Input Mesh Research Articles

Stress‐Aligned Hexahedral Lattice Structures

AbstractMaintaining the maximum stiffness of components with as little material as possible is an overarching objective in computational design and engineering. It is well‐established that in stiffness‐optimal designs, material is aligned with orthogonal principal stress directions. In the limit of material volume, this alignment forms micro‐structures resembling quads or hexahedra. Achieving a globally consistent layout of such orthogonal micro‐structures presents a significant challenge, particularly in three‐dimensional settings. In this paper, we propose a novel geometric algorithm for compiling stress‐aligned hexahedral lattice structures. Our method involves deforming an input mesh under load to align the resulting stress field along an orthogonal basis. The deformed object is filled with a hexahedral grid, and the deformation is reverted to recover the original shape. The resulting stress‐aligned mesh is used as basis for a final hollowing procedure, generating a volume‐reduced stiff infill composed of hexahedral micro‐structures. We perform quantitative comparisons with structural optimization and hexahedral meshing approaches and demonstrate the superior mechanical performance of our designs with finite element simulation experiments.

Deep neural helmholtz operators for 3D elastic wave propagation and inversion

Summary Numerical simulations of seismic wave propagation in heterogeneous 3D media are central to investigating subsurface structures and understanding earthquake processes, yet are computationally expensive for large problems. This is particularly problematic for full waveform inversion, which typically involves numerous runs of the forward process. In machine learning there has been considerable recent work in the area of operator learning, with a new class of models called neural operators allowing for data-driven solutions to partial differential equations. Recent works in seismology have shown that when neural operators are adequately trained, they can significantly shorten the compute time for wave propagation. However, the memory required for the 3D time domain equations may be prohibitive. In this study, we show that these limitations can be overcome by solving the wave equations in the frequency domain, also known as the Helmholtz equations, since the solutions for a set of frequencies can be determined in parallel. The 3D Helmholtz neural operator is 40 times more memory-efficient than an equivalent time-domain version. We employ a Helmholtz neural operator for 2D and 3D elastic wave modeling, achieving two orders of magnitude acceleration compared to a baseline spectral element method. The neural operator accurately generalizes to variable velocity structures and can be evaluated on denser input meshes than used in the training simulations. We also show that when solving for wavefields strictly on the surface, the accuracy can be significantly improved via a graph neural operator layer. In leveraging automatic differentiation, the proposed method can serve as an alternative to the adjoint-state approach for 3D full-waveform inversion, reducing the computation time by a factor of 350.

EasySkinning: Target-oriented skinning by mesh contraction and curve editing

Skinning, a critical process in animation that defines how bones influence the vertices of a 3D character model, significantly impacts the visual effect in animation production. Traditional methods are time-intensive and skill-dependent, whereas automatic techniques lack in flexibility and quality. Our research introduces EasySkinning, a user-friendly system applicable to complex meshes. This method comprises three key components: rigid weight initialization through Voronoi contraction, precise weight editing via curve tools, and smooth weight solving for reconstructing target deformations. EasySkinning begins by contracting the input mesh inwards to the skeletal bones, which improves vertex-to-bone mappings, particularly in intricate mesh areas. We also design intuitive curve-editing tools, allowing users to define more precise bone influential regions. The final stage employs advanced deformation algorithms for smooth weight solving, crucial for achieving desired animations. Through extensive experiments, we demonstrate that EasySkinning not only simplifies the creation of high-quality skinning weights but also consistently outperforms existing automatic and interactive skinning methods.

Neural shape completion for personalized Maxillofacial surgery

In this paper, we investigate the effectiveness of shape completion neural networks as clinical aids in maxillofacial surgery planning. We present a pipeline to apply shape completion networks to automatically reconstruct complete eumorphic 3D meshes starting from a partial input mesh, easily obtained from CT data routinely acquired for surgery planning. Most of the existing works introduced solutions to aid the design of implants for cranioplasty, i.e. all the defects are located in the neurocranium. In this work, we focus on reconstructing defects localized on both neurocranium and splanchnocranium. To this end, we introduce a new dataset, specifically designed for this task, derived from publicly available CT scans and subjected to a comprehensive pre-processing procedure. All the scans in the dataset have been manually cleaned and aligned to a common reference system. In addition, we devised a pre-processing stage to automatically extract point clouds from the scans and enrich them with virtual defects. We experimentally compare several state-of-the-art point cloud completion networks and identify the two most promising models. Finally, expert surgeons evaluated the best-performing network on a clinical case. Our results show how casting the creation of personalized implants as a problem of shape completion is a promising approach for automatizing this complex task.

Smooth Bijective Projection in a High-order Shell

We propose a new structure called a higher-order shell, which is composed of a set of triangular prisms. Each triangular prism is enveloped by three Bézier triangles (top, middle, and bottom) and three side surfaces, each of which is trimmed from a bilinear surface. Moreover, we define a continuous vector field to smoothly and bijectively transfer attributes between two surfaces inside the shell. Since the higher-order shell has several hard construction constraints, we apply an interior-point strategy to robustly and automatically construct a high-order shell for an input mesh. Specifically, the strategy starts from a valid linear shell with a small thickness. Then, the shell is optimized until the specified thickness is reached, where explicit checks ensure that the constraints are always satisfied. We extensively test our method on more than 8300 models, demonstrating its robustness and performance. Compared to state-of-the-art methods, our bijective projection is smoother, and the space between the shell and input mesh is more uniform.

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.

Adaptively Isotropic Remeshing Based on Curvature Smoothed Field.

With the development of 3D digital geometry technology, 3D triangular meshes are becoming more useful and valuable in industrial manufacturing and digital entertainment. A high quality triangular mesh can be used to represent a real world object with geometric and physical characteristics. While anisotropic meshes have advantages of representing shapes with sharp features (such as trimmed surfaces) more efficiently and accurately, isotropic meshes allow more numerically stable computations. When there is no anisotropic mesh requirement, isotropic triangles are always a good choice. In this paper, we propose a remeshing method to convert an input mesh into an adaptively isotropic one based on a curvature smoothed field (CSF). With the help of the CSF, adaptively isotropic remeshing can retain the curvature sensitivity, which enables more geometric features to be kept, and avoid the occurrence of obtuse triangles in the remeshed model as much as possible. The remeshed triangles with locally isotropic property benefit various geometric processes such as neighbor-based feature extraction and analysis. The experimental results show that our method achieves better balance between geometric feature preservation and mesh quality improvement compared to peers. We provide the implementation codes of our resampling method at github.com/vvvwo/Adaptively-Isotropic-Remeshing.

Scalar Field Prediction on Meshes Using Interpolated Multi-Resolution Convolutional Neural Networks

Abstract Scalar fields, such as stress or temperature fields, are often calculated in shape optimization and design problems in engineering. For complex problems where shapes have varying topology and cannot be parametrized, data-driven scalar field prediction can be faster than traditional finite element methods. However, current data-driven techniques to predict scalar fields are limited to a fixed grid domain, instead of arbitrary mesh structures. In this work, we propose a method to predict scalar fields on arbitrary meshes. It uses a convolutional neural network whose feature maps at multiple resolutions are interpolated to node positions before being fed into a multilayer perceptron to predict solutions to partial differential equations at mesh nodes. The model is trained on finite element von Mises stress fields, and once trained it can estimate stress values at each node on any input mesh. Two shape datasets are investigated, and the model has strong performance on both, with a median R-squared value of 0.91. We also demonstrate the model on a temperature field in a heat conduction problem, where its predictions have a median R-squared value of 0.99. Our method provides a potential flexible alternative to finite element analysis in engineering design contexts. Code and datasets are available at: https://github.com/kevinferg/sfp-cnn.

Interactive reverse engineering of CAD models

Reverse engineering Computer-Aided Design (CAD) models based on the original geometry is a valuable and challenging research problem that has numerous applications across various tasks. However, previous approaches have often relied on excessive manual interaction, leading to limitations in reconstruction speed. To mitigate this issue, in this study, we approach the reconstruction of a CAD model by sequentially constructing geometric primitives (such as vertices, edges, loops, and faces) and performing Boolean operations on the generated CAD modules. We address the complex reconstruction problem in four main steps. Firstly, we use a plane to cut the input mesh model and attain a loop cutting line, ensuring accurate normals. Secondly, the cutting line is automatically fitted to edges using primitive information and connected to form a primitive loop. This eliminates the need for time-consuming manual selection of each endpoint and significantly accelerates the reconstruction process. Subsequently, we construct the loop of primitives as a chunked CAD model through a series of CAD procedural operations, including extruding, lofting, revolving, and sweeping. Our approach incorporates an automatic height detection mechanism to minimize errors that may arise from manual designation of the extrusion height. Finally, by merging Boolean operations, these CAD models are assembled together to closely approximate the target geometry. We conduct a comprehensive evaluation of our algorithm using a diverse range of CAD models from both the Thingi10K dataset and real-world scans. The results validate that our method consistently delivers accurate, efficient, and robust reconstruction outcomes while minimizing the need for manual interactions. Furthermore, our approach demonstrates superior performance compared to competing methods, especially when applied to intricate geometries.

Generated realistic noise and rotation-equivariant models for data-driven mesh denoising

3D mesh denoising is a crucial pre-processing step in many graphics applications. However, existing data-driven mesh denoising models, primarily trained on synthetic white noise, are less effective when applied to real-world meshes with the noise of complex intensities and distributions. Moreover, how to comprehensively capture information from input meshes and apply suitable denoising models for feature-preserving mesh denoising remains a critical and unresolved challenge. This paper presents a rotation-Equivariant model-based Mesh Denoising (EMD) model and a Realistic Mesh Noise Generation (RMNG) model to address these issues. Our EMD model leverages rotation-equivariant features and self-attention weights of geodesic patches for more effective feature extraction, thereby achieving SOTA denoising results. The RMNG model, based on the Generative Adversarial Networks (GANs) architecture, generates massive amounts of realistic noisy and noiseless mesh pairs data for data-driven mesh denoising model training, significantly benefiting real-world denoising tasks. To address the smooth degradation and loss of sharp edges commonly observed in captured meshes, we further introduce varying levels of Laplacian smoothing to input meshes during the paired training data generation, endowing the trained denoising model with feature recovery capabilities. Experimental results demonstrate the superior performance of our proposed method in preserving fine-grained features while removing noise on real-world captured meshes.

Skeleton based tetrahedralization of surface meshes

We propose a new method for generating tetrahedralizations for 3D surface meshes. The method builds upon a segmentation of the mesh that forms a rooted skeleton structure. Each segment in the structure is fitted with a stamp - a predefined basic shape with a regular and well-defined topology. After molding each stamp to the shape of the segment it is assigned to, we connect the segments with a layer of tetrahedra using a new approach to stitching two triangulated surfaces with tetrahedra. Our method not only generates a tetrahedralization with regular topology mimicking a bone-like structure with tissue being grouped around it, but also achieves running times that would allow for real-time usages. The running time of the method is closely correlated with the density of the input mesh which allows for controlling the expected time by decreasing the vertex count while still preserving the general shape of the object.

Feature-preserving shrink wrapping with adaptive alpha

Recent advancements in shrink-wrapping-based mesh approximation have shown tremendous advantages for non-manifold defective meshes. However, these methods perform unsatisfactorily when maintaining the regions with sharp features and rich details of the input mesh. We propose an adaptive shrink-wrapping method based on the recent Alpha Wrapping technique, offering improved feature preservation while handling defective inputs. The proposed approach comprises three main steps. First, we compute a new sizing field with the capability to assess the discretization density of non-manifold defective meshes. Then, we generate a mesh feature skeleton by projecting input feature lines onto the offset surface, ensuring the preservation of sharp features. Finally, an adaptive wrapping approach based on normal projection is applied to preserve the regions with sharp features and rich details simultaneously. By conducting experimental tests on various datasets including Thingi10k, ABC, and GrabCAD, we demonstrate that our method exhibits significant improvements in mesh fidelity compared to the Alpha Wrapping method, while maintaining the advantage of manifold property inherited from shrink-wrapping methods.

A task-driven network for mesh classification and semantic part segmentation

Given the rapid advancements in geometric deep-learning techniques, there has been a dedicated effort to create mesh-based convolutional operators that act as a link between irregular mesh structures and widely adopted backbone networks. Despite the numerous advantages of Convolutional Neural Networks (CNNs) over Multi-Layer Perceptrons (MLPs), mesh-oriented CNNs often require intricate network architectures to tackle irregularities of a triangular mesh. These architectures not only demand that the mesh be manifold and watertight but also impose constraints on the abundance of training samples. In this paper, we note that for specific tasks such as mesh classification and semantic part segmentation, large-scale shape features play a pivotal role. This is in contrast to the realm of shape correspondence, where a comprehensive understanding of 3D shapes necessitates considering both local and global characteristics. Inspired by this key observation, we introduce a task-driven neural network architecture that seamlessly operates in an end-to-end fashion. Our method takes as input mesh vertices equipped with the heat kernel signature (HKS) and dihedral angles between adjacent faces. Notably, we replace the conventional convolutional module, commonly found in ResNet architectures, with MLPs and incorporate Layer Normalization (LN) to facilitate layer-wise normalization. Our approach, with a seemingly straightforward network architecture, demonstrates an accuracy advantage. It exhibits a marginal 0.1% improvement in the mesh classification task and a substantial 1.8% enhancement in the mesh part segmentation task compared to state-of-the-art methodologies. Moreover, as the number of training samples decreases to 1/50 or even 1/100, the accuracy advantage of our approach becomes more pronounced. In summary, our convolution-free network is tailored for specific tasks relying on large-scale shape features and excels in the situation with a limited number of training samples, setting itself apart from state-of-the-art methodologies.

Localized Evaluation for Constructing Discrete Vector Fields.

Topological abstractions offer a method to summarize the behavior of vector fields, but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. We couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.

Robust Coarse Cage Construction With Small Approximation Errors.

We propose a robust and automatic method to construct manifold cages for 3D triangular meshes. The cage contains hundreds of triangles to tightly enclose the input mesh without self-intersections. To generate such cages, our algorithm consists of two phases: (1) construct manifold cages satisfying the tightness, enclosing, and intersection-free requirements and (2) reduce mesh complexities and approximation errors without violating the enclosing and intersection-free requirements. To theoretically make the first stage have those properties, we combine the conformal tetrahedral meshing and tetrahedral mesh subdivision. The second step is a constrained remeshing process using explicit checks to ensure that the enclosing and intersection-free constraints are always satisfied. Both phases use a hybrid coordinate representation, i.e., rational numbers and floating point numbers, combined with exact arithmetic and floating point filtering techniques to guarantee the robustness of geometric predicates with a favorable speed. We extensively test our method on a data set of over 8500 models, demonstrating robustness and performance. Compared to other state-of-the-art methods, our method possesses much stronger robustness.

Skeleton Extraction for Articulated Objects With the Spherical Unwrapping Profiles.

Embedding unified skeletons into unregistered scans is fundamental to finding correspondences, depicting motions, and capturing underlying structures among the articulated objects in the same category. Some existing approaches rely on laborious registration to adapt a predefined LBS model to each input, while others require the input to be set to a canonical pose, e.g., T-pose or A-pose. However, their effectiveness is always influenced by the water-tightness, face topology, and vertex density of the input mesh. At the core of our approach lies a novel unwrapping method, named SUPPLE (Spherical UnwraPping ProfiLEs), which maps a surface into image planes independent of mesh topologies. Based on this lower-dimensional representation, a learning-based framework is further designed to localize and connect skeletal joints with fully convolutional architectures. Experiments demonstrate that our framework yields reliable skeleton extractions across a broad range of articulated categories, from raw scans to online CADs.

Local geometry-perceptive mesh convolution with multi-ring receptive field

Learning 3D mesh representation is necessary for many computer vision and graphic tasks. Recently, some works have studied convolution methods for directly processing input meshes. However, these methods are usually weak in extracting local geometry information because of the disadvantages such as isotropic filter, neglect of mesh topology, and a small convolution field. In this paper, we introduce a local geometry-perceptive mesh convolution, which pays attention to mesh irregular structures for efficiently capturing geometry features in a multi-ring receptive field. Specifically, we define each template node’s dynamic neighbor-attention weights used in multiple attention aggregation operations for obtaining local mesh change information of different vertices in the multi-ring field. After each aggregation, a shared anisotropic filter maps the catenation of each new vertex and its neighbors for extracting geometry features of the current ring. Then, complete local geometry features of each vertex in its large local field are obtained by summing the mapped results of each aggregation. Moreover, the position features of each vertex are added to its local geometry features to get the final representation vector of the vertex. We demonstrate the proposed mesh convolution method’s strong ability in modeling 3D mesh shapes.

Learning the Geodesic Embedding with Graph Neural Networks

We present GEGNN, a learning-based method for computing the approximate geodesic distance between two arbitrary points on discrete polyhedra surfaces with constant time complexity after fast precomputation. Previous relevant methods either focus on computing the geodesic distance between a single source and all destinations, which has linear complexity at least or require a long precomputation time. Our key idea is to train a graph neural network to embed an input mesh into a high-dimensional embedding space and compute the geodesic distance between a pair of points using the corresponding embedding vectors and a lightweight decoding function. To facilitate the learning of the embedding, we propose novel graph convolution and graph pooling modules that incorporate local geodesic information and are verified to be much more effective than previous designs. After training, our method requires only one forward pass of the network per mesh as precomputation. Then, we can compute the geodesic distance between a pair of points using our decoding function, which requires only several matrix multiplications and can be massively parallelized on GPUs. We verify the efficiency and effectiveness of our method on ShapeNet and demonstrate that our method is faster than existing methods by orders of magnitude while achieving comparable or better accuracy. Additionally, our method exhibits robustness on noisy and incomplete meshes and strong generalization ability on out-of-distribution meshes. The code and pretrained model can be found on https://github.com/IntelligentGeometry/GeGnn.

Feature‐Preserving Offset Mesh Generation from Topology‐Adapted Octrees

AbstractWe introduce a reliable method to generate offset meshes from input triangle meshes or triangle soups. Our method proceeds in two steps. The first step performs a Dual Contouring method on the offset surface, operating on an adaptive octree that is refined in areas where the offset topology is complex. Our approach substantially reduces memory consumption and runtime compared to isosurfacing methods operating on uniform grids. The second step improves the output Dual Contouring mesh with an offset‐aware remeshing algorithm to reduce the normal deviation between the mesh facets and the exact offset. This remeshing process reconstructs concave sharp features and approximates smooth shapes in convex areas up to a user‐defined precision. We show the effectiveness and versatility of our method by applying it to a wide range of input meshes. We also benchmark our method on the Thingi10k dataset: watertight and topologically 2‐manifold offset meshes are obtained for 100% of the cases.

SRL-assisted AFM: Generating planar unstructured quadrilateral meshes with supervised and reinforcement learning-assisted advancing front method

High-quality mesh generation is the foundation of accurate finite element analysis. Due to the vast interior vertices search space and complex initial boundaries, mesh generation for complicated domains requires substantial manual processing and has long been considered the most challenging and time-consuming bottleneck of the entire modeling and analysis process. In this paper, we present a novel computational framework named “SRL-assisted AFM” for meshing planar geometries by combining the advancing front method with neural networks that select reference vertices and update the front boundary using “policy networks”. These deep neural networks are trained using a unique pipeline that combines supervised learning with reinforcement learning to iteratively improve mesh quality. First, we generate different initial boundaries by randomly sampling points in a square domain and connecting them sequentially. These boundaries are used for obtaining input meshes and extracting training datasets in the supervised learning module. We then iteratively improve the reinforcement learning model performance with reward functions designed for special requirements, such as improving the mesh quality and controlling the number and distribution of extraordinary points. Our proposed supervised learning neural networks achieve an accuracy higher than 98% on predicting commercial software. The final reinforcement learning neural networks automatically generate high-quality quadrilateral meshes for complex planar domains with sharp features and boundary layers.