More Like this

1. Surface Networks

   Kostrikov, Ilya; Jiang, Zhongshi; Panozzo, Daniele; Zorin, Denis; Bruna, Joan (June 2018, IEEE Conference on Computer Vision and Pattern Recognition)

   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 curva- ture directions — this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models 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. 

   more » « less
2. Geometric analysis of the generalized surface quasi-geostrophic equations

   https://doi.org/10.1007/s00208-024-02867-z  

   Bauer, Martin; Heslin, Patrick; Misiołek, Gerard; Preston, Stephen_C (April 2024, Mathematische Annalen)

   Abstract We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach 

   more » « less
3. RXMesh: a GPU mesh data structure

   https://doi.org/10.1145/3450626.3459748  

   Mahmoud, Ahmed_H; Porumbescu, Serban_D; Owens, John_D (July 2021, ACM Transactions on Graphics)

   We propose a new static high-performance mesh data structure for triangle surface meshes on the GPU. Our data structure is carefully designed for parallel execution while capturing mesh locality and confining data access, as much as possible, within the GPU's fast shared memory. We achieve this by subdividing the mesh intopatchesand representing these patches compactly using a matrix-based representation. Our patching technique is decorated withribbons, thin mesh strips around patches that eliminate the need to communicate between different computation thread blocks, resulting in consistent high throughput. We call our data structureRXMesh: Ribbon-matriX Mesh. We hide the complexity of our data structure behind a flexible but powerful programming model that helps deliver high performance by inducing load balance even in highly irregular input meshes. We show the efficacy of our programming model on common geometry processing applications---mesh smoothing and filtering, geodesic distance, and vertex normal computation. For evaluation, we benchmark our data structure against well-optimized GPU and (single and multi-core) CPU data structures and show significant speedups. 

   more » « less
4. Computational Design of Self‐Actuated Surfaces by Printing Plastic Ribbons on Stretched Fabric

   https://doi.org/10.1111/cgf.14489  

   Jourdan, David; Skouras, Mélina; Vouga, Etienne; Bousseau, Adrien (May 2022, Computer Graphics Forum)

   Abstract We introduce a new mechanism for self‐actuating deployable structures, based on printing a dense pattern of closely‐spaced plastic ribbons on sheets of pre‐stretched elastic fabric. We leverage two shape‐changing effects that occur when such an assembly is printed and allowed to relax: first, the incompressible plastic ribbons frustrate the contraction of the fabric back to its rest state, forcing residual strain in the fabric and creating intrinsic curvature. Second, the differential compression at the interface between the plastic and fabric layers yields abilayer effectin the direction of the ribbons, making each ribbon buckle into an arc at equilibrium state and creating extrinsic curvature. We describe an inverse design tool to fabricate low‐cost, lightweight prototypes of freeform surfaces using the controllable directional distortion and curvature offered by this mechanism. The core of our method is a parameterization algorithm that bounds surface distortions along and across principal curvature directions, along with a pattern synthesis algorithm that covers a surface with ribbons to match the target distortions and curvature given by the aforementioned parameterization. We demonstrate the flexibility and accuracy of our method by fabricating and measuring a variety of surfaces, including nearly‐developable surfaces as well as surfaces with positive and negative mean curvature, which we achieve thanks to a simple hardware setup that allows printing on both sides of the fabric. 

   more » « less
5. A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry Processing

   https://doi.org/10.1145/3673652  

   King, Nathan; Su, Haozhe; Aanjaneya, Mridul; Ruuth, Steven; Batty, Christopher (June 2024, ACM Transactions on Graphics)

   Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in\(\mathbb {R}^2 \)or\(\mathbb {R}^3 \), such as curves or surfaces. Suchmanifold PDEsoften involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold’s interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of theclosest point method(CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM’s finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver that can scale to tens of millions of degrees of freedom, allowing PDEs to be solved on more complex manifolds. We demonstrate our method’s convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations. 

   more » « less