带法向约束的隐式B样条曲线重构PIA方法EI北大核心CSCD

为使隐式曲线能够更好地拟合散乱数据点及其几何特征,提出一种带法向约束的隐式曲线重构渐进迭代(progressive and iterative approximation,PIA)方法.首先,基于隐式B样条函数提出有效的曲线拟合模型;其次,通过加入偏移数据点来消除额外零水平集,同时加入法向项来控制曲线的法向误差;最后,经多次优化迭代得到高精度的拟合曲线.在配置为2.6 GHz英特尔处理器,内存为16 GB的电脑上采用MATLAB实现编程.经多条不同形态封闭曲线拟合的实验结果表明,与隐式PIA(implicit PIA,I-PIA)方法和T样条曲线重构方法相比,从数据点精度和法向误差以及收敛速度3个评价指标进行评估,该方法能够在保证数据点精度的前提下,有效地降低法向误差,并具有更快的收敛速度.此外,实例结果也表明该方法具备鲁棒性.     

隐式B样条曲线拟合的加权PIA算法

为了使拟合数据点的曲线生成速度更快、误差更小,提出一种隐式B样条曲线拟合数据点的加权PIA算法.首先,用待拟合数据点以及给定法向量生成偏移点集.然后,通过偏移点集构造差分向量,从而得到需要调整的误差控制系数,为了使迭代效率更高,在迭代过程中对误差控制系数做加权处理.最后,用最新的控制系数矩阵得到拟合数据点的曲线.文中5个数值算例采用均匀节点序列,实验结果表明,在相同迭代次数下,相对于I-PIA算法,该算法得到的拟合曲线误差值更小,曲线能更好保特征.     

集多种特性的三次三角伪B样条

目的 为了同时解决传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,提出了一类集多种特性的三次三角伪B样条。方法 首先构造了一组带两个参数的三次三角伪B样条基函数,然后在此基础上定义了相应的参数伪B样条曲线,并讨论了该曲线的特性及光顺性问题,最后研究了相应的代数伪B样条,并给出了最优代数伪B样条的确定方法。结果 参数伪B样条曲线不仅满足C²连续,而且无需求解方程系统即可自动插值于给定的型值点。当型值点保持不变时,插值曲线的形状还可通过自带的两个参数进行调控。在适当条件下,该参数伪B样条曲线可精确表示圆弧、椭圆弧、星形线等常见的工程曲线。相应的代数伪B样条具有参数伪B样条曲线类似的性质,利用最优代数伪B样条可获得满意的插值效果。结论 所提出的伪B样条同时解决了传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,是一种实用的曲线造型方法。     

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.     

带形状控制的自由曲线曲面参数样条

目的 样条曲线曲面的构造是工程制图中的一个重要部分。针对双曲抛物面上参数样条曲线的构造,在已有的研究基础上提出了一种样条方法使曲线曲面可以任意地逼近一个多边形或者一个网格。方法 在标准四面体内构造一个双曲抛物面,在该曲面上以基函数参数化的方法定义一种带形状参数的参数样条曲线曲面,样条基函数通过将双曲抛物面的有理参数化进行限定,生成单参数有理样条基函数。详细研究了样条的保形性及其端点性质。结果 样条曲线具有一个可变的形状控制因子,可以对曲线进行调整,能以任意精度逼近这个控制四边形或网格。对空间节点列,利用该样条可以生成G²-连续空间曲线,同样对于空间网格可以构造G²-连续的拟合曲面,它所对应的基函数可以是有理形式。结论 实验结果表明,本文在笔者已有的研究基础上提出的参数样条曲线可以通过重心坐标系变换适应为任意的四边形,除了空间四面体内的样条曲线,四面体退化成四边形同样可实现。     

C3连续的单位四元数插值样条曲线

目的 构造一类C³连续的单位四元数插值样条曲线,证明它的插值性和连续性,并把它应用于刚体关键帧动画设计中。方法 利用R³空间中插值样条曲线的5次多项式调配函数的累和形式构造了S³空间中单位四元数插值样条曲线,它不仅能精确通过一系列给定的方向,而且能生成C³连续的朝向曲线。结果 与Nielson的单位四元数均匀B样条插值曲线的迭代构造方法相比,所提方法避免了为获取四元数B样条曲线控制顶点对非线性方程组迭代求解的过程,提高了运算效率;与单位四元数代数三角混合插值样条曲线的构造方法（Su方法）相比,所提方法只用到多项式基,运算速度更快。本例中创建关键帧动画所需的时间与Nielson方法和Su方法相比平均下降了73%和33%。而且,相比前两种方法,所提方法产生的四元数曲线连续性更高,由C²连续提高到C³连续,这意味着动画中刚体的朝向变化更加自然。结论 仿真结果表明,本文方法对刚体关键帧动画设计是有效的,对实时性和流畅性要求高的动画设计场合尤为适用。     

隐式B-样条曲线重建的直接Greville纵标法

提出了一种以隐式B-样条曲线为表达形式,基于直接Greville纵标的曲线重建方法。根据点云建立有向距离场,并作为B-样条函数的Greville纵标,然后根据高影响区内的平均代数误差优化Greville纵标;得到一个隐式B-样条函数,该函数的零点集即为重建曲线。该方法具有模型简单,重建速度快,无多余分支,无需手工调节任何参数的优点。实验结果证实了该直接法的效率明显高于点拟合法和普通场拟合法,以几何误差为准则的精度亦优于普通场拟合方法。     

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．     

网格修补的特征配准方法

目的 网格重建和编辑会产生几何特征缺失的模型,填补这些空洞具有重要的意义。为了克服复杂曲面修补中网格融合难以配准的问题,提出了环驱动球坐标结合基于曲率及法向ICP(iterative closest point)迭代配准的网格修补方法。方法 首先用户查找合适的源网格面片放入空洞处周围;然后对目标网格空洞环建立B样条曲线,将带修补网格包边界置于B样条曲线上,构架环驱动球坐标,将源网格变形初步配准目标网格空洞周围领域;最后使用Laplacian光顺并基于网格曲率及法向进行ICP迭代配准,使源网格与目标网格光滑拼接融合。结果 该方法能够有效修补网格空洞缺失的细节特征,并且拼接处光滑连续。 结论 环驱动球坐标配准避免了网格变形的包围网格笼子构造,再通过ICP迭代精确配准网格,和以往的网格修补方法相比,该方法能够很好地修补网格空洞处细节特征。     

三次B样条插值的网格拼接和融合

目的 网格模型的拼接和融合是3维模型编辑的一个重要方面。为了提高3维模型之间拼接曲面的精度和效率,提出一种基于三次均匀B样条曲线曲面的网格融合方法。方法 首先,利用协变分析和数据驱动方法在目标模型上选定融合区域、确定要融合模型的大小及方向;其次,根据选定的3维网格模型,确定待拼接区域的边界,识别并记录边界点集,利用三次B样条插值边界点集;然后,对边界曲线进行双三次B样条曲面插值得到拼接区域连续曲面,并以此作为两模型拼接时的过渡面;最后,对拼接区域重采样,并对其三角化,以实现网格模型的无缝光滑拼接和融合。结果 为了验证本文方法对3维模型拼接的有效性,选取4组不同的模型,分别对其使用本文提出的融合拼接方法进行实验,对前两组模型的拼接效果进行了对比试验,实验结果表明,本文方法可以达到很好的拼接效果,对于融合区域以外的部分能够保持源模型的细节特征,拼接部分的过渡区域光顺平滑,拼接后的模型完整性佳。在运行时间相差0.05 s内,与数据驱动的建模方法相比,本文方法可以处理的节点数至少多2 000个,面片数至少多5 000个。结论 本文方法能够适用于具有任何边界的模型,在选取模型时,对于模型的形状、大小、拓扑结构等的要求较低,适用于新模型的快速建造,因此,该算法可应用于医学、商业广告、动画娱乐以及几何建模和制造等较为广阔的应用领域。     

FasTFit: A fast T-spline fitting algorithm

T-spline has been recently developed to represent objects of arbitrary shapes using a smaller number of control points than the conventional NURBS or B-spline representations in computer aided design, computer graphics, and reverse engineering. However, existing methods for fitting a T-spline over a point cloud are slow. By shifting away from the conventional iterative fit-and-refine paradigm, we present a novel split-connect-fit algorithm to more efficiently perform the T-spline fitting. Through adaptively dividing a point cloud into a set of B-spline patches, we first discover a proper topology of T-spline control points, i.e., the T-mesh. We then connect these B-spline patches into a single T-spline surface with different continuity options between neighboring patches according to the data. The T-spline control points are initialized from their correspondences in the B-spline patches, which are refined by using a conjugate gradient method. In experiments using several types of large-sized point clouds, we demonstrate that our algorithm is at least an order of magnitude faster than state-of-the-art algorithms while provides comparable or better results in terms of quality and conciseness.     

任意NUBS曲线的小波分析和造型技术

为了对任意NUBS曲线进行精确的分解和重构，提出了半正交B样条小波分解和重构的新算法，同时给出了处理非均匀B样条曲线的非整数阶分辨率的小波分解和重构算法，并实现了任意非均匀B样条曲线的多分辨率表示，对于任意非均匀B样条或NUBS曲线，无论它有多少个控制点，均可以对它进行半正交分解和重构，而不受控制点数必须等于2＋3的限制，从这个意义上讲，该方法不仅可以实现连续分辨率水平（continuous-resolutionlevel)的非均匀B样条曲线造型，还可以对非均匀B样条和NURBS曲线进行精确的分解和重构，这对于B样条曲线曲面的多分辨率造型与显示具有重大应用价值。     

Fitting unorganized point clouds with active implicit B-spline curves

In computer-aided geometric design and computer graphics, fitting point clouds with a smooth curve (known as curve reconstruction) is a widely investigated problem. In this paper, we propose an active model to solve the curve reconstruction problem, where the point clouds are approximated by an implicit B-spline curve, i.e., the zero set of a bivariate tensor-product B-spline function. We minimize the geometric distance between the point clouds and the implicit B-spline curve and an energy term (or smooth term) which helps to extrude the possible extra branches of the implicit curve. In each step of the iteration, the trust region algorithm in optimization theory is applied to solve the corresponding minimization problem. We also discuss the proper choice of the initial shape of the approximation curve. Examples are provided to illustrate the effectiveness and robustness of our algorithm. The examples show that the proposed algorithm is capable of handling point clouds with complicated topologies.     

Contour curve reconstruction from cloud data for rapid prototyping

In this study, a method for generation of sectional contour curves directly from cloud point data is given. This method computes contour curves for rapid prototyping model generation via adaptive slicing, data points reducing and B-spline curve fitting. In this approach, first a cloud point data set is segmented along the component building direction to a number of layers. The points are projected to the mid-plane of the layer to form a 2-dimensional (2D) band of scattered points. These points are then utilized to construct a boundary curve. A number of points are picked up along the band and a B-spline curve is fitted. Then points are selected on the B-spline curve based on its discrete curvature. These are the points used as centers for generation of circles with a user-define radius to capture a piece of the scattered band. The geometric center of the points lying within these circles is treated as a control point for a B-spline curve fitting that represents a boundary contour curve. The advantage of this method is simplicity and insensitivity to common small inaccuracies. Two experimental results are included to demonstrate the effectiveness and applicability of the proposed method.     

空间曲线基于内在几何量的高质量采样和B样条拟合

为解决均匀参数采样在许多情况下得到质量不高的采样点,进而生成不理想的B样条拟合曲线,提出空间曲线基于内在几何量的均匀采样方法,以获得给定总数且具有代表性的采样点.首先定义基于弧长、曲率和挠率加权组合的特征函数,通过调整组合参数更好匹配不同的曲线形状;然后提出空间曲线基于内在几何量的自适应采样方法,迭代生成满足给定距离阈值的采样点.采用最大绝对误差和均方根误差作为评价指标,与均匀弧长采样方法和基于弧长和曲率平均的均匀采样方法进行对比,并通过实例进行验证.结果表明,文中方法在采样质量和B样条拟合结果上获得明显改善.     

B-spline surface fitting based on adaptive knot placement using dominant columns

By expanding the idea of B-spline curve fitting using dominant points (Park and Lee 2007 [13]), we propose a new approach to B-spline surface fitting to rectangular grid points, which is based on adaptive knot placement using dominant columns along u- and v-directions. The approach basically takes approximate B-spline surface lofting which performs adaptive multiple B-spline curve fitting along and across rows of the grid points to construct a resulting B-spline surface. In multiple B-spline curve fitting, rows of points are fitted by compatible B-spline curves with a common knot vector whose knots are computed by averaging the parameter values of dominant columns selected from the points. We address how to select dominant columns which play a key role in realizing adaptive knot placement and thereby yielding better surface fitting. Some examples demonstrate the usefulness and quality of the proposed approach.     

Shape modeling based on specifying the initial B-spline curve and scaled BFGS optimization method

In this paper, we consider the problem of fitting the B-spline curves to a set of ordered points, by finding the control points and the location parameters. The presented method takes two main steps: specifying initial B-spline curve and optimization. The method determines the number and the position of control points such that the initial B-spline curve is very close to the target curve. The proposed method introduces a length parameter in which this allows us to adjust the number of the control points and increases the precision of the initial B-spline curve. Afterwards, the scaled BFGS algorithm is used to optimize the control points and the foot points simultaneously and generates the final curve. Furthermore, we present a new procedure to insert a new control point and repeat the optimization method, if it is necessary to modify the fitting accuracy of the generated B-spline fitting curve. Associated examples are also offered to show that the proposed approach performs accurately for complex shapes with a large number of data points and is able to generate a precise fitting curve with a high degree of approximation.