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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 coarseintrinsic triangulationof the input domain. In the spirit of thequadric 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.

 

Physical systems ranging from elastic bodies to kinematic linkages are defined on high-dimensional configuration spaces, yet their typical low-energy configurations are concentrated on much lower-dimensional subspaces. This work addresses the challenge of identifying such subspaces automatically: given as input an energy function for a high-dimensional system, we produce a low-dimensional map whose image parameterizes a diverse yet low-energy submanifold of configurations. The only additional input needed is a single seed configuration for the system to initialize our procedure; no dataset of trajectories is required. We represent subspaces as neural networks that map a low-dimensional latent vector to the full configuration space, and propose a training scheme to fit network parameters to any system of interest. This formulation is effective across a very general range of physical systems; our experiments demonstrate not only nonlinear and very low-dimensional elastic body and cloth subspaces, but also more general systems like colliding rigid bodies and linkages. We briefly explore applications built on this formulation, including manipulation, latent interpolation, and sampling. 

We introduce a new general-purpose approach to deep learning on three-dimensional surfaces based on the insight that a simple diffusion layer is highly effective for spatial communication. The resulting networks are automatically robust to changes in resolution and sampling of a surface—a basic property that is crucial for practical applications. Our networks can be discretized on various geometric representations, such as triangle meshes or point clouds, and can even be trained on one representation and then applied to another. We optimize the spatial support of diffusion as a continuous network parameter ranging from purely local to totally global, removing the burden of manually choosing neighborhood sizes. The only other ingredients in the method are a multi-layer perceptron applied independently at each point and spatial gradient features to support directional filters. The resulting networks are simple, robust, and efficient. Here, we focus primarily on triangle mesh surfaces and demonstrate state-of-the-art results for a variety of tasks, including surface classification, segmentation, and non-rigid correspondence. 

This course provides a first introduction to intrinsic triangulations and their use in mesh processing algorithms. As geometric data becomes more ubiquitous, e.g., in applications such as augmented reality or machine learning, there is a pressing need to develop algorithms that work reliably on low-quality data. Intrinsic triangulations provide a powerful framework for these problems, by de-coupling the mesh used to encode geometry from the one used for computation. The basic shift in perspective is to encode the geometry of a mesh not in terms of ordinary vertex positions, but instead only in terms of edge lengths. Intrinsic triangulations have a long history in mathematics, but only in recent years have been applied to practical geometric computing. The course begins by giving motivation for intrinsic triangulations in terms of recent problems in computer graphics, followed by an interactive coding session where participants can make first contact with the idea of intrinsic meshes. We then give some mathematical background, and describe key data structures (overlay, signpost, normal coordinates). Using this machinery, we translate algorithms from computational geometry and scientific computing into cutting-edge algorithms for curved surfaces. For instance, we look at mesh parameterization, vector field processing, finding geodesics, solving partial differential equations (PDEs), and more. We also discuss processing of nonmanifold meshes and point clouds; participants can explore these algorithms via interactive demos. We conclude with a discussion of open questions and opportunities for future work. 

We describe a discrete Laplacian suitable for any triangle mesh, including those that are nonmanifold or nonorientable (with or without boundary). Our Laplacian is a robust drop‐in replacement for the usual cotan matrix, and is guaranteed to have nonnegative edge weights on both interior and boundary edges, even for extremely poor‐quality meshes. The key idea is to build what we call a “tufted cover” over the input domain, which has nonmanifold vertices but manifold edges. Since all edges are manifold, we can flip to an intrinsic Delaunay triangulation; our Laplacian is then the cotan Laplacian of this new triangulation. This construction also provides a high‐quality point cloud Laplacian, via a nonmanifold triangulation of the point set. We validate our Laplacian on a variety of challenging examples (including all models from Thingi10k), and a variety of standard tasks including geodesic distance computation, surface deformation, parameterization, and computing minimal surfaces.

 

This paper develops a global variational approach to cutting curved surfaces so that they can be flattened into the plane with low metric distortion. Such cuts are a critical component in a variety of algorithms that seek to parameterize surfaces over flat domains, or fabricate structures from flat materials. Rather than evaluate the quality of a cut solely based on properties of the curve itself (e.g., its length or curvature), we formulate a flow that directly optimizes the distortion induced by cutting and flattening. Notably, we do not have to explicitly parameterize the surface in order to evaluate the cost of a cut, but can instead integrate a simple evolution equation defined on the cut curve itself. We arrive at this flow via a novel application of shape derivatives to the Yamabe equation from conformal geometry. We then develop an Eulerian numerical integrator on triangulated surfaces, which does not restrict cuts to mesh edges and can incorporate user-defined data such as importance or occlusion. The resulting cut curves can be used to drive distortion to arbitrarily low levels, and have a very different character from cuts obtained via purely discrete formulations. We briefly explore potential applications to computational design, as well as connections to space filling curves and the problem of uniform heat distribution.