Pub Date : 2025-02-01DOI: 10.1016/j.cagd.2025.102413
Thomas Takacs
In this study we consider domains that are composed of an infinite sequence of self-similar rings and corresponding finite element spaces over those domains. The rings are parameterized using piecewise polynomial or tensor-product B-spline mappings of degree q over quadrilateral meshes. We then consider finite element discretizations which, over each ring, are mapped, piecewise polynomial functions of degree p. Such domains that are composed of self-similar rings may be created through a subdivision scheme or from a scaled boundary parameterization.
We study approximation properties over such recursively parameterized domains. The main finding is that, for generic isoparametric discretizations (i.e., where ), the approximation properties always depend only on the degree of polynomials that can be reproduced exactly in the physical domain and not on the degree p of the mapped elements. Especially, in general, -errors converge at most with the rate , where h is the mesh size, independent of the degree . This has implications for subdivision based isogeometric analysis, which we will discuss in this paper.
{"title":"Approximation properties over self-similar meshes of curved finite elements and applications to subdivision based isogeometric analysis","authors":"Thomas Takacs","doi":"10.1016/j.cagd.2025.102413","DOIUrl":"10.1016/j.cagd.2025.102413","url":null,"abstract":"<div><div>In this study we consider domains that are composed of an infinite sequence of self-similar rings and corresponding finite element spaces over those domains. The rings are parameterized using piecewise polynomial or tensor-product B-spline mappings of degree <em>q</em> over quadrilateral meshes. We then consider finite element discretizations which, over each ring, are mapped, piecewise polynomial functions of degree <em>p</em>. Such domains that are composed of self-similar rings may be created through a subdivision scheme or from a scaled boundary parameterization.</div><div>We study approximation properties over such recursively parameterized domains. The main finding is that, for generic isoparametric discretizations (i.e., where <span><math><mi>p</mi><mo>=</mo><mi>q</mi></math></span>), the approximation properties always depend only on the degree of polynomials that can be reproduced exactly in the physical domain and not on the degree <em>p</em> of the mapped elements. Especially, in general, <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-errors converge at most with the rate <span><math><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <em>h</em> is the mesh size, independent of the degree <span><math><mi>p</mi><mo>=</mo><mi>q</mi></math></span>. This has implications for subdivision based isogeometric analysis, which we will discuss in this paper.</div></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"116 ","pages":"Article 102413"},"PeriodicalIF":1.3,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143402940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1016/j.cagd.2024.102402
Ciyang Zhou, Xingce Wang, Zhongke Wu
The thin membranes and shells in nature are heterogeneous. They are widely used in surgical simulation, biological techniques, and computer animation. The corresponding surface deformable models can implement dynamic simulations of thin membranes and shells in nature, while most surface deformable models are isotropic and cannot represent thin membranes and shells in nature accurately. Therefore, we propose a novel physically-based heterogeneous deformable surface model. By utilizing the same B-spline basis functions or the parameter space of surfaces' geometric representations, we implement material modeling and propose the representations of surfaces with material variations with composite or continuous material functions. Then, we propose a novel physically-based elastic deformable surface model that constructs infinitesimal elements in the parameter space and employs elasticity to analyze their deformation. The corresponding elastic potential energy function is only related to surfaces' continuous representations, and our model avoids the computation error caused by meshes' quality and large rotation of points' frames. We employ isogeometric analysis to solve the dynamic equations derived from our surface model. To demonstrate the validity and reality of our model, several comparison experiments are designed. The corresponding results are in line with expectations and consistent with physical laws.
自然界中的薄膜和外壳是异质的。它们被广泛应用于外科模拟、生物技术和计算机动画。相应的表面可变形模型可以对自然界中的薄膜和薄壳进行动态模拟，而大多数表面可变形模型都是各向同性的，不能准确地表现自然界中的薄膜和薄壳。因此，我们提出了一种基于物理的新型异质可变形表面模型。通过利用相同的 B 样条基函数或曲面几何表示的参数空间，我们实现了材料建模，并提出了用复合材料函数或连续材料函数表示具有材料变化的曲面。然后，我们提出了一种基于物理的新型弹性可变形曲面模型，该模型在参数空间中构建了无穷小元素，并利用弹性分析其变形。相应的弹性势能函数只与曲面的连续表示相关，我们的模型避免了因网格质量和点框架的大幅旋转造成的计算误差。我们采用等距分析法来求解由曲面模型导出的动态方程。为了证明我们模型的有效性和现实性，我们设计了几个对比实验。相应的结果符合预期，并与物理规律相一致。
{"title":"A novel heterogeneous deformable surface model based on elasticity","authors":"Ciyang Zhou, Xingce Wang, Zhongke Wu","doi":"10.1016/j.cagd.2024.102402","DOIUrl":"10.1016/j.cagd.2024.102402","url":null,"abstract":"<div><div>The thin membranes and shells in nature are heterogeneous. They are widely used in surgical simulation, biological techniques, and computer animation. The corresponding surface deformable models can implement dynamic simulations of thin membranes and shells in nature, while most surface deformable models are isotropic and cannot represent thin membranes and shells in nature accurately. Therefore, we propose a novel physically-based heterogeneous deformable surface model. By utilizing the same B-spline basis functions or the parameter space of surfaces' geometric representations, we implement material modeling and propose the representations of surfaces with material variations with composite or continuous material functions. Then, we propose a novel physically-based elastic deformable surface model that constructs infinitesimal elements in the parameter space and employs elasticity to analyze their deformation. The corresponding elastic potential energy function is only related to surfaces' continuous representations, and our model avoids the computation error caused by meshes' quality and large rotation of points' frames. We employ isogeometric analysis to solve the dynamic equations derived from our surface model. To demonstrate the validity and reality of our model, several comparison experiments are designed. The corresponding results are in line with expectations and consistent with physical laws.</div></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"115 ","pages":"Article 102402"},"PeriodicalIF":1.3,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.cagd.2024.102401
Ada Šadl Praprotnik , Aleš Vavpetič , Emil Žagar
The optimal one-sided parametric polynomial approximants of a circular arc are considered. More precisely, the approximant must be entirely in or out of the underlying circle of an arc. The natural restriction to an arc's approximants interpolating boundary points is assumed. However, the study of approximants, which additionally interpolate corresponding tangent directions and curvatures at the boundary of an arc, is also considered. Several low-degree polynomial approximants are studied in detail. When several solutions fulfilling the interpolation conditions exist, the optimal one is characterized, and a numerical algorithm for its construction is suggested. Theoretical results are demonstrated with several numerical examples and a comparison with general (i.e. non-one-sided) approximants are provided.
{"title":"Optimal one-sided approximants of circular arc","authors":"Ada Šadl Praprotnik , Aleš Vavpetič , Emil Žagar","doi":"10.1016/j.cagd.2024.102401","DOIUrl":"10.1016/j.cagd.2024.102401","url":null,"abstract":"<div><div>The optimal one-sided parametric polynomial approximants of a circular arc are considered. More precisely, the approximant must be entirely in or out of the underlying circle of an arc. The natural restriction to an arc's approximants interpolating boundary points is assumed. However, the study of approximants, which additionally interpolate corresponding tangent directions and curvatures at the boundary of an arc, is also considered. Several low-degree polynomial approximants are studied in detail. When several solutions fulfilling the interpolation conditions exist, the optimal one is characterized, and a numerical algorithm for its construction is suggested. Theoretical results are demonstrated with several numerical examples and a comparison with general (i.e. non-one-sided) approximants are provided.</div></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"115 ","pages":"Article 102401"},"PeriodicalIF":1.3,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-18DOI: 10.1016/j.cagd.2024.102393
Weiming Wang , Shan Li , Li Yang , Jiepeng Liu , Yi Xia , Ligang Liu
Additive manufacturing (AM) technology enables the fabrication of three-dimensional objects with complex shapes and has been extensively applied in various industries. AM is a layer-wise fabrication process where a variety of factors affect manufacturing performance and product quality. One of the most important factor is the thermal distortion, which is caused by the high temperature gradients in the fabrication process. The thermal distortion is also influenced by the support structure of the printing object, and this distortion varies depending on the chosen build orientation. Additionally, for the overhang regions, extra supports are required for printing and will be removed in the post-processing. For the 3D printing process, the thermal distortion and support requirements are interconnected and linked to build orientation. To investigate suitable build orientation, the thermal distortion and support are quantified, and a multi-objective build orientation optimization method is proposed to obtain representative orientations. Based on the proposed method, 9 typical 3D shapes are evaluated. In addition, the single-objective build orientation optimization problem is studied and compared, and the influence of slicing layers per stage on the simulation accuracy and efficiency is discussed. The effectiveness and applicability of the method are verified, and representative directions can be obtained for different fabrication purposes.
{"title":"Build orientation optimization considering thermal distortion in additive manufacturing","authors":"Weiming Wang , Shan Li , Li Yang , Jiepeng Liu , Yi Xia , Ligang Liu","doi":"10.1016/j.cagd.2024.102393","DOIUrl":"10.1016/j.cagd.2024.102393","url":null,"abstract":"<div><div>Additive manufacturing (AM) technology enables the fabrication of three-dimensional objects with complex shapes and has been extensively applied in various industries. AM is a layer-wise fabrication process where a variety of factors affect manufacturing performance and product quality. One of the most important factor is the thermal distortion, which is caused by the high temperature gradients in the fabrication process. The thermal distortion is also influenced by the support structure of the printing object, and this distortion varies depending on the chosen build orientation. Additionally, for the overhang regions, extra supports are required for printing and will be removed in the post-processing. For the 3D printing process, the thermal distortion and support requirements are interconnected and linked to build orientation. To investigate suitable build orientation, the thermal distortion and support are quantified, and a multi-objective build orientation optimization method is proposed to obtain representative orientations. Based on the proposed method, 9 typical 3D shapes are evaluated. In addition, the single-objective build orientation optimization problem is studied and compared, and the influence of slicing layers per stage on the simulation accuracy and efficiency is discussed. The effectiveness and applicability of the method are verified, and representative directions can be obtained for different fabrication purposes.</div></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"114 ","pages":"Article 102393"},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-17DOI: 10.1016/j.cagd.2024.102389
Maodong Pan , Ruijie Zou , Bert Jüttler
The simple mesh refinement algorithm of Groiss et al. (2023) generates T-meshes admitting Reachable Minimally supported (RM) B-splines that possess the property of local linear independence and form a non-negative partition of unity. The construction was first presented for the bilinear case and has later been extended to -smooth splines of degree . The present paper is devoted to algorithms and data structures for RMB-splines. We prove that the memory consumption of the data structures for representing a T-mesh and the associated RMB-splines is linear with respect to the mesh size, and we describe the details of the underlying refinement algorithm. Moreover, we introduce a novel evaluation algorithm for RMB-spline surfaces, which is based solely on repeated convex combinations of the control points, thereby generalizing de Boor's algorithm for tensor-product splines. Numerical experiments are included to demonstrate the advantageous behavior of the proposed data structures and algorithms with respect to their efficiency. We observe that the total computational time (which includes also error estimation and spline coefficient computation) scales roughly linearly with the number of degrees of freedom for the meshes considered.
Groiss 等人（2023 年）提出的简单网格细化算法可生成 T 型网格，允许具有局部线性独立特性的可达到最小支持（RM）B 样条，并形成非负的统一分割。该构造最初是针对双线性情况提出的，后来被扩展到 p=2s+1 度的 Cs 平滑样条曲线。本文主要介绍人民币样条曲线的算法和数据结构。我们证明了表示 T 形网格和相关人民币样条曲线的数据结构的内存消耗与网格大小呈线性关系，并描述了底层细化算法的细节。此外，我们还介绍了人民币样条曲线曲面的新型评估算法，该算法完全基于控制点的重复凸组合，从而推广了 de Boor 的张量乘积样条曲线算法。我们通过数值实验证明了所提出的数据结构和算法在效率方面的优势。我们发现，对于所考虑的网格，总计算时间（还包括误差估计和样条系数计算）与自由度数大致成线性关系。
{"title":"Algorithms and data structures for Cs-smooth RMB-splines of degree 2s + 1","authors":"Maodong Pan , Ruijie Zou , Bert Jüttler","doi":"10.1016/j.cagd.2024.102389","DOIUrl":"10.1016/j.cagd.2024.102389","url":null,"abstract":"<div><div>The simple mesh refinement algorithm of <span><span>Groiss et al. (2023)</span></span> generates T-meshes admitting Reachable Minimally supported (RM) B-splines that possess the property of local linear independence and form a non-negative partition of unity. The construction was first presented for the bilinear case and has later been extended to <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span>-smooth splines of degree <span><math><mi>p</mi><mo>=</mo><mn>2</mn><mi>s</mi><mo>+</mo><mn>1</mn></math></span>. The present paper is devoted to algorithms and data structures for RMB-splines. We prove that the memory consumption of the data structures for representing a T-mesh and the associated RMB-splines is linear with respect to the mesh size, and we describe the details of the underlying refinement algorithm. Moreover, we introduce a novel evaluation algorithm for RMB-spline surfaces, which is based solely on repeated convex combinations of the control points, thereby generalizing de Boor's algorithm for tensor-product splines. Numerical experiments are included to demonstrate the advantageous behavior of the proposed data structures and algorithms with respect to their efficiency. We observe that the total computational time (which includes also error estimation and spline coefficient computation) scales roughly linearly with the number of degrees of freedom for the meshes considered.</div></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"114 ","pages":"Article 102389"},"PeriodicalIF":1.3,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-16DOI: 10.1016/j.cagd.2024.102394
Xiaoge He , Yuanpeng Liu , Jun Zhou , Yuqi Zhang , Jun Wang
This paper addresses the problem of recognizing multiple objects and multiple instances from point clouds. Whereas existing methods utilize descriptors on 3D fields or pointwise voting to achieve this task, our framework takes advantage of both descriptor-based and voting-based schemes to realize more robust and efficient prediction. Specifically, we propose a novel and robust descriptor called an orientation-enhanced fast point feature histogram (OE-FPFH) to describe points in both the object model and scene, and further to build the correspondence set. The OE-FPFH integrates an orientation vector through mining the geometric tensor of the local structure of a surface point, which is more representative than the original FPFH descriptor. To improve voting efficiency, we devise a novel single-point voting mechanism (SPVM), which constructs a unique local reference frame (LRF) on a single point using the orientation vector. The SPVM takes as input the corresponding point set and can generate a pose candidate for each correspondence. The process is realized by matching LRFs from two corresponding points. All pose candidates are subsequently divided into clusters and aggregated using the K-means clustering algorithm to deduce the poses for different objects or instances in the scene. Experiments on three challenging datasets demonstrate that our method is effective, efficient, and robust to occlusions and multiple instances.
{"title":"Efficient object recognition under cluttered scenes via descriptor-based matching and single point voting","authors":"Xiaoge He , Yuanpeng Liu , Jun Zhou , Yuqi Zhang , Jun Wang","doi":"10.1016/j.cagd.2024.102394","DOIUrl":"10.1016/j.cagd.2024.102394","url":null,"abstract":"<div><div>This paper addresses the problem of recognizing multiple objects and multiple instances from point clouds. Whereas existing methods utilize descriptors on 3D fields or pointwise voting to achieve this task, our framework takes advantage of both descriptor-based and voting-based schemes to realize more robust and efficient prediction. Specifically, we propose a novel and robust descriptor called an orientation-enhanced fast point feature histogram (OE-FPFH) to describe points in both the object model and scene, and further to build the correspondence set. The OE-FPFH integrates an orientation vector through mining the geometric tensor of the local structure of a surface point, which is more representative than the original FPFH descriptor. To improve voting efficiency, we devise a novel single-point voting mechanism (SPVM), which constructs a unique local reference frame (LRF) on a single point using the orientation vector. The SPVM takes as input the corresponding point set and can generate a pose candidate for each correspondence. The process is realized by matching LRFs from two corresponding points. All pose candidates are subsequently divided into clusters and aggregated using the <em>K</em>-means clustering algorithm to deduce the poses for different objects or instances in the scene. Experiments on three challenging datasets demonstrate that our method is effective, efficient, and robust to occlusions and multiple instances.</div></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"114 ","pages":"Article 102394"},"PeriodicalIF":1.3,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-15DOI: 10.1016/j.cagd.2024.102392
Kai Hormann , Claudio Mancinelli
Subdivision schemes are used to generate smooth curves by iteratively refining an initial control polygon. The simplest such schemes are corner cutting schemes, which specify two distinct points on each edge of the current polygon and connect them to get the refined polygon, thus cutting off the corners of the current polygon. While de Boor (1987) shows that this process always converges to a Lipschitz continuous limit curve, no matter how the points on each edge are chosen, Gregory and Qu (1996) discover that the limit curve is continuously differentiable under certain constraints. We extend these results and show that the limit curve can even be curvature continuous for specific sequences of cut ratios.
{"title":"Curvature continuous corner cutting","authors":"Kai Hormann , Claudio Mancinelli","doi":"10.1016/j.cagd.2024.102392","DOIUrl":"10.1016/j.cagd.2024.102392","url":null,"abstract":"<div><div>Subdivision schemes are used to generate smooth curves by iteratively refining an initial control polygon. The simplest such schemes are corner cutting schemes, which specify two distinct points on each edge of the current polygon and connect them to get the refined polygon, thus cutting off the corners of the current polygon. While <span><span>de Boor (1987)</span></span> shows that this process always converges to a Lipschitz continuous limit curve, no matter how the points on each edge are chosen, <span><span>Gregory and Qu (1996)</span></span> discover that the limit curve is continuously differentiable under certain constraints. We extend these results and show that the limit curve can even be curvature continuous for specific sequences of cut ratios.</div></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"114 ","pages":"Article 102392"},"PeriodicalIF":1.3,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142437820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-09DOI: 10.1016/j.cagd.2024.102390
Nannan Li , Xinyuan Li , Jun Zhou , Dong Jiang , Jian Liu , Hong Qin
Normal estimation has been one of the key tasks in point cloud analysis, while it is challenging when facing with severe noises or complex regions. The challenges mainly come from the selection of supporting points for estimation, that is, improper selections of points and points' scale will lead to insufficient information, loss of details, etc. To this end, this paper proposes one feature-centric fitting scheme, GeoHi-GNN, by learning geometry-aware hierarchical graph representation for fitting weights estimation. The main functional module is the continuously conducted Hierarchically Geometric-aware (HG) module, consisting of two core operations, namely, the graph node construction (GNC) and the geometric-aware dynamic graph convolution (GDGC). GNC aims to aggregate the feature information onto a smaller number of nodes, providing global-to-local information while avoiding the interferences from noises in larger scales. With these nodes distributed in different scales, GDGC dynamically updates the node features regarding to both intrinsic feature and extrinsic geometric information. Finally, the hierarchical graphical features are cascaded to estimate the weights for supporting points in the surface fitting. Through the extensive experiments and comprehensive comparisons with the state-of-the-arts, our scheme has exhibited many attractive advantages such as being geometry-aware and robust, empowering further applications like more accurate surface reconstruction.
{"title":"GeoHi-GNN: Geometry-aware hierarchical graph representation learning for normal estimation","authors":"Nannan Li , Xinyuan Li , Jun Zhou , Dong Jiang , Jian Liu , Hong Qin","doi":"10.1016/j.cagd.2024.102390","DOIUrl":"10.1016/j.cagd.2024.102390","url":null,"abstract":"<div><div>Normal estimation has been one of the key tasks in point cloud analysis, while it is challenging when facing with severe noises or complex regions. The challenges mainly come from the selection of supporting points for estimation, that is, improper selections of points and points' scale will lead to insufficient information, loss of details, etc. To this end, this paper proposes one feature-centric fitting scheme, GeoHi-GNN, by learning geometry-aware hierarchical graph representation for fitting weights estimation. The main functional module is the continuously conducted Hierarchically Geometric-aware (HG) module, consisting of two core operations, namely, the graph node construction (GNC) and the geometric-aware dynamic graph convolution (GDGC). GNC aims to aggregate the feature information onto a smaller number of nodes, providing global-to-local information while avoiding the interferences from noises in larger scales. With these nodes distributed in different scales, GDGC dynamically updates the node features regarding to both intrinsic feature and extrinsic geometric information. Finally, the hierarchical graphical features are cascaded to estimate the weights for supporting points in the surface fitting. Through the extensive experiments and comprehensive comparisons with the state-of-the-arts, our scheme has exhibited many attractive advantages such as being geometry-aware and robust, empowering further applications like more accurate surface reconstruction.</div></div>","PeriodicalId":55226,"journal":{"name":"Computer Aided Geometric Design","volume":"114 ","pages":"Article 102390"},"PeriodicalIF":1.3,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-24DOI: 10.1016/j.cagd.2024.102391
Cédric Gérot , Loïc Barthe , Neil A. Dodgson , Malcolm A. Sabin
The quality of a subdivision scheme in the vicinity of a vertex or a face-centre is related to the eigenstructure of the subdivision matrix. When the scheme has the appropriate symmetries, a common technique, based on discrete Fourier transform, builds small complex matrices that ease the numerical analysis of the eigenelements using in particular their Fourier index. But the numerical analysis of the eigenelements remains difficult when matrix entries involve complex numbers and unknowns, for example, in cases where we are tuning a scheme. We present techniques to build similar small matrices, still associated with a Fourier index and whose eigenstructure is simply related to the full matrix, but which are real. They extend the known techniques to schemes which rotate the lattice and with vertices which do not lie topologically on symmetry axes of the studied vicinity of vertex or face centre. Our techniques make it easier to tune these subdivision schemes. We illustrate it with the analysis of the so-called Simplest Scheme at the centre of an n-sided face.
顶点或面中心附近细分方案的质量与细分矩阵的特征结构有关。当方案具有适当的对称性时，一种基于离散傅里叶变换的常用技术可以建立小的复数矩阵，从而方便地利用其傅里叶指数对特征元素进行数值分析。但是，当矩阵条目涉及复数和未知数时，例如我们正在调整方案时，对等元的数值分析仍然很困难。我们提出了建立类似小矩阵的技术，这些小矩阵仍与傅立叶指数相关，其特征结构与全矩阵简单相关，但都是实数。这些技术将已知技术扩展到了旋转晶格的方案，以及顶点在拓扑上不位于所研究的顶点或面中心附近的对称轴上的方案。我们的技术使调整这些细分方案变得更容易。我们通过分析 n 边面中心的所谓最简单方案来说明这一点。
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Pub Date : 2024-09-18DOI: 10.1016/j.cagd.2024.102388
Csaba Bálint, Róbert Bán, Gábor Valasek
This paper presents a new method for the computation of the generalized Voronoi diagram of planar polygons. First, we show that the vertices of the cut locus can be computed efficiently. This is achieved by enumerating the tripoints of the polygon, a superset of the cut locus vertices. This is the set of all points that are of equal distance to three distinct topological entities. Then our algorithm identifies and connects the appropriate tripoints to form the cut locus vertex connectivity graph, where edges define linear or parabolic boundary segments between the Voronoi regions, resulting in the generalized Voronoi diagram. Our proposed method is validated on complex polygon soups. We apply the algorithm to represent the exact signed distance function of the polygon by augmenting the Voronoi regions with linear and radial functions, calculating the cut locus both inside and outside.
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