Triangle Meshes Research Articles

Curved Image Triangulation Based on Differentiable Rendering

AbstractImage triangulation methods, which decompose an image into a series of triangles, are fundamental in artistic creation and image processing. This paper introduces a novel framework that integrates cubic Bézier curves into image triangulation, enabling the precise reconstruction of curved image features. Our developed framework constructs a well‐structured curved triangle mesh, effectively preventing overlaps between curves. A refined energy function, grounded in differentiable rendering, establishes a direct link between mesh geometry and rendering effects and is instrumental in guiding the curved mesh generation. Additionally, we derive an explicit gradient formula with respect to mesh parameters, facilitating the adaptive and efficient optimization of these parameters to fully leverage the capabilities of cubic Bézier curves. Through experimental and comparative analyses with state‐of‐the‐art methods, our approach demonstrates a significant enhancement in both numerical accuracy and visual quality.

An interface tracking method with triangle edge cuts

This paper introduces a volume-conserving interface tracking algorithm on unstructured triangle meshes. We propose to discretize the interface via triangle edge cuts which represent the intersections between the interface and the triangle mesh edges using a compact 6 numbers per triangle. This enables an efficient implicit representation of the sub-triangle polygonal material regions without explicitly storing connectivity information. Moreover, we propose an efficient advection algorithm for this interface representation that is based on geometric queries and does not require an optimization process. This advection algorithm is extended via an area correction step that enforces volume-conservation of the materials. We demonstrate the efficacy of our method on a variety of advection problems on a triangle mesh and compare its performance to existing interface tracking methods including VOF and MOF.

Fast spline collision detection (FSCD) algorithm for solving multiple contacts in real-time

Collision detection has been widely studied in the last decades. While plenty of solutions exist, certain simulation scenarios are still challenging when permanent contact and deformable bodies are involved. In this paper, we introduce a novel approach based on volumetric splines that is applicable to complex deformable tubes, such as in the simulation of colonoscopy and other endoscopies. The method relies on modeling radial control points, extracting surface information from a triangle mesh, and storing the volume information around a spline path. Such information is later used to compute the intersection between the object surfaces under the assumption of spatial coherence between neighboring splines. We analyze the method’s performance in terms of both speed and accuracy, comparing it with previous works. Results show that our method solves collisions between complex meshes with over 300k triangles, generating over 1,000 collisions per frame between objects while maintaining an average time of under 1ms without compromising accuracy.

Mesh Convolution With Continuous Filters for 3-D Surface Parsing.

Geometric feature learning for 3-D surfaces is critical for many applications in computer graphics and 3-D vision. However, deep learning currently lags in hierarchical modeling of 3-D surfaces due to the lack of required operations and/or their efficient implementations. In this article, we propose a series of modular operations for effective geometric feature learning from 3-D triangle meshes. These operations include novel mesh convolutions, efficient mesh decimation, and associated mesh (un)poolings. Our mesh convolutions exploit spherical harmonics as orthonormal bases to create continuous convolutional filters. The mesh decimation module is graphics processing unit (GPU)-accelerated and able to process batched meshes on-the-fly, while the (un)pooling operations compute features for upsampled/downsampled meshes. We provide an open-source implementation of these operations, collectively termed Picasso. Picasso supports heterogeneous mesh batching and processing. Leveraging its modular operations, we further contribute a novel hierarchical neural network for perceptual parsing of 3-D surfaces, named PicassoNet++. It achieves highly competitive performance for shape analysis and scene segmentation on prominent 3-D benchmarks. The code, data, and trained models are available at https://github.com/EnyaHermite/Picasso.

Piecewise Developable Modeling via Implicit Neural Deformation and Feature-Guided Cutting.

We propose a novel and automatic method to model shapes using a small set of discrete developable patches. Central to our approach is using implicit neural shape representation that makes our algorithm independent of tessellation and allows us to obtain the Gaussian curvature of each point analytically. With this powerful representation, we first deform the input shape to be an almost developable shape with clear and sparse salient feature curves. Then, we convert the deformed implicit field to a triangle mesh, which is further cut to disk topology along parts of the sparse feature curves. Finally, we achieve the resulting piecewise developable mesh by alternatingly optimizing discrete developability, enforcing manufacturability constraints, and merging patches. The feasibility and practicability of our method are demonstrated over various shapes. Compared to the state-of-the-art methods, our method achieves a better tradeoff between the number of developable patches and the approximation error.

Mesh Parameterization Meets Intrinsic Triangulations

AbstractA parameterization of a triangle mesh is a realization in the plane so that all triangles have positive signed area. Triangle mesh parameterizations are commonly computed by minimizing a distortion energy, measuring the distortions of the triangles as they are mapped into the parameter domain. It is assumed that the triangulation is fixed and the triangles are mapped affinely. We consider a more general setup and additionally optimize among the intrinsic triangulations of the piecewise linear input geometry. This means the distortion energy is computed for the same geometry, yet the space of possible parameterizations is enlarged. For minimizing the distortion energy, we suggest alternating between varying the parameter locations of the vertices and intrinsic flipping. We show that this process improves the mapping for different distortion energies at moderate additional cost. We also find intrinsic triangulations that are better starting points for the optimization of positions, offering a compromise between the full optimization approach and exploiting the additional freedom of intrinsic triangulations.

Differentiable Geodesic Distance for Intrinsic Minimization on Triangle Meshes

Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

NeurCADRecon: Neural Representation for Reconstructing CAD Surfaces by Enforcing Zero Gaussian Curvature

Despite recent advances in reconstructing an organic model with the neural signed distance function (SDF), the high-fidelity reconstruction of a CAD model directly from low-quality unoriented point clouds remains a significant challenge. In this paper, we address this challenge based on the prior observation that the surface of a CAD model is generally composed of piecewise surface patches, each approximately developable even around the feature line. Our approach, named NeurCADRecon , is self-supervised, and its loss includes a developability term to encourage the Gaussian curvature toward 0 while ensuring fidelity to the input points (see the teaser figure). Noticing that the Gaussian curvature is non-zero at tip points, we introduce a double-trough curve to tolerate the existence of these tip points. Furthermore, we develop a dynamic sampling strategy to deal with situations where the given points are incomplete or too sparse. Since our resulting neural SDFs can clearly manifest sharp feature points/lines, one can easily extract the feature-aligned triangle mesh from the SDF and then decompose it into smooth surface patches, greatly reducing the difficulty of recovering the parametric CAD design. A comprehensive comparison with existing state-of-the-art methods shows the significant advantage of our approach in reconstructing faithful CAD shapes.

A Heat Method for Generalized Signed Distance

We introduce a method for approximating the signed distance function (SDF) of geometry corrupted by holes, noise, or self-intersections. The method implicitly defines a completed version of the shape, rather than explicitly repairing the given input. Our starting point is a modified version of the heat method for geodesic distance, which diffuses normal vectors rather than a scalar distribution. This formulation provides robustness akin to generalized winding numbers (GWN) , but provides distance function rather than just an inside/outside classification. Our formulation also offers several features not common to classic distance algorithms, such as the ability to simultaneously fit multiple level sets, a notion of distance for geometry that does not topologically bound any region, and the ability to mix and match signed and unsigned distance. The method can be applied in any dimension and to any spatial discretization, including triangle meshes, tet meshes, point clouds, polygonal meshes, voxelized surfaces, and regular grids. We evaluate the method on several challenging examples, implementing normal offsets and other morphological operations directly on imperfect curve and surface data. In many cases we also obtain an inside/outside classification dramatically more robust than the one obtained provided by GWN.

Laplacian2Mesh: Laplacian-Based Mesh Understanding.

Geometric deep learning has sparked a rising interest in computer graphics to perform shape understanding tasks, such as shape classification and semantic segmentation. When the input is a polygonal surface, one has to suffer from the irregular mesh structure. Motivated by the geometric spectral theory, we introduce Laplacian2Mesh, a novel and flexible convolutional neural network (CNN) framework for coping with irregular triangle meshes (vertices may have any valence). By mapping the input mesh surface to the multi-dimensional Laplacian-Beltrami space, Laplacian2Mesh enables one to perform shape analysis tasks directly using the mature CNNs, without the need to deal with the irregular connectivity of the mesh structure. We further define a mesh pooling operation such that the receptive field of the network can be expanded while retaining the original vertex set as well as the connections between them. Besides, we introduce a channel-wise self-attention block to learn the individual importance of feature ingredients. Laplacian2Mesh not only decouples the geometry from the irregular connectivity of the mesh structure but also better captures the global features that are central to shape classification and segmentation. Extensive tests on various datasets demonstrate the effectiveness and efficiency of Laplacian2Mesh, particularly in terms of the capability of being vulnerable to noise to fulfill various learning tasks.

A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains

We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.

A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry Processing

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. Such manifold PDEs often 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 the closest 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.

Procedural generation of geometric patterns for thin shell fabrication

This paper addresses the design of surface shells as assemblies of tileable, flat geometric patterns with predictable performance in response to mechanical stimuli. We design a family of tileable and fabricable patterns represented as triangle meshes, which can be assembled for creating surface tessellations. First, a regular recursive subdivision of the planar space generates different geometric configurations for candidate patterns, having interesting and varied aesthetic properties. Then, a refinement step addresses manufacturability by solving for non-manifold configurations and sharp angles that would produce disconnected or weak patterns. We devise a strategy for creating continuous variations on the geometry of individual patterns, in both aesthetics and behavior, to enrich the catalog of available designs. Finally, we simulate our patterns to evaluate their mechanical response when loaded in different scenarios targeting out-of-plane bending. Through a simple browsing interface, we show that our patterns span a variety of different bending behaviors. The result is a catalog of patterns with varied aesthetics and predefined mechanical behavior, to use for the direct design of mechanical metamaterials. To assess the feasibility of our design-to-fabricate approach, we show fabricated 3D objects with different curvatures, and compare physical and simulated experiments.

Retargeting of facial model for unordered dense point cloud

Facial retargeting is a widely used technique in the game and film industries that replicates the expressions of a source facial model onto a target model. Existing methods for facial retargeting rely on either hand-crafted uniform triangle meshes or sparse points obtained from motion capture(mocap). In this paper, we propose an end-to-end facial retargeting algorithm that copies facial expressions from unordered dense point clouds onto the target model. First, a corresponding building method based on bi-harmonic function is introduced to ensure that the template model and a cluster of point clouds share the same triangle topology. Second, a deformation transferring method is presented to transfer the calculated deformation onto the target model. Several experiments are conducted on the SIAT-3DFE dataset to demonstrate the accuracy and efficiency of our method.

Spatio-Temporal Hierarchical Data Structure Based on Sparse Voxel Octrees for Representation of the Geometry of Time-Variable Three-Dimensional Scenes

Abstract The paper deals with the problematics of representation of the geometry of three-dimensional scenes in computer graphics using domain-specific hierarchical data structures based on octant trees or directed acyclic graphs as the means of space saving binary representation of this information. Several hierarchical data structures were introduced in this field in past decades. In this paper we are moving this problematic to the field of the representation of three-dimensional scenes which are rather dynamic in time, so their geometry is changing on the regular basis. We are investigating the influence of this time-variability on the size of the related hierarchical data structure on binary level. We are comparing the approach in which each time step is considered as the stand-alone static three-dimensional scene and therefore is represented by its own hierarchical data structure with the approach when this time-variable scene is considered as the four dimensional space and spatio-temporal sparse voxel octree hierarchical data structure is used for the representation of the geometry of this scene. We tested both approaches on three different scenes which were obtained by voxelization into two different resolutions of polygonal surface models represented originally as the mesh of triangles in Wavefront Object file format. Results of the tests show that spatio-temporal sparse voxel octree is better solution, when we obtained from 1.73 to 2.08 more compact binary representation, in comparison to the solution when time frames were encoded separately as the set of sparse voxel octrees.

Development of Node-centered finite-volume diffusion method on triangle mesh for detonation propagation simulation in insensitive high explosives

This study presents a numerical simulation of detonation waves in insensitive high explosives (IHE). A direct numerical simulation (DNS) method of detonation waves propagation was developed. It solves two-dimensional reactive Euler equations by using a semi-discrete node-centered finite-volume (NCFV) scheme on triangle mesh. Employing ZND model analytical solution as the initial condition, the upper and lower boundary conditions were designed as local sonic equilibrium conditions. The DNS method was validated using steady detonation wave propagation experimental results for tri-amino-tri-nitro-benzene (TATB) based explosives. The two-dimensional steady detonation propagation of the circular arc experiments was completed using a non-embedded technique (electric and optical fibre probe velocimetry). The comparison results demonstrate that the numerical method can provide a good prediction of the pseudo-steady-state detonation wave front propagation and the angular speed.

Numerical simulation of acoustic waves propagation by finite element method based on optimized matrices

Abstract Based on the wave equation, scholars worldwide have proposed various methods for numerical simulation of seismic wave propagation in underground and surface media. The finite element method offers a unique advantage in accurately depicting the undulating surfaces and steep palaeoburial hills with its triangular mesh. However, its computational efficiency cannot meet our needs while lots of memories are occupied. To address this, we optimized and improved the critical Mass matrix and Stiffness matrix of spatial discretization of the acoustic wave equation. We first fully utilized the flexibility of triangles to fit different undulating terrains, then reorganized the numbering of triangle mesh nodes and elements to reduce the bandwidth of the matrices, and then used optimized matrices for solving. The Crank-Nicolson scheme was adopted for time discretization, and the Perfectly Matched Layer condition was utilized to eliminate false waves reflected from the boundary. The numerical experiments with simple and significant fluctuation models proved that this method can accelerate computational efficiency while ensuring computational accuracy.

Parallel Topology-aware Mesh Simplification on Terrain Trees

We address the problem of performing a topology-aware simplification algorithm on a compact and distributed data structure for triangle meshes, the Terrain trees. Topology-aware operators have been defined to coarsen a Triangulated Irregular Network (TIN) without affecting the topology of its underlying terrain, i.e., without modifying critical features of the terrain, such as pits, saddles, peaks, and their connectivity. However, their scalability is limited for large-scale meshes. Our proposed algorithm uses a batched processing strategy to reduce both the memory and time requirements of the simplification process, and thanks to the spatial decomposition on the basis of Terrain trees, it can be easily parallelized. Also, since a Terrain tree after the simplification process becomes less compact and efficient, we propose an efficient post-processing step for updating hierarchical spatial decomposition. Our experiments on real-world TINs, derived from topographic and bathymetric LiDAR data, demonstrate the scalability and efficiency of our approach. Specifically, topology-aware simplification on Terrain trees uses 40% less memory and half the time compared to the most compact and efficient connectivity-based data structure for TINs. Furthermore, the parallel simplification algorithm on the Terrain trees exhibits a 12× speedup with an OpenMP implementation. The quality of the output mesh is not significantly affected by the distributed and parallel simplification strategy of Terrain trees, and we obtain similar quality levels compared to the global baseline method.

Boundary SPH for Robust Particle–Mesh Interaction in Three Dimensions

This paper introduces an algorithm to tackle the boundary condition (BC) problem, which has long persisted in the numerical and computational treatment of smoothed particle hydrodynamics (SPH). Central to the BC problem is a need for an effective method to reconcile a numerical representation of particles with 2D or 3D geometry. We describe and evaluate an algorithmic solution—boundary SPH (BSPH)—drawn from a novel twist on the mesh-based boundary method, allowing SPH particles to interact (directly and implicitly) with either convex or concave 3D meshes. The method draws inspiration from existing works in graphics, particularly discrete signed distance fields, to determine whether particles are intersecting or submerged with mesh triangles. We evaluate the efficacy of BSPH through application to several simulation environments of varying mesh complexity, showing practical real-time implementation in Unity3D and its high-level shader language (HLSL), which we test in the parallelization of particle operations. To examine robustness, we portray slip and no-slip conditions in simulation, and we separately evaluate convex and concave meshes. To demonstrate empirical utility, we show pressure gradients as measured in simulated still water tank implementations of hydrodynamics. Our results identify that BSPH, despite producing irregular pressure values among particles close to the boundary manifolds of the meshes, successfully prevents particles from intersecting or submerging into the boundary manifold. Average FPS calculations for each simulation scenario show that the mesh boundary method can still be used effectively with simple simulation scenarios. We additionally point the reader to future works that could investigate the effect of simulation parameters and scene complexity on simulation performance, resolve abnormal pressure values along the mesh boundary, and test the method’s robustness on a wider variety of simulation environments.

Editing mesh sequences with varying connectivity

Time-varying connectivity of triangle mesh sequences leads to substantial difficulties in their processing. Unlike editing sequences with constant connectivity, editing sequences with varying connectivity requires addressing the problem of temporal correspondence between the frames of the sequence. We present a method for time-consistent editing of triangle mesh sequences with varying connectivity using sparse temporal correspondence, which can be obtained using existing methods. Our method includes a deformation model based on the usage of the sparse temporal correspondence, which is suitable for the temporal propagation of user-specified deformations of the edited surface with respect to the shape and true topology of the surface while preserving the individual connectivity of each frame. Since there is no other method capable of comparable types of editing on time-varying meshes, we compare our method and the proposed deformation model with a baseline approach and demonstrate the benefits of our framework.