Computer Aided Geometric Design最新文献

Mesh denoising is a crucial research topic in geometric processing, as it is widely used in reverse engineering and 3D modeling. The main objective of denoising is to eliminate noise while preserving sharp features. In this paper, we propose a novel denoising method called Attention Enhanced Dual Mesh Denoise (ADMD), which is based on a graph neural network and attention mechanism. ADMD simulates the two-stage denoising method by using a new training strategy and total variation (TV) regular term to enhance feature retention. Our experiments have demonstrated that ADMD can achieve competitive or superior results to state-of-the-art methods for noise CAD models, non-CAD models, and real-scanned data. Moreover, our method can effectively handle large mesh models with different-scale noisy situations and prevent model shrinking after mesh denoising.

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.

To construct a parametric polynomial curve for interpolating a set of data points, the interpolation accuracy and shape of the constructed curve are influenced by two principal factors: parameterization of the data points (computing a node for each data point) and interpolation method. A new method of computing nodes for a set of data points was proposed. In this paper, the functional relationship between data points and corresponding nodes in cubic polynomials was established. Using this functional relationship, a functional cubic polynomial with one degree of freedom can pass through four adjacent data points. The degree of the freedom can be represented by two adjacent node intervals can be obtained by minimizing the cubic terms of the cubic polynomial. Since each node is computed in different node spaces, a method for constructing a quadratic curve is presented, which transforms all the quadratic curves into a unified form to compute nodes. Nodes computed using the new method exhibit quadratic polynomial precision, i.e., if the set of data point is taken from a quadratic polynomial $F\left(t\right)$, the nodes by the new method are used to construct a interpolation curve, an interpolation method reproducing quadratic polynomial gives quadratic polynomial $F\left(t\right)$. The primary advantage of the proposed method is that the constructed curve has a shape described by data points. Another advantage of the new method is that the nodes computed by it have affine invariance. The experimental results indicate that the curve constructed by the nodes using the new method has a better interpolation accuracy and shape compared to that constructed using other methods.

This paper introduces two improved variants of the least squares progressive-iterative approximation (LSPIA) by leveraging momentum techniques. Specifically, based on the Polyak's and Nesterov's momentum techniques, the proposed methods utilize the previous iteration information to update the control points. We name these two methods PmLSPIA and NmLSPIA, respectively. The introduction of momentum enhances the determination of the search directions, leading to a significant improvement in convergence rate. The geometric interpretations of PmLSPIA and NmLSPIA are elucidated, providing insights into the underlying principles of these accelerated algorithms. Rigorous convergence analyses are conducted, revealing that both PmLSPIA and NmLSPIA exhibit faster convergence than LSPIA. Numerical results further validate the efficacy of the proposed algorithms.

Implicit representations possess many merits when dealing with geometries with certain properties, such as small holes, reentrant corners and other complex details. Truncated hierarchical B-splines (THB-splines) has recently emerged as a novel tool in many fields including design and analysis due to its local refinement ability. In this paper, we propose an adaptive collocation method with weighted extended THB-splines (WETHB-splines) on implicit domains. We modify the classification strategy for the WETHB-basis, and the centers of the supports of inner THB-splines on each level are chosen to be collocation points. We also use weighted collocation in the transition regions, in order to enrich information concerning the hierarchical basis. In contrast to the traditional WEB-collocation method, the proposed approach possesses much higher convergence rate. To show the efficiency and superiority of the proposed method, numerical examples in two and three dimensions are performed to solve Poisson's equations.

In digital orthodontic treatment, the high-precision reconstruction of complete teeth, encompassing both the crown and the actual root, plays a pivotal role. Current mainstream techniques, prioritizing the high resolution of intraoral scanned models (IOS), are confined to using IOS data for orthodontic treatments. However, the lack of root information in the IOS data may lead to complications such as dehiscence. In contrast, Cone Beam Computed Tomography (CBCT) data encompasses comprehensive dental information with roots. Nonetheless, the radiative character of CBCT scans renders patients unsuitable for repeated examinations in a short time. In addition, lower scanning precision of CBCT leads to suboptimal teeth segmentation outcomes, hindering the accurate representation of dental occlusal relationships. Therefore, in order to fully utilize the complementarity between dental multimodal data, we propose a method for high-precision 3D teeth model reconstruction based on IOS and CBCT, which mainly consists of global rigid registration and local nonrigid registration. Specifically, we extract the priori information of dental arch curves for coarse alignment to provide a good initial position for the Iterative Closest Point (ICP) algorithm, and design a conformal parameterization method for a single tooth to effectively obtain the point correspondence between IOS and CBCT crowns. The rough crown of the CBCT will gradually fit towards the IOS through iterative optimization of nonrigid registration. The experimental results show that our method robustly fuses the advantageous features of IOS and CBCT. The 3D teeth model reconstructed by our method contains the high-precision crown of IOS and the real root of CBCT, which can be effectively used in clinical orthodontic treatment.

We propose a simple yet effective method to interpolate high-order meshes. Given two manifold high-order triangular (or tetrahedral) meshes with identical connectivity, our goal is to generate a continuum of curved shapes with as little distortion as possible in the mapping from the source mesh to the interpolated mesh. Our algorithm contains two steps: (1) linearly blend the pullback metric of the identity mapping and the input mapping between two Bézier elements on a set of sampling points; (2) project the interpolated metric into the metric space between Bézier elements using the Newton method for nonlinear optimization. We demonstrate the feasibility and practicability of the method for high-order meshes through extensive experiments in both 2D and 3D.

The structural design of 3D auxetic linkages is a burgeoning field in digital manufacturing. This article presents a novel algorithm for designing 3D auxetic linkage structures based on Kirigami principles to address existing limitations. The 3D input model is initially mapped to a 2D space using conformal mapping based on the BFF method. This is followed by 2D re-meshing using an equilateral triangle mesh. Subsequently, a 3D topological mesh of the auxetic linkage is calculated through inverse mapping based on directed area. We then introduce new basic rotating and non-rotating units, employing them as the initial structure of the 3D auxetic linkage in accordance with Kirigami techniques. Lastly, a deformation energy function is defined to optimize the shape of the rotating units. The vertex coordinates of the non-rotating units are updated according to the optimized positions of the rotating units, thereby generating an optimal 3D auxetic linkage structure. Experimental results validate the effectiveness and accuracy of our algorithm. Quantitative analyses of structural porosity and optimization accuracy, as well as comparisons with related works, indicate that our algorithm yields structures with smaller shape errors.

Establishing accurate and representative matches is a crucial step in addressing the point cloud registration problem. A commonly employed approach involves detecting keypoints with salient geometric features and subsequently mapping these keypoints from one frame of the point cloud to another. However, methods within this category are hampered by the repeatability of the sampled keypoints. In this paper, we introduce a saliency-guided transformer, referred to as D3Former, which entails the joint learning of repeatable Dense Detectors and feature-enhanced Descriptors. The model comprises a Feature Enhancement Descriptor Learning (FEDL) module and a Repetitive Keypoints Detector Learning (RKDL) module. The FEDL module utilizes a region attention mechanism to enhance feature distinctiveness, while the RKDL module focuses on detecting repeatable keypoints to enhance matching capabilities. Extensive experimental results on challenging indoor and outdoor benchmarks demonstrate that our proposed method consistently outperforms state-of-the-art point cloud matching methods. Notably, tests on 3DLoMatch, even with a low overlap ratio, show that our method consistently outperforms recently published approaches such as RoReg and RoITr. For instance, with the number of extracted keypoints reduced to 250, the registration recall scores for RoReg, RoITr, and our method are 64.3%, 73.6%, and 76.5%, respectively.

This paper addresses the challenge of representing geodesic distance fields on triangular meshes in a piecewise linear manner. Unlike general scalar fields, which often assume piecewise linear changes within each triangle, geodesic distance fields pose a unique difficulty due to their non-differentiability at ridge points, where multiple shortest paths may exist. An interesting observation is that the geodesic distance field exhibits an approximately linear change if each triangle is further decomposed into sub-regions by the ridge curve. However, computing the geodesic ridge curve is notoriously difficult. Even when using exact algorithms to infer the ridge curve, desirable results may not be achieved, akin to the well-known medial-axis problem. In this paper, we propose a two-stage algorithm. In the first stage, we employ Dijkstra's algorithm to cut the surface open along the dual structure of the shortest path tree. This operation allows us to extend the surface outward (resembling a double cover but with distinctions), enabling the discovery of longer geodesic paths in the extended surface. In the second stage, any mature geodesic solver, whether exact or approximate, can be employed to predict the real ridge curve. Assuming the fast marching method is used as the solver, despite its limitation of having a single marching direction in a triangle, our extended surface contains multiple copies of each triangle, allowing various geodesic paths to enter the triangle and facilitating ridge curve computation. We further introduce a simple yet effective filtering mechanism to rigorously ensure the connectivity of the output ridge curve. Due to its merits, including robustness and compatibility with any geodesic solver, our algorithm holds great potential for a wide range of applications. We demonstrate its utility in accurate geodesic distance querying and high-fidelity visualization of geodesic iso-lines.