B样条快速求值算法及其在数控插补中的应用

通过对B样条的de Boor-Cox定义式分析,给出了一种基于向量扩展的B样条基函数快速求值算法。该算法能够将k次B样条非零值计算效率提高2k+1倍。该算法用于数控实时插补中的B样条曲线求值求导运算时,可获得比de Boor算法更高的计算效率。     

双二次B-样条插值图像缩放

双线性和各种双三次插值方法是图像缩放中常用方法,但是双二次插值函数却很少被人提起。本文提出了一种基于双二次B-样条局部插值的图像缩放方法,该算法在图像局部重构过程中对称地采用了4×4采样点,并通过对该函数进行重采样来实现图像的缩放,避免了二次函数在图像重构与采样中的相位失真问题,此算法是一个局部性算法,易于扩展。实验结果表明,本文算法得到的图像的峰值信噪比(PSNR)、MISSIM值比双线性插值、双三次卷积、Catmull-Rom三次插值、Dodgson插值算法都要好,接近于最好的双三次B-样条算法,视觉效果虽然不如双三次B-样条插值算法,但优于Dodgson方法,计算时间比双三次B-样条减少了近三分之一。由于该算法没有对图像边缘特征进行特殊处理,对于一些细节纹理比较丰富的图像,将进一步研究。     

周期B样条基函数系数的并行算法

在现有周期B样条插值方法中,需要用迭代算法确定B样条基函数系数。针对现有方法的不足,建立B样条基函数系数的并行算法。首先构造周期区域的正交B样条基,得出正交B样条基函数系数的并行算法;进一步利用正交B样条基函数系数与B样条基函数系数的关系,得出B样条基函数系数的并行算法;最后推导二阶、三阶、四阶周期插值B样条基函数系数及插值点函数值的显式算式。实验证明了该方法在实现B样条基函数系数快速并行算法的同时保持了B样条基函数简单的函数关系。     

B样条曲面的双向插值造型算法

基于Coons-Gordon造型原理,研究了插值两族相交截面线采样点的B样条曲面双向插值造型算法。参数化各采样点并计算每条截面线的节点矢量,估算每条截面线对应的曲面参数,根据每条截面线的节点分布以及另一族截面线对应的曲面参数统一节点矢量。分别插值两族截面线采样点及其公共点得到三张B样条曲面,其布尔和即为插值两族截面线采样点的B样条插值曲面。实例表明,得到的双向插值曲面控制顶点数少,光顺性好。     

Local control of interval tension using weighted splines

Cubic spline interpolation and B-spline sums are useful and powerful tools in computer aided design. These are extended by weighted cubic splines which have tension controls that allow the user to tighten or loosen the curve on intervals between interpolation points. The weighted spline is a C¹ piecewise cubic that minimizes a variational problem similar to one that a C² cubic spline minimizes. A B-spline like basis is constructed for weighted splines where each basis function is nonnegative and nonzero only on four intervals. The basis functions sum up identically to one, thus curves generated by summing control points multiplied by the basis functions have the convex hull property. Different weights are built into the basis functions so that the control point curves are piecewise cubics with local control of interval tension. If all weights are equal, then the weighted spline is the C² cubic spline and the basis functions are B-splines.     

基于四次B样条的曲线逼近算法

考虑到插值算法增减节点困难,传统逼近算法精度不够等缺点,有文献提出一种基于三次B样条的曲线逼近算法。该算法通过迭代逼近,提高了计算速度与精度。在系统研究此算法的基础上,将该算法推广到四次B样条,使其具有三阶可导性,并给出该算法收敛性的理论证明。最后用该算法对常用函数进行逼近效果实验。结果表明,所提出的四次B样条的曲线逼近算法收敛速度更快,且能够满足更高精度的实际工业生产需要。     

函数的分段有理二次B样条插值

通过对函数进行合理分割,给出函数分段三角形凸包的概念。提出了以分段区间端点的两条切线确定控制多边形的方案。详细地讨论了函数的分段参数有理二次B样条插值算法。插值函数保持了原始函数的一些重要几何性质、如单调性、凹凸性、G1连续性。数值实验表明,算法提供了函数近似表示的一条有效途径。     

Reducing control points in lofted B-spline surface interpolation using common knot vector determination

A new algorithm for reducing control points in lofted surface interpolation to rows of data points is presented in this paper. The key step of surface lofting is to obtain a set of compatible B-spline curves interpolating each row. Given a set of points and their parameterization, a necessary and sufficient condition is proposed to determine the existence of interpolating B-spline curves defined on a given knot vector. Based on this condition, we first properly construct a common knot vector that guarantees the existence of interpolating B-spline curves to each row of points. Then we calculate a set of interpolating B-spline curves defined on the common knot vector by energy minimization. Using this method, fewer control points are employed while maintaining a visually pleasing shape of the lofted surface. Several experimental results demonstrate the usability and quality of the proposed method.     

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。     

基于误差控制的自适应3次B样条曲线插值

针对现有曲线插值算法不能有效压缩型值点的缺陷,研究了一种自适应三次B样条曲线插值算法。从型值点序列中选用最少的点插值一条初始曲线,基于提出的点到曲线的最小距离计算方法,分别计算各非插值点对应的插值误差,并从中提取最大插值误差。若最大误差大于给定的误差阈值,则将其对应的型值点加入插值型值点序列,重新插值曲线,直到最大插值误差满足误差要求。与现有曲线插值算法相比,该算法可以在保证插值精度的前提下有效压缩数据量。     

密集散乱测量数据点的B样条曲面拟合研究

回顾了密集散乱数量数据点面拟合研究发展情况,针对异形边界自由曲面密集散乱测量数据点,提出一种B样条曲面多步拟合算法,其中涉及边界插值B样条曲面生成、Hardy′s双二次局部插值、规则网格数据点B样条曲面最小二乘拟合等关键技术,通过一个工程实例,对文中提出的B样条曲面多步拟合算法进行了实验验证。     

An Algorithm for Direct Multiplication of B-Splines

B-spline multiplication, that is, finding the coefficients of the product B-spline of two given B-splines is useful as an end result, in addition to being an important prerequisite component to many other symbolic computation operations on B-splines. Algorithms for B-spline multiplication standardly use indirect approaches such as nodal interpolation or computing the product of each set of polynomial pieces using various bases. The original direct approach is complicated. B-spline blossoming provides another direct approach that can be straightforwardly translated from mathematical equation to implementation; however, the algorithm does not scale well with degree or dimension of the subject tensor product B-splines. To addresses the difficulties mentioned heretofore, we present the Sliding Windows Algorithm (SWA), a new blossoming based algorithm for the multiplication of two B-spline curves, two B-spline surfaces, or any two general multivariate B-splines.      

Simple algorithms and architectures for B-spline interpolation

It is proved that the Toeplitz binary value matrix inversion associated with mth-order B-spline interpolation can be implemented using only 2(m+1) additions. Pipelined architectures are developed for real-time B-spline interpolation based on simple running average filters. It is shown that an ideal interpolating function, which is approximated by a truncated sinc function with M half cycles, can be implemented using B-splines with M+2 multiplies. With insignificant loss of performance, the coefficients at the knots of the truncated sinc function can be approximated using coefficients which are powers of two. The resulting implementation requires only M+4m+6 additions. It is believed that the truncated sinc function approximated by zero-order B-spline functions actually achieves the best visual performance     

基于最小变量的B-样条移动机器人路径规划研究

为解决移动机器人在避障时的曲线优化问题,提出了基于最小变量的B-样条路径规划方法。对该方法从数学模型上进行了推导,指出了该方法相对于其它B-样条方法的优点,并对该方法进行了优化,给出了相应的优化算法。研究表明：具有最小变量的B-样条函数比只用B-样条函数定义的曲线具有更优化的线性约束,其曲线具有更好的光滑性。     

两类B样条模糊系统及其应用

能够同时逼近函数及其导函数的模糊系统在应用中具有重要意义.本文利用B样条函数作为推理前件,得到了两类能够同时逼近函数及其导函数的B样条模糊系统.其中第一类B样条模糊系统是插值系统且对函数及其一阶导函数分别具有二阶和一阶逼近精度,第二类B样条模糊系统是拟插值系统且对函数及其一阶、二阶导函数均具有二阶逼近精度.最后,将这两类模糊系统应用到一级倒立摆的稳定控制中,仿真结果表明利用这两类模糊系统设计的控制器是可行的,且具有一定的鲁棒性.     

图像配准中B样条滤波和GPVE的融合插值算法

针对B样条GPVE插值法和B样条滤波法产生归一化互信息(NMI)测度伪极值点的原因,结合它们导致联合直方图聚散程度的互补效果,提出了各阶B样条滤波和GPVE的融合插值算法。通过图像之间的刚体配准实验,从测度曲线光滑性能和极值点数目方面,对比分析了一阶、三阶、五阶B样条滤波法,B样条GPVE插值法和新提出的融合插值算法。实验结果表明,提出的各阶B样条融合插值算法的配准性能都优于对应阶次B样条滤波法和B样条GPVE插值法。     

Linearized controller design for the output probability density functions of non-Gaussian stochastic systems

1 Introduction There are many practical systems that require the control of the shape of the output probability den- sity function rather than just their mean values and variances. These systems are seen in papermaking processes[1,2], chemical engineering, material science, combustion ?ame distribution systems and food pro- cessing industries. For example, in chemical engineer- ing the control of particle size distribution has al- ways been regarded as an important area of research[3], whilst …     

基于JPS改进的移动机器人路径规划算法

针对跳点搜索（jump point search,JPS）路径规划算法在大尺度复杂场景下存在内存资源消耗较大、路径结果平滑度较低且路径过于靠近障碍物等问题,提出融合安全势场等级函数与优化Floyd算法的改进JPS算法。首先建立了安全等级函数对栅格地图中的栅格状态进行重新赋值构建安全等级地图;然后改进了启发式函数,引入目标与主方向两项偏置函数项结合安全等级函数项,进一步减少对称性搜索带来的时间消耗,改善了所规划路径的安全程度。其次通过添加二次平滑算法流程优化了Floyd算法;最后结合B-spline样条插值法,进一步提高了改进算法所规划路径的平滑程度。仿真实验验证了改进优化算法在内存资源消耗、路径长度、路径平滑程度以及路径安全程度都有显著提升。     

高阶连续的形状可调三角多项式曲线曲面

目的目前使用的B样条曲线曲面存在着高连续阶与高局部调整性两者无法兼而有之的不足,且B样条曲线曲面的形状被控制顶点和节点向量唯一确定,这些因素影响着B样条方法的几何设计效果与方便性。本文旨在克服这种局限,以期构造具有高次B样条方法的高连续阶,低次B样条方法的高局部调整性,以及有理B样条方法权因子决定的形状调整性的曲线曲面。方法在三角函数空间上构造了一组含参数的调配函数,进而定义具有与3次B样条曲线曲面相同结构的新曲线与张量积曲面。结果新曲线曲面继承了B样条方法的凸包性、对称性、几何不变性等诸多性质。不同的是,同样是基于4点分段,3次均匀B样条曲线C2连续,而对于等距节点,在一般情况下,新曲线C5连续,当参数取特殊值时可达C7连续。新曲线在C5连续的情况下存在1个形状参数,能较好地调整曲线的形状同时又无须改变控制顶点。另外,将形状参数设为特定值,新曲线可以自动插值给定点列。新曲面具有与新曲线相应的优点。结论在强局部性下实现高阶连续性的形状可调分段组合曲线曲面,为高阶光滑曲线曲面的设计提供了可能,并且新曲线实现了逼近与插值的统一表示,能较好地应用于工程实际。调配函数的构造方法具有一般性,可用相同方式构造其他具有类似性质的调配函数。     

Point-tangent/point-normal B-spline curve interpolation by geometric algorithms

We introduce a novel method to interpolate a set of data points as well as unit tangent vectors or unit normal vectors at the data points by means of a B-spline curve interpolation technique using geometric algorithms. The advantages of our algorithm are that it has a compact representation, it does not require the magnitudes of the tangent vectors or normal vectors, and it has C² continuity. We compare our method with the conventional curve interpolation methods, namely, the standard point interpolation method, the method introduced by Piegl and Tiller, which interpolates points as well as the first derivatives at every point, and the piecewise cubic Hermite interpolation method. Examples are provided to demonstrate the effectiveness of the proposed algorithms.