基于B样条整体最小二乘的非线性多功能传感器信号重构方法

提出一种基于B样条整体最小二乘(Total least squares,TLS)的非线性多功能传感器信号重构新方法.该方法利用B样条基函数直接构建描述多功能传感器传递函数反函数的张量积B样条曲面;采用TLS求解超定方程组以获得稳定的控制系数估计.以二输入二输出多功能传感器模型为实验对象,在两种非线性情况下对多功能传感器的输入信号进行了重构,重构相对误差分别为0.162％和1.043％,并与常用重构方法进行了对比分析.理论和实验表明,B样条TLS重构方法对非线性多功能传感器传递函数的反函数具有良好的逼近性能,在信号重构中表现出较好的有效性.     

Bézier曲线间最近距离的计算方法

针对Bézier曲线间最近距离计算问题,提出一种简捷、可靠的计算方法.该方法以Bernstein多项式算术运算为工具,建立Bézier曲线间最近距离的计算模型;然后充分利用Bézier曲面的凸包性质和de Casteljau分割算法进行求解.该方法几何意义明确,能有效地避免迭代初始值的选择和非线性方程组的求解,并可进一步推广应用于计算Bézier曲线/曲面间的最近距离.实验结果表明,该方法简捷、可靠且容易实现,与Newton-Raphson方法的融合可进一步提高该方法的运行速度.     

推广Dirichlet 方法用于B 样条极小曲面设计

为弥补当前NURBS系统无法有效设计工程所急需的B样条极小曲面的缺陷,将构造Bézier极小曲面的Dirichlet方法成功地推广到了B样条极小曲面设计.提出了插值控制网格边界的B样条曲面模型,运用B样条基函数的求导公式及求值割角算法,将计算极小曲面内部控制顶点的问题转化为一个线性方程组的求解,从而避免了强非线性问题所导致的困惑,极大地提高了运算效率.最后,用大量实例对理论和算法进行了验证.     

一种新的均匀样条曲线曲面设计方法

本文根据均匀B样条基函数的de Boor-Cox递推公式提出了一种新的样条曲线曲面设计方法。该方法从满足正性、局部支柱性和权性的初始基函数出发，可构造出具有高阶低次或低次高阶的多项式样条基函数和多种函数类型的样条函数。给出了设计这种样条曲线曲面的几种方法和实例，并对基函数的连续可微性进行了证明。该样条基函数和样条曲线曲面具有和均匀B样条类似的几何性质，且均匀B样条是其特例，可用于曲线曲面的几何造型和样条插值。     

B样条乘积计算的区间跳跃算法及应用

利用B样条基函数节点区间的对应关系,首先给出了B样条基函数间的转换矩阵的计算方法,进而给出了计算B样条乘积的区间跳跃算法。该算法仅需计算部分节点区间上的转换矩阵,因此称其为区间跳跃算法。这一方法解决了分段多项式与B样条曲线乘积的计算问题,可应用到B样条曲线的升阶、曲面间光滑拼接等问题中。通过算例验证了该方法计算简捷、易于实现。     

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．     

解析曲面与自由曲面混合模型的约束优化重建方法

首先阐述了平面、球面、柱面、锥面、B样条曲面的表达方式和相应的点到曲面的距离函数;然后给出了解析曲面与自由曲面保持连续性所需满足的约束方程;带约束的拟合问题通过惩罚函数方法来求解.实例表明:该方法可以应用于工业零件的反求设计中.     

无序B样条曲线的曲面拟合算法

针对目前无序曲线拟合算法不能控制拟合误差的问题,提出了利用B样条曲面拟合4条边界线及一组无序B样条曲线的算法.首先由边界曲线得到初始曲面,并将曲线曲面写成分段Bezier形式;然后借鉴曲面蒙皮的思想,得到关于待拟合曲面的方程组,并对相邻的Bézier曲面施加C1连续约束;接着利用SVD以及能量优化来求得唯一的拟合曲面;最后在曲线曲面距离最大处插入节点,重复求解过程,直到误差满足要求.实验结果表明,与已有算法相比,该算法可以得到满足用户误差要求的、光滑的拟合曲面,且具有更好的数值稳定性.     

基于小波的准均匀B样条曲线曲面变分造型

变分造型是自由曲线曲面交互设计中常用的技术之一，在传统的变分造型技术中，一般将此类问题转化为对控制顶点的优化求解问题，但当基函数多进，B样条的局部性常会导致待求解系统的性态不好，求解效率降低，对此，文中应用小波方法解决准均匀B样条曲线曲面变分造型中的优化求解问题，由于小波基具有多分辨特性，该方法可以大大提高迭代速度。     

等几何分析中的复杂物理区域投影算子及误差分析

针对等几何分析中复杂物理区域上的偏微分方程求解问题,提出了多片参数域上双三次样条投影映射的方法.首先基于多片参数化,构造了复杂物理域上的一个投影映射;其次对于物理域上的光滑函数,讨论了该投影映射的逼近误差,理论分析表明该投影映射可达到最优逼近阶;最后基于投影映射的思想,给出了一类适用于基于等几何分析在复杂物理区域上二阶椭圆方程求解的样条空间.数值算例的结果表明,该方法求解的逼近误差阶可达到最优.     

Variational Modeling of Free-form Curve and Surface Using Wavelet Basis Functions

Variational modeling approach is often used to interactively design free-form curves and surfaces. Traditionally, a variational problem can be transformed to the optimization of control points. Unfortunately, as the number of basis functions grows, the local support property of B-spline often makes the computation of an optimization system time-consuming. To solve this problem, wavelet basis instead of B-spline basis is used to represent the curves or surfaces. Because the wavelet basis is a hierarchical basis with multiresolution property, the coarse wavelet basis can be used to describe the overall shape of the curves/surfaces, while the finer wavelet basis used to describe the details of the curves/surfaces. Thus, the computing speed of the optimization system can be raised greatly.     

Fast Evaluation Algorithm for NURBS Surfaces

Through a precise recursion of B-spline bases and the resursive expression of the derivatives of rational surfaces, this paper presents an efficient algorithm for the calculation of NURBS surfaces and all their directional derivatives. The algorithm requires less storage and proves to be stable.     

有理B样条曲面的区间隐式化

提出有理B样条曲面的区间隐式化方法,即对一个有理B样条曲面,寻求包含给定的曲面的区间隐式B样条曲面,使得区间隐式B样条曲面的"厚度"尽量小,同时尽量避免出现多余分支.该问题等价于求区间隐式B样条曲面的2个边界曲面.针对该问题建立一个最优化模型并求解.     

形状可调的5次组合样条及其参数选择

目的 为了克服3次参数B样条在形状调整与局部性方面的不足,提出带参数的5次多项式组合样条。方法 首先构造一组带参数的5次多项式基函数;然后采用与3次B样条曲线相同的组合方式定义带参数的5次多项式组合样条曲线,并讨论基于能量优化法的5次组合样条曲线参数最佳取值问题;最后定义相应的组合样条曲面,并研究利用粒子群算法求解曲面的最佳参数取值。结果 5次组合样条不仅继承了3次B样条的诸多性质,而且还比3次B样条具有更强的局部性及形状可调性。由于5次组合样条仍为多项式模型,因此方程结构相对较为简单,符合实际工程的需要。利用能量优化法可获得光顺的5次组合样条曲线与曲面。结论 所提出5次多项式组合样条克服了3次参数B样条在形状调整与局部性方面的不足,是一种实用的自由曲线曲面造型方法。     

Lofted B-spline surface interpolation by linearly constrained energy minimization

This paper proposes a new approach for lofted B-spline surface interpolation to serial contours, where the number of points varies from contour to contour. The approach first finds a common knot vector consisting of fewer knots that contain enough degrees of freedom to guarantee the existence of a B-spline curve interpolating each contour. Then, it computes from the contours a set of compatible B-spline curves defined on the knot vector by adopting B-spline curve interpolation based on linearly constrained energy minimization. Finally, it generates a B-spline surface interpolating the curves via B-spline surface lofting. As the energy functional is quadratic, the energy minimization problem leads to that of solving a linear system. The proposed approach is efficient in computation and can realize more efficient data reduction than previous approaches while providing visually pleasing B-spline surfaces. Moreover, the approach works well on measured data with noise. Some experimental results demonstrate its usefulness and quality.     

Systolic architecture for B-spline surfaces

B-spline surfaces are among the most commonly used types of surfaces for modeling objects in computer graphics and CAD applications. One of the time consuming operations in B-spline surface generation is that of inversion. An efficient algorithm is proposed for solving this problem. This algorithm is implemented on a systolic architecture in order to facilitate fast interactive surface design.     

Constructing iterative non-uniform B-spline curve and surface to fit data points

In this paper, based on the idea of profit and loss modification, we presentthe iterative non-uniform B-spline curve and surface to settle a key problem in computeraided geometric design and reverse engineering, that is, constructing the curve (surface)fitting (interpolating) a given ordered point set without solving a linear system. We startwith a piece of initial non-uniform B-spline curve (surface) which takes the given point setas its control point set. Then by adjusting its control points gradually with iterative formula,we can get a group of non-uniform B-spline curves (surfaces) with gradually higherprecision. In this paper, using modern matrix theory, we strictly prove that the limit curve(surface) of the iteration interpolates the given point set. The non-uniform B-spline curves(surfaces) generated with the iteration have many advantages, such as satisfying theNURBS standard, having explicit expression, gaining locality, and convexity preserving,etc     

Quadratic B-spline curve interpolation

Traditional approach in performing even-degree B-spline curve/surface interpolation would generate undesired results. In this paper, we show that the problem is with the selection of interpolation parameter values, not with even-degree B-spline curves and surfaces themselves. We prove this by providing a new approach to perform quadratic B-spline curve interpolation. This approach generates quadratic B-spline curves whose quality is comparable to that of cubic interpolating B-spline curves. This makes quadratic B-spline curves better choices than cubic B-spline curves in some applications in graphics and geometric modeling, since it is cheaper to render/subdivide a quadratic curve and it is easier to find the intersection of two quadratic curves.     

A novel method for 3D reconstruction: Division and merging of overlapping B-spline surfaces

B-spline surfaces, extracted from scanned sensor data, are usually required to represent objects in inspection, surveying technology, metrology and reverse engineering tasks. In order to express a large object with a satisfactory accuracy, multiple scans, which generally lead to overlapping patches, are always needed due to, inter-alia, practical limitations and accuracy of measurements, uncertainties in measurement devices, calibration problems as well as skills of the experimenter. In this paper, we propose an action sequence consisting of division and merging. While the former divides a B-spline surface into many patches with corresponding scanned data, the latter merges the scanned data and its overlapping B-spline surface patch. Firstly, all possible overlapping cases of two B-spline surfaces are enumerated and analyzed from a view of the locations of the projection points of four corners of one surface in the interior of its overlapping surface. Next, the general division and merging methods are developed to deal with all overlapping cases, and a simulated example is used to illustrate aforementioned detailed procedures. In the sequel, two scans obtained from a three-dimensional laser scanner are simulated to express a large house with B-spline surfaces. The simulation results show the efficiency and efficacy of the proposed method. In this whole process, storage space of data points is not increased with new obtained overlapping scans, and none of the overlapping points are discarded which increases the representation accuracy. We believe the proposed method has a number of potential applications in the representation and expression of large objects with three-dimensional laser scanner data.