through extensive qualitative and quantitative evaluations, along with user studies.

This paper presents a unified differentiable boolean operator for implicit solid shape modeling using Constructive Solid Geometry (CSG). Traditional CSG relies on min, max operators to perform boolean operations on implicit shapes. But because these boolean operators are discontinuous and discrete in the choice of operations, this makes optimization over the CSG representation challenging. Drawing inspiration from fuzzy logic, we present a unified boolean operator that outputs a continuous function and is differentiable with respect to operator types. This enables optimization of both the primitives and the boolean operations employed in CSG with continuous optimization techniques, such as gradient descent. We further demonstrate that such a continuous boolean operator allows modeling of both sharp mechanical objects and smooth organic shapes with the same framework. Our proposed boolean operator opens up new possibilities for future research toward fully continuous CSG optimization., Comment: SIGGRAPH'24

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.

Volume-preserving hyperelastic materials are widely used to model near-incompressible materials such as rubber and soft tissues. However, the numerical simulation of volume-preserving hyperelastic materials is notoriously challenging within this regime due to the non-convexity of the energy function. In this work, we identify the pitfalls of the popular eigenvalue clamping strategy for projecting Hessian matrices to positive semi-definiteness during Newton's method. We introduce a novel eigenvalue filtering strategy for projected Newton's method to stabilize the optimization of Neo-Hookean energy and other volume-preserving variants under high Poisson's ratio (near 0.5) and large initial volume change. Our method only requires a single line of code change in the existing projected Newton framework, while achieving significant improvement in both stability and convergence speed. We demonstrate the effectiveness and efficiency of our eigenvalue projection scheme on a variety of challenging examples and over different deformations on a large dataset., Comment: SIGGRAPH 2024 (Conference track). Project page: https://www.cs.columbia.edu/cg/abs-psd/

Traditional character animation specializes in characters with a rigidly articulated skeleton and a bipedal/quadripedal morphology. This assumption simplifies many aspects for designing physically based animations, like locomotion, but comes with the price of excluding characters of arbitrary deformable geometries. To remedy this, our framework makes use of a spatio-temporal actuation subspace built off of the natural vibration modes of the character geometry. The resulting actuation is coupled to a reduced fast soft body simulation, allowing us to formulate a locomotion optimization problem that is tractable for a wide variety of high resolution deformable characters., Comment: 15 pages, 14 figures

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., Comment: SIGGRAPH 2023

Neural implicit fields have recently emerged as a useful representation for 3D shapes. These fields are commonly represented as neural networks which map latent descriptors and 3D coordinates to implicit function values. The latent descriptor of a neural field acts as a deformation handle for the 3D shape it represents. Thus, smoothness with respect to this descriptor is paramount for performing shape-editing operations. In this work, we introduce a novel regularization designed to encourage smooth latent spaces in neural fields by penalizing the upper bound on the field's Lipschitz constant. Compared with prior Lipschitz regularized networks, ours is computationally fast, can be implemented in four lines of code, and requires minimal hyperparameter tuning for geometric applications. We demonstrate the effectiveness of our approach on shape interpolation and extrapolation as well as partial shape reconstruction from 3D point clouds, showing both qualitative and quantitative improvements over existing state-of-the-art and non-regularized baselines.

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., Comment: 13 pages, 27 figures, SIGGRAPH 2021

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., Comment: Eurographics Symposium on Geometry Processing

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 tailormade for visualization and optimize an operator separately for computation., Comment: 16 pages, 28 figures

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., Comment: 16 pages

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 l1-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., Comment: 10 pages, 28 figures, SIGGRAPH Asia 2019

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 for spectral 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 operator-sensitive sampling, shape matching, and graph pooling for convolutional neural networks., Comment: 13 pages, 30 figures. ACM Transactions on Graphics (SIGGRAPH) 2019

Many machine learning image classifiers are vulnerable to adversarial attacks, inputs with perturbations designed to intentionally trigger misclassification. Current adversarial methods directly alter pixel colors and evaluate against pixel norm-balls: pixel perturbations smaller than a specified magnitude, according to a measurement norm. This evaluation, however, has limited practical utility since perturbations in the pixel space do not correspond to underlying real-world phenomena of image formation that lead to them and has no security motivation attached. Pixels in natural images are measurements of light that has interacted with the geometry of a physical scene. As such, we propose the direct perturbation of physical parameters that underly image formation: lighting and geometry. As such, we propose a novel evaluation measure, parametric norm-balls, by directly perturbing physical parameters that underly image formation. One enabling contribution we present is a physically-based differentiable renderer that allows us to propagate pixel gradients to the parametric space of lighting and geometry. Our approach enables physically-based adversarial attacks, and our differentiable renderer leverages models from the interactive rendering literature to balance the performance and accuracy trade-offs necessary for a memory-efficient and scalable adversarial data augmentation workflow.