基于隐式T样条的曲面重构算法

提出隐式T样条曲面，将T网格从二维推广到三维情形，同时利用八叉树及其细分过程，从无结构散乱点数据集构造T网格，利用曲面拟合模型将曲面重构问题转化为最优化问题；然后基于隐式T样条曲面将最优化问题通过矩阵形式表述，依据最优化原理将该问题转化成线性方程组，通过求解线性方程组解决曲面重构问题；最后结合计算实例进行讨论．该方法能较好地解决曲面重构问题，与传统张量B样条函数相比，能效地减少未知控制系数与计算量．     

隐式T样条实现封闭曲面重建

为了简化法向偏差约束条件和优化光滑能量项,提出一种隐式T样条曲面重建算法.首先利用八叉树及其细分过程从采样点集构造三维T网格,以确定每个控制系数对应的混合函数;然后基于隐式T样条曲面建立目标函数,利用偏移曲面点集控制法向,采用广义交叉检验(GCV)方法估计最优光滑项系数,并依据最优化原理将该问题转化为线性方程组求解得到控制系数,从而实现三角网格曲面到光滑曲面的重建.在误差较大的区域插入控制系数进行T网格局部修正,使得重建曲面达到指定精度.该算法使重建曲面C1连续条件得到松弛,同时给出最优的光顺项系数估计,较好地解决了封闭曲面的重建问题.实例结果表明,文中算法逼近精度高,运算速度快,仿真结果逼真.     

海量点云曲面增量拓扑重建EI北大核心CSCD

针对现有的曲面重建算法难以兼顾大规模采样数据的重建效率与重建曲面拓扑正确性的问题,提出一种基于局部Delaunay网格剖分的曲面增量重建算法.该算法采用波前扩展的策略,通过波前环的扩张、分裂、重叠面片的消除等步骤,将局部重建过程传播至每个样点的邻近区域,获得插值于采样点集的二维定向流形网格曲面,实现整个采样点集的增量拓扑重建;在曲面局部重建过程中,分别基于局部区域的Cocone算法与二维投影点集的Delaunay网格剖分方法重建曲面的尖锐区域与平坦区域,其中局部区域重建曲面网格的边界的正确性由区域之外的少量辅助样点保护.实验结果表明,文中算法具有较高的重建效率,适用于封闭和非封闭海量点云数据的重建;且在采样密度符合要求的情况下,重建的网格曲面与原表面拓扑同构.     

海量点云曲面增量拓扑重建

针对现有的曲面重建算法难以兼顾大规模采样数据的重建效率与重建曲面拓扑正确性的问题,提出一种基于局部Delaunay网格剖分的曲面增量重建算法.该算法采用波前扩展的策略,通过波前环的扩张、分裂、重叠面片的消除等步骤,将局部重建过程传播至每个样点的邻近区域,获得插值于采样点集的二维定向流形网格曲面,实现整个采样点集的增量拓扑重建;在曲面局部重建过程中,分别基于局部区域的Cocone算法与二维投影点集的Delaunay网格剖分方法重建曲面的尖锐区域与平坦区域,其中局部区域重建曲面网格的边界的正确性由区域之外的少量辅助样点保护.实验结果表明,文中算法具有较高的重建效率,适用于封闭和非封闭海量点云数据的重建;且在采样密度符合要求的情况下,重建的网格曲面与原表面拓扑同构.     

基于SOM的散乱数据点集的B样条曲面重建

利用自组织映射神经网络(SOM)技术对散乱数据点集进行B样条曲面重建时,往往存在网络学习时间过长和学习效果不理想等问题。提出了一种新的神经元初始化方法和分块学习算法,该算法首先运用主元素分析方法(PCA)对散乱数据进行分块,将拓扑结构为四边形的输出层神经元初始化在每块散乱数据的最小二乘平面上进行网络学习和训练,将分块学习得到的各网格曲面拼接成一个整体;然后对该整体网格曲面的边界和内部单独学习,得到一张逼近待重建曲面的双线性B样条曲面;最后对该B样条曲面误差进行了修正。实例证明,该算法可以明显地减少SOM网络学习时间,并改善网络学习效果。     

逆向工程的关键技术及其应用研究

全面论述了逆向工程的应用领域;讨论了坐标点的测量技术,说明了各种测量方法的特点和适用范围;阐述了逆向工程中曲面重建技术及存在的主要问题,提出了一种用Ｂ样条曲面重构复杂雕塑曲面的算法。该算法首先利用部分测量数据构造张量积的Ｂ样条网格曲面,通过确定适当的检查点,并计算检查点到所构造的Ｂ样条网格曲面间的距离,从而控制重构曲面的构造精度;文章最后给出了与通用ＣＡＤ／ＣＡＭ集成一体的曲面自动重建模块的主要功能。     

带法向约束的隐式T样条曲线重构

目的 隐式曲线能够描述复杂的几何形状和拓扑结构,而传统的隐式B样条曲线的控制网格需要大量多余的控制点满足拓扑约束。有些情况下,获取的数据点不仅包含坐标信息,还包含相应的法向约束条件。针对这个问题,提出了一种带法向约束的隐式T样条曲线重建算法。方法 结合曲率自适应地调整采样点的疏密,利用二叉树及其细分过程从散乱数据点集构造2维T网格;基于隐式T样条函数提出了一种有效的曲线拟合模型。通过加入偏移数据点和光滑项消除额外零水平集,同时加入法向项减小曲线的法向误差,并依据最优化原理将问题转化为线性方程组求解得到控制系数,从而实现隐式曲线的重构。在误差较大的区域进行T网格局部细分,提高重建隐式曲线的精度。结果 实验在3个数据集上与两种方法进行比较,实验结果表明,本文算法的法向误差显著减小,法向平均误差由10^-3数量级缩小为10^-4数量级,法向最大误差由10^-2数量级缩小为10^-3数量级。在重构曲线质量上,消除了额外零水平集。与隐式B样条控制网格相比,3个数据集的T网格的控制点数量只有B样条网格的55.88%、39.80%和47.06%。结论 本文算法能在保证数据点精度的前提下,有效降低法向误差,消除了额外的零水平集。与隐式B样条曲线相比,本文方法减少了控制系数的数量,提高了运算速度。     

残缺网格模型的快速B样条曲面重建

针对残缺的三角网格模型,提出一种将网格模型的散乱数据点转化为有序阵列点再进行B样条曲面快速重建的算法.首先确定最小二乘平面上的一个矩形参数域,再构造出一个平面阵列点列,并部分映射到三维网格上;然后利用空间阵列点的邻域信息估计4个角点的空间坐标,并构造径向基函数曲面,用于补充空间阵列点列中残缺的数据;最后利用有序点列拟合的高效性构造B样条曲面.实验结果表明:该算法速度快、拟合精度高、鲁棒性强,重建的曲面具有良好的光顺性和可延伸性,适用于逆向工程中对经过数据分割后的网格模型的自由曲面重建.     

任意散乱点集的B-样条曲面重建

为了解决工业设计中复杂形体的曲面造型问题,提出了一种张量积型的低阶B-样条曲面重建算法。先将采集到的任意拓扑形状的散乱数据点进行三次不同的参数化得到四边形控制网格,然后再采用张量积型的双二次、双三次B-样条进行拟合,在拟合的过程中采用距离函数来控制拟合误差,得到光滑的曲面。运用该方法,直接对初始散乱点集进行重建,方法简单易实施,重建效率高并且重建后的样条曲面自然满足切平面连续。与以往的方法相比,该方法在逆向工程中可以在保证连续性的情况下,得到精准的结果曲面,提高了曲面造型的质量和效率。     

针对密集点云的快速曲面重建算法

为了能够快速地从高密度散乱点云生成三角形网格曲面,提出一种针对散乱点云的曲面重建算法.首先通过逐层外扩建立原始点云的近似网格曲面,然后对近似网格曲面进行二次剖分生成最终的精确曲面;为了能够处理噪声点云,在剖分过程中所有网格曲面顶点都通过层次B样条进行了优化.相比于其他曲面重建方法,该算法剖分速度快,且能够保证点云到所生成的三角网格曲面的距离小于预先设定容限.实验结果表明,文中算法能够有效地实现高密度散乱点云的三角剖分,且其剖分速度较已有算法有大幅提高.     

Comparison of the meccano method with standard mesh generation techniques

The meccano method is a novel and promising mesh generation technique for simultaneously creating adaptive tetrahedral meshes and volume parameterizations of a complex solid. The method combines several former procedures: a mapping from the meccano boundary to the solid surface, a 3-D local refinement algorithm and a simultaneous mesh untangling and smoothing. In this paper we present the main advantages of our method against other standard mesh generation techniques. We show that our method constructs meshes that can be locally refined using the Kossaczky bisection rule and maintaining a high mesh quality. Finally, we generate volume T-mesh for isogeometric analysis, based on the volume parameterization obtained by the method.     

Parameterization of computational domain in isogeometric analysis: Methods and comparison

Parameterization of computational domain plays an important role in isogeometric analysis as mesh generation in finite element analysis. In this paper, we investigate this problem in the 2D case, i.e., how to parametrize the computational domains by planar B-spline surface from the given CAD objects (four boundary planar B-spline curves). Firstly, two kinds of sufficient conditions for injective B-spline parameterization are derived with respect to the control points. Then we show how to find good parameterization of computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of planar B-spline parameterization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of planar B-spline parameterization. By using this method, the resulted parameterization has no self-intersections, and the isoparametric net has good uniformity and orthogonality. After introducing a posteriori error estimation for isogeometric analysis, we propose r-refinement method to optimize the parameterization by repositioning the inner control points such that the estimated error is minimized. Several examples are tested on isogeometric heat conduction problem to show the effectiveness of the proposed methods and the impact of the parameterization on the quality of the approximation solution. Comparison examples with known exact solutions are also presented.     

Reconstructing regular meshes from points

We propose an algorithm for reconstructing regular meshes from unorganized point clouds. At first, a nearly isometric point parameterization is computed using only the location of the points. A mesh, composed of nearly equilateral triangles, is later created using a regular sampling pattern. This approach produces meshes with high visual quality and suitable for use with applications such as finite element analysis, which tend to impose strong constraints on the regularity of the input mesh. Geometric properties, such as local connectivity and surface features, are identified directly from the points and are stored independent of the resulting mesh. This decoupling preserves most details and allows more flexibility for meshing. The resulting parameterization supports several direct applications, such as texturing and bump mapping. In addition, novel boundary identification and cut parameterization algorithms are proposed to overcome the difficulties caused by cuts, non-closed surfaces and possible self-overlapping parameter patches. We demonstrate the effectiveness of our approach by reconstructing regular meshes from real datasets, such as a human colon obtained from CT scan and objects digitized using laser scanners.     

A new method for T-spline parameterization of complex 2D geometries

We present a new strategy, based on the idea of the meccano method and a novel T-mesh optimization procedure, to construct a T-spline parameterization of 2D geometries for the application of isogeometric analysis. The proposed method only demands a boundary representation of the geometry as input data. The algorithm obtains, as a result, high quality parametric transformation between 2D objects and the parametric domain, the unit square. First, we define a parametric mapping between the input boundary of the object and the boundary of the parametric domain. Then, we build a T-mesh adapted to the geometric singularities of the domain to preserve the features of the object boundary with a desired tolerance. The key of the method lies in defining an isomorphic transformation between the parametric and physical T-mesh finding the optimal position of the interior nodes by applying a new T-mesh untangling and smoothing procedure. Bivariate T-spline representation is calculated by imposing the interpolation conditions on points sited both in the interior and on the boundary of the geometry. The efficacy of the proposed technique is shown in several examples. Also we present some results of the application of isogeometric analysis in a geometry parameterized with this technique.     

Feature-preserving T-mesh construction using skeleton-based polycubes

This paper presents a novel algorithm which uses skeleton-based polycube generation to construct feature-preserving T-meshes. From the skeleton of the input model, we first construct initial cubes in the interior. By projecting corners of interior cubes onto the surface and generating a new layer of boundary cubes, we split the entire interior domain into different cubic regions. With the splitting result, we perform octree subdivision to obtain T-spline control mesh or T-mesh. Surface features are classified into three groups: open curves, closed curves and singularity features. For features without introducing new singularities like open or closed curves, we preserve them by aligning to the parametric lines during subdivision, performing volumetric parameterization from frame field, or modifying the skeleton. For features introducing new singularities, we design templates to handle them. With a valid T-mesh, we calculate rational trivariate T-splines and extract Bézier elements for isogeometric analysis.     

Template-based 3D model fitting using dual-domain relaxation

We introduce a template fitting method for 3D surface meshes. A given template mesh is deformed to closely approximate the input 3D geometry. The connectivity of the deformed template model is automatically adjusted to facilitate the geometric fitting and to ascertain high quality of the mesh elements. The template fitting process utilizes a specially tailored Laplacian processing framework, where in the first, coarse fitting stage we approximate the input geometry with a linearized biharmonic surface (a variant of LS-mesh), and then the fine geometric detail is fitted further using iterative Laplacian editing with reliable correspondence constraints and a local surface flattening mechanism to avoid foldovers. The latter step is performed in the dual mesh domain, which is shown to encourage near-equilateral mesh elements and significantly reduces the occurrence of triangle foldovers, a well-known problem in mesh fitting. To experimentally evaluate our approach, we compare our method with relevant state-of-the-art techniques and confirm significant improvements of results. In addition, we demonstrate the usefulness of our approach to the application of consistent surface parameterization (also known as cross-parameterization).     

Partwise cross-parameterization via nonregular convex hull domains

In this paper, we propose a novel partwise framework for cross-parameterization between 3D mesh models. Unlike most existing methods that use regular parameterization domains, our framework uses nonregular approximation domains to build the cross-parameterization. Once the nonregular approximation domains are constructed for 3D models, different (and complex) input shapes are transformed into similar (and simple) shapes, thus facilitating the cross-parameterization process. Specifically, a novel nonregular domain, the convex hull, is adopted to build shape correspondence. We first construct convex hulls for each part of the segmented model, and then adopt our convex-hull cross-parameterization method to generate compatible meshes. Our method exploits properties of the convex hull, e.g., good approximation ability and linear convex representation for interior vertices. After building an initial cross-parameterization via convex-hull domains, we use compatible remeshing algorithms to achieve an accurate approximation of the target geometry and to ensure a complete surface matching. Experimental results show that the compatible meshes constructed are well suited for shape blending and other geometric applications.     

GeoBrush: Interactive Mesh Geometry Cloning

We propose a method for interactive cloning of 3D surface geometry using a paintbrush interface, similar to the continuous cloning brush popular in image editing. Existing interactive mesh composition tools focus on atomic copy‐and‐paste of preselected feature areas, and are either limited to copying surface displacements, or require the solution of variational optimization problems, which is too expensive for an interactive brush interface. In contrast, our GeoBrush method supports real‐time continuous copying of arbitrary high‐resolution surface features between irregular meshes, including topological handles. We achieve this by first establishing a correspondence between the source and target geometries using a novel generalized discrete exponential map parameterization. Next we roughly align the source geometry with the target shape using Green Coordinates with automatically‐constructed cages. Finally, we compute an offset membrane to smoothly blend the pasted patch with C continuity before stitching it into the target. The offset membrane is a solution of a bi‐harmonic PDE, which is computed on the GPU in real time by exploiting the regular parametric domain. We demonstrate the effectiveness of GeoBrush with various editing scenarios, including detail enrichment and completion of scanned surfaces.     

Watermarking 3D mesh by spherical parameterization

In this paper, a robust 3D trianglular mesh watermarking algorithm is presented by applying spherical parameterization. First, we transform the coordinate signals of the 3D triangular mesh into spherical signals using a global spherical parameterization and an even sampling scheme. Then, spherical harmonic transformation is used to generate some data for embedding watermarks. As a result, the watermarks can be embedded in the Fourier-frequency domain of the original mesh. Experimental results show that our watermarking algorithm is robust since watermarks can be extracted without mesh alignment or re-meshing under a variety of attacks, including noise addition, crop, filtering, enhancement, rotation, translation, scale and re-sampling.     

Quasi-isometric parameterization for texture mapping

In this paper, we present a new 3D triangular mesh parameterization method that is computationally efficient and yields minimized distance errors. The method has four steps. Firstly, multidimensional scaling (MDS) is used to flatten each submesh consisting of one vertex and its direct neighbours on the 3D triangular mesh. Secondly, an optimal method is used to compute the linear reconstructing weights of each vertex with respect to its neighbours. Thirdly, a spectral decomposition method is used to obtain initial 2D parameterization coordinates. Fourthly, the initial coordinates are rotated and scaled to minimize the distance errors. It is demonstrated that this method can be used for texture mapping. Analyses and examples show the effectiveness of this parameterization method compared with alternatives.