Computer Aided Geometric Design最新文献

In this paper, we address the computation of the symmetries of polynomial (and thus also rational) planar vector fields using elements from Computer Algebra. We show that they can be recovered from the symmetries of the roots of an associated univariate complex polynomial which is constructed as a generator of a certain elimination ideal. Computing symmetries of the roots of the auxiliary polynomial is a task considerably simpler than the original problem, which can be done efficiently working with classical Computer Algebra tools. Special cases, in which the group of symmetries of the polynomial roots is infinite, are separately considered and investigated. The presented theory is complemented by illustrative examples. The main steps of the procedure for investigating the symmetries of a given polynomial vector field are summarized in a flow chart for clarity.

In recent years, there has been growing interest in the representation of volumes within the field of geometric modeling (GM). While polygonal patches for surface modeling have been extensively studied, there has been little focus on the representation of polyhedral volumes. Inspired by the polygonal representation of the Generalized Bézier (GB) patch proposed by Várady et al. (2016), this paper introduces a novel method for polyhedral volumetric modeling called the Generalized Bézier (GB) volume.

GB volumes are defined over simple convex polyhedra using generalized barycentric coordinates (GBCs), with the control nets which are a direct generalization of those of tensor-product Bézier volumes. GB volumes can be smoothly connected to adjacent tensor-product Bézier or GB volumes with $G^1$ or $G^2$ continuity. Besides, when the parametric polyhedron becomes a prism, the GB volume also degenerates into a tensor-product form. We provide some practical examples to demonstrate the advantages of GB volumes. Suggestions for future work are also discussed.

The conventional wisdom in point cloud analysis predominantly explores 3D geometries. It is often achieved through the introduction of intricate learnable geometric extractors in the encoder or by deepening networks with repeated blocks. However, these methods contain a significant number of learnable parameters, resulting in substantial computational costs and imposing memory burdens on CPU/GPU. Moreover, they are primarily tailored for object-level point cloud classification and segmentation tasks, with limited extensions to crucial scene-level applications, such as autonomous driving. To this end, we introduce PointeNet, an efficient network designed specifically for point cloud analysis. PointeNet distinguishes itself with its lightweight architecture, low training cost, and plug-and-play capability, while also effectively capturing representative features. The network consists of a Multivariate Geometric Encoding (MGE) module and an optional Distance-aware Semantic Enhancement (DSE) module. MGE employs operations of sampling, grouping, pooling, and multivariate geometric aggregation to lightweightly capture and adaptively aggregate multivariate geometric features, providing a comprehensive depiction of 3D geometries. DSE, designed for real-world autonomous driving scenarios, enhances the semantic perception of point clouds, particularly for distant points. Our method demonstrates flexibility by seamlessly integrating with a classification/segmentation head or embedding into off-the-shelf 3D object detection networks, achieving notable performance improvements at a minimal cost. Extensive experiments on object-level datasets, including ModelNet40, ScanObjectNN, ShapeNetPart, and the scene-level dataset KITTI, demonstrate the superior performance of PointeNet over state-of-the-art methods in point cloud analysis. Notably, PointeNet outperforms PointMLP with significantly fewer parameters on ModelNet40, ScanObjectNN, and ShapeNetPart, and achieves a substantial improvement of over 2% in $3DA{P}_{R40}$ for PointRCNN on KITTI with a minimal parameter cost of 1.4 million. Code is publicly available at https://github.com/lipeng-gu/PointeNet.

Computer-Aided Design (CAD) software remains a pivotal tool in modern engineering and manufacturing, driving the design of a diverse range of products. In this work, we introduce VQ-CAD, the first CAD generation model based on Denoising Diffusion Probabilistic Models. This model utilizes a vector quantized diffusion model, employing multiple hierarchical codebooks generated through VQ-VAE. This integration not only offers a novel perspective on CAD model generation but also achieves state-of-the-art performance in 3D CAD model creation in a fully automatic fashion. Our model is able to recognize and incorporate implicit design constraints by simply forgoing traditional data augmentation. Furthermore, by melding our approach with CLIP, we significantly simplify the existing design process, directly generate CAD command sequences from initial design concepts represented by text or sketches, capture design intentions, and ensure designs adhere to implicit constraints.

Although significant advances have been made in the field of multi-view 3D reconstruction using implicit neural field-based methods, existing reconstruction methods overlook the estimation of the material information (e.g. the base color, albedo, roughness, and metallic) during the learning process. In this paper, we propose a novel differentiable rendering framework, named as material NueS (M-NeuS), for simultaneously achieving precise surface reconstruction and competitive material estimation. For surface reconstruction, we perform multi-view geometry optimization by proposing an enhanced-low-to-high frequency encoding registration strategy (EFERS) and a second-order interpolated signed distance function (SI-SDF) for precise details and outline reconstruction. For material estimation, inspired by the NeuS, we first propose a volume-rendering-based material estimation strategy (VMES) to estimate the base color, albedo, roughness, and metallic accurately. And then, different from most material estimation methods that need ground-truth geometric priors, we use the geometry information reconstructed in the surface reconstruction stage and the directions of incidence from different viewpoints to model a neural light field, which can extract the lighting information from image observations. Next, the extracted lighting and the estimated base color, albedo, roughness, and metallic are optimized by the physics-based rendering equation. Extensive experiments demonstrate that our M-NeuS can not only reconstruct more precise geometry surface than existing state-of-the-art (SOTA) reconstruction methods but also can estimate competitive material information: the base color, albedo, roughness, and metallic.

One of the main challenges of indoor scene synthesis is preserving the functionality of synthesized scenes to create practical and usable indoor environments. Function groups exhibit the capability of balancing the global structure and local scenes of an indoor space. In this paper, we propose a function-centric indoor scene synthesis framework, named FuncScene. Our key idea is to use function groups as an intermedium to connect the local scenes and the global structure, thus achieving a coarse-to-fine indoor scene synthesis while maintaining the functionality and practicality of synthesized scenes. Indoor scenes are synthesized by first generating function groups using generative models and then instantiating by searching and matching the specific function groups from a dataset. The proposed framework also makes it easier to achieve multi-level generation control of scene synthesis, which was challenging for previous works. Extensive experiments on various indoor scene synthesis tasks demonstrate the validity of our method. Qualitative and quantitative evaluations show the proposed framework outperforms the existing state-of-the-art.

The generalization of barycentric coordinates to arbitrary simple polygons with more than three vertices has been a subject of study for a long time. Among the different constructions proposed, mean value coordinates have emerged as a popular choice, particularly due to their suitability for the non-convex setting. Since their introduction, they have found applications in numerous fields, and several equivalent formulas for their evaluation have been presented in the literature. However, so far, there has been no study regarding their numerical stability. In this paper, we aim to investigate the numerical stability of the algorithms that compute mean value coordinates. We show that all the known methods exhibit instability in some regions of the domain. To address this problem, we introduce a new formula for computing mean value coordinates, explain how to implement it, and formally prove that our new algorithm provides a stable evaluation of mean value coordinates. We validate our results through numerical experiments.

In motion planning, two-dimensional (2D) splinegons are typically used to represent the contours of 2D objects. In general, a 2D splinegon must be pre-decomposed to support rapid queries of the shortest paths or visibility. Herein, we propose a new region decomposition strategy, known as the Voronoi-based decomposition (VBD), based on the Voronoi diagram of curved boundary-segment generators (either convex or concave). The number of regions in the VBD is O(n+$c_1$). Compared with the well-established horizontal visibility decomposition (HVD), whose complexity is O(n+$c_2$), the VBD decomposition generally contains less regions because $c_1$≤$c_2$, where n is the number of the vertices of the input splinegon, and $c_1$ and $c_2$ are the number of inserted vertices at the boundary. We systematically discuss the usage of VBD. Based on the VBD, the shortest path tree (SPT) can be computed in linear time. Statistics show that the VBD performs faster than HVD in SPT computations. Additionally, based on the SPT, we design algorithms that can rapidly compute the visibility between two points, the visible area of a point/line-segment, and the shortest path between two points.

In recent years, with the rapid development of the computer aided design and computer graphics, a large number of 3D models have emerged, making it a challenge to quickly find models of interest. As a concise and informative representation of 3D models, shape descriptors are a key factor in achieving effective retrieval. In this paper, we propose a novel global descriptor for 3D models that incorporates both geometric and topological information. We refer to this descriptor as the persistent heat kernel signature descriptor (PHKS). Constructed by concatenating our isometry-invariant geometric descriptor with topological descriptor, PHKS possesses high recognition ability, while remaining insensitive to noise and can be efficiently calculated. Retrieval experiments of 3D models on the benchmark datasets show considerable performance gains of the proposed method compared to other descriptors based on HKS and advanced topological descriptors.

In this paper, we propose a novel surface reconstruction method for unoriented points by establishing and solving a nonlinear equation system. By treating normals as unknown parameters and imposing the conditions that the implicit field is constant and its gradients parallel to the normals on the input point cloud, we establish a nonlinear equation system involving the oriented normals. To simplify the system, we transform it into a 0-1 integer programming problem solely focusing on orientation by incorporating inconsistent oriented normal information through PCA. We solve the simplified problem using flipping-based iterative algorithms and propose two novel criteria for flipping based on theoretical analysis.

Extensive experiments on renowned datasets demonstrate that our flipping-based method with wavelet surface reconstruction achieves state-of-the-art results in orientation and reconstruction. Furthermore, it exhibits linear computational and storage complexity by leveraging the orthogonality and compact support properties of wavelet bases. The source code is available at https://github.com/mayueji/FISR_code.