Approximation and estimation bounds for free knot splines

The fitting to data by splines has long been known to improve dramatically if the knots can be adjusted adaptively. To demonstrate the quality of the obtained free knot spline, it is essential to characterize its generalization ability. By utilizing the powerful techniques of the empirical process and approximation theory to address the estimation and approximation error bounds, respectively, the generalization ability of the free knot spline learning strategy is successfully characterized. We show that the Pseudo-dimension of free knot splines is essentially a linear function of the number of knots. A class of rather general loss functions is considered here and the squared loss is specially treated for its excellent property. We also provide some numerical results to demonstrate the utility of these theoretical results in guiding the process of choosing the appropriate knot numbers through the training data to avoid the overfitting/underfitting problem.     

Surface reconstruction using bivariate simplex splines on Delaunay configurations

Recently, a new bivariate simplex spline scheme based on Delaunay configuration has been introduced into the geometric computing community, and it defines a complete spline space that retains many attractive theoretic and computational properties. In this paper, we develop a novel shape modeling framework to reconstruct a closed surface of arbitrary topology based on this new spline scheme. Our framework takes a triangulated set of points, and by solving a linear least-square problem and iteratively refining parameter domains with newly added knots, we can finally obtain a continuous spline surface satisfying the requirement of a user-specified error tolerance. Unlike existing surface reconstruction methods based on triangular B-splines (or DMS splines), in which auxiliary knots must be explicitly added in advance to form a knot sequence for construction of each basis function, our new algorithm completely avoids this less-intuitive and labor-intensive knot generating procedure. We demonstrate the efficacy and effectiveness of our algorithm on real-world, scattered datasets for shape representation and computing.     

Elitist clonal selection algorithm for optimal choice of free knots in B-spline data fitting

Data fitting with B-splines is a challenging problem in reverse engineering for CAD/CAM, virtual reality, data visualization, and many other fields. It is well-known that the fitting improves greatly if knots are considered as free variables. This leads, however, to a very difficult multimodal and multivariate continuous nonlinear optimization problem, the so-called knot adjustment problem. In this context, the present paper introduces an adapted elitist clonal selection algorithm for automatic knot adjustment of B-spline curves. Given a set of noisy data points, our method determines the number and location of knots automatically in order to obtain an extremely accurate fitting of data. In addition, our method minimizes the number of parameters required for this task. Our approach performs very well and in a fully automatic way even for the cases of underlying functions requiring identical multiple knots, such as functions with discontinuities and cusps. To evaluate its performance, it has been applied to three challenging test functions, and results have been compared with those from other alternative methods based on AIS and genetic algorithms. Our experimental results show that our proposal outperforms previous approaches in terms of accuracy and flexibility. Some other issues such as the parameter tuning, the complexity of the algorithm, and the CPU runtime are also discussed.     

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.     

B-Spline Curve Approximation by Utilizing Big Bang-Big Crunch Method

The location of knot points and estimation of the number of knots are undoubtedly known as one of the most difficult problems in B-Spline curve approximation. In the literature, different researchers have been seen to use more than one optimization algorithm in order to solve this problem. In this paper, Big Bang-Big Crunch method (BB-BC) which is one of the evolutionary based optimization algorithms was introduced and then the approximation of B-Spline curve knots was conducted by this method. The technique of reverse engineering was implemented for the curve knot approximation. The detection of knot locations and the number of knots were randomly selected in the curve approximation which was performed by using BB-BC method. The experimental results were carried out by utilizing seven different test functions for the curve approximation. The performance of BB-BC algorithm was examined on these functions and their results were compared with the earlier studies performed by the researchers. In comparison with the other studies, it was observed that though the number of the knot in BB-BC algorithm was high, this algorithm approximated the B-Spline curves at the rate of minor error.     

基于极小化二阶导矢确定节点

构造参数拟合曲线的关键问题之一是为每个数据点指定一个参数值（节点）。提出了一种确定节点的新方法。对于每个数据点,新方法由相邻的三个数据点构造一条二次多项式曲线,二次曲线的节点通过极小化其二阶导矢的平方确定。两个相邻数据点间的节点区间由两条二次曲线确定。为使节点计算公式能有效反映出相邻数据点的变化情况,新方法改进了修正弦长方法并应用于节点计算。新方法是一个局部化方法,因此适合于曲线曲面的交互设计。实验结果说明,新方法比其他节点计算方法有效。     

基于差分进化算法的B 样条曲线曲面拟合

应用B 样条曲线曲面拟合内在形状带有间断或者尖点的数据时,最小二乘法得到的 拟合结果往往在间断和尖点处误差较大,原因在于最小二乘法将拟合函数B 样条的节点固定。本 文在利用3 次B 样条曲线和曲面拟合数据时,应用差分进化算法设计出一种能够自适应地设置B 样条节点的方法,同时对节点的数量和位置进行优化,使得B 样条拟合曲线曲面在间断和尖点处 产生拟多重节点,实现高精度地拟合采样于带有间断或尖点的曲线和曲面数据。     

Automatic knot adjustment using an artificial immune system for B-spline curve approximation

Reverse engineering transforms real parts into engineering concepts or models. First, sampled points are mapped from the object’s surface by using tools such as laser scanners or cameras. Then, the sampled points are fitted to a free-form surface or a standard shape by using one of the geometric modeling techniques. The curves on the surface have to be modeled before surface modeling. In order to obtain a good B-spline curve model from large data, the knots are usually respected as variables. A curve is then modeled as a continuous, nonlinear and multivariate optimization problem with many local optima. For this reason it is very difficult to reach a global optimum. In this paper, we convert the original problem into a discrete combinatorial optimization problem like in Yoshimoto et al. [F. Yoshimoto, M. Moriyama, T. Harada, Automatic knot placement by a genetic algorithm for data fitting with a spline, in: Proceedings of the International Conference on Shape Modeling and Applications, IEEE Computer Society Press, 1999, pp. 162-169] and Sarfraz et al. [M. Sarfraz, S.A. Raza, Capturing outline of fonts using genetic algorithm and splines, in: Fifth International Conference on Information Visualisation (IV’01), 2001, pp. 738-743]. Then, we suggest a new method that solves the converted problem by artificial immune systems. We think the candidates of the locations of knots as antibodies. We define the affinity measure benefit from Akaike’s Information Criterion (AIC). The proposed method determines the appropriate location of knots automatically and simultaneously. Furthermore, we do not need any subjective parameter or good population of initial location of knots for a good iterative search. Some examples are also given to demonstrate the efficiency and effectiveness of our method.     

Data fitting with a spline using a real-coded genetic algorithm

To obtain a good approximation for data fitting with a spline, frequently we have to deal with knots as variables. The problem to be solved then becomes a continuous nonlinear and multivariate optimization problem with many local optima. Therefore, it is difficult to obtain the global optimum. In this paper, we propose a method for solving this problem by using a real-coded genetic algorithm. Our method can treat not only data with a smooth underlying function, but also data with an underlying function having discontinuous points and/or cusps. We search for the best model among candidate models by using the Bayes Information Criterion (BIC). With this, we can appropriately determine the number and locations of knots automatically and simultaneously. Five examples of data fitting are given to show the performance of our method.     

一种B样条表面插值控制点的缩减方法

研究了从给定节点向量中选择节点进行B样条曲线插值的方法，并将此方法应用到行数据点不相同的B样条曲面插值，得到了一个通过对行节点矢量调整传递的曲面插值方法，理论分析和实验表明该方法可大量减少曲面控制点的数目．     

高精度三次参数样条曲线的构造

构造参数样条曲线的关键是选取节点，该文讨论了GC^2三次参数样条曲线需满足的连续性方程，提出了构造GC^2三次参数样条曲线的新方法，在讨论了平面有序五点确定一组三次多项式函数曲线，平面有序六点唯一确定一条三次多项式函数曲线的基础上，提出了计算相邻两区间上的节点的算法，构造的插值曲线具有三次多项式函数精,该文还以实例对新方法与其它方法构造的插值曲线的精度进行了比较。     

B样条曲线节点插入和升阶的统一方法

利用 B样条基转换矩阵的有关结果 ,本文给出了 B样条曲线插入节点和升阶的统一方法及算法 .本文方法建立在严密的数学背景上 ,以简洁严谨的递推公式给出其算法的数学模型 ,相应的算法效率高且易实现 ,算法的时间复杂性为 O((k- k 1) kn) ,其中 k,k分别为升阶前后曲线的阶 ,n k 1为插入节点和升阶后的节点数 .而且 ,本文方法及算法使用灵活 ,适用范围广 ,可用于同时插入任意个相同的或不同的节点并升任意阶 ,也可用于只插入节点或只升阶 .Cohen等的 Oslo算法、升阶方法都是本文方法的特例 ,而且本文方法效率更高     

带参数的多结点样条

多结点样条函数是在通常样条函数中引入更多的附加结点，其优越性表现在使插值过程无须求解任何方程组，而且有局部性，对多结点样条函数做进一步研究，构造了一类带参数的多结点样条基本函数．该类函数不仅保持了一般多结点样条函数的优点，而且由于参数的引进，使得基数型的插值公式可形成一族，可以根据实际问题的需要在函数(曲线)族中作出最优选择．文中研究的带参数的多结点样条函数，除了能用于表达平滑的数据及几何造型之外。尤其能适应波动较大、频率较高的数据拟合问题，有助于解决信号处理及非规则几何造型的一些问题。     

Suboptimal Robot Joint Interpolation Within User-Specified Knot Tolerances

Approximation of a desired robot path can be accomplished by interpolating a curve through a sequence of joint-space knots. A smooth interpolated trajectory can be realized by using trigonometric splines. But, sometimes the joint trajectory is not required to exactly pass through the given knots. The knots may rather be centers of tolerances near which the trajectory is required to pass. In this article, we optimize trigonometric splines through a given set of knots subject to user-specified knot tolerances. The contribution of this article is the straightforward way in which intermediate constraints (i.e., knot angles) are incorporated into the parameter optimization problem. Another contribution is the exploitation of the decoupled nature of trigonometric splines to reduce the computational expense of the problem. The additional freedom of varying the knot angles results in a lower objective function and a higher computational expense compared to the case in which the knot angles are constrained to exact values. The specific objective functions considered are minimum jerk and minimum torque. In the minimum jerk case, the optimization problem reduces to a quadratic programming problem. Simulation results for a two-link manipulator are presented to support the results of this article.     

Spherical DCB-spline surfaces with hierarchical and adaptive knot insertion

This paper develops a novel surface fitting scheme for automatically reconstructing a genus-0 object into a continuous parametric spline surface. A key contribution for making such a fitting method both practical and accurate is our spherical generalization of the Delaunay configuration B-spline (DCB-spline), a new non-tensor-product spline. In this framework, we efficiently compute Delaunay configurations on sphere by the union of two planar Delaunay configurations. Also, we develop a hierarchical and adaptive method that progressively improves the fitting quality by new knot-insertion strategies guided by surface geometry and fitting error. Within our framework, a genus-0 model can be converted to a single spherical spline representation whose root mean square error is tightly bounded within a user-specified tolerance. The reconstructed continuous representation has many attractive properties such as global smoothness and no auxiliary knots. We conduct several experiments to demonstrate the efficacy of our new approach for reverse engineering and shape modeling.     

基于二进制GA的B样条重构曲线节点优化

带法向约束的自由曲线曲面重构在光学反射面设计中起至关重要的作用.本文为解决法向约束下的曲线重构问题提出了一种优化方案,使得重构出的曲线在逼近数据点的同时,亦能满足相应法向约束.首先,利用惩罚函数的方法将带法向约束的优化问题转化为无约束的优化问题.然后,引入二进制编码的遗传算法（GA）,建立合适的适应度函数,自适应产生优化节点向量,如此迭代进化,直到产生令人满意的重构曲线为止.考虑到节点向量非递减的特性,而遗传算法在寻找最优节点向量的过程中有可能打乱节点向量的顺序,所以在建立适应度函数的时候将变量调整为无序有界变量.通过与传统最小二乘方法和粒子群智能优化方法的比较,本文方案在解决带法向条件约束的曲线重构问题上优势明显,且对于任意形状的曲线重构都行之有效.     

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.     

Approximate merging of B-spline curves via knot adjustment and constrained optimization

This paper addresses the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. The basic idea of the approach is to find the conditions for precise merging of two B-spline curves, and perturb the control points of the curves by constrained optimization subject to satisfying these conditions. To obtain a merged curve without superfluous knots, we present a new knot adjustment algorithm for adjusting the end k knots of a kth order B-spline curve without changing its shape. The more general problem of merging curves to pass through some target points is also discussed.     

Free knot splines for logistic models and threshold selection

The logistic regression model has been in use in statistical analysis for many years. The paper introduces a spline model to remove the linear restriction on logit function. By considering knot locations as free variables, spline approximation of data is improved. The number of knots and the degree of the spline functions can still be determined by using a model selection procedure. Moreover, a knot, seen as a free parameter for a piecewise linear spline, represents a break point in the logit function which may be interpreted as a threshold value. This method is applied to a clinical trial for an in vitro fertilization program.     

均匀B样条曲线的降阶

文中给出均匀 B样条曲线退化的充要条件 ,证明了满足退化条件的 k阶均匀 B样条曲线可以表示为 (k- 1)阶均匀 B样条曲线 ,并给出相应的表示方法 .在此基础上 ,利用约束优化方法 ,提出均匀 B样条曲线的一种降阶方法 .而传统方法从升阶的反问题考虑降阶 ,要求被降阶的 B样条曲线的所有节点都是多重节点 ,从而无法处理均匀B样条曲线 .